# Nanocrystal Programmable Assembly Beyond Hard Spheres (or Shapes) and Other
(Simple) Potentials

Alex Travesset

###### Abstract

Ligands are the key to almost any strategy in the assembly of programmable
nanocrystals (or nanoparticles) and must be accurately considered in any
predictive model. Hard Spheres (or Shapes) provide the simplest and yet quite
successful approach to assembly, with remarkable sophisticated predictions
verified in experiments. There are, however, many situations where hard
spheres/shapes predictions fail. This prompts three important questions: In
what situations should hard spheres/shapes models be expected to work? and
when they do not work, Is there a general model that successfully corrects
hard sphere/shape predictions? and given other successful models where ligands
are included explicitly, and of course, numerical simulations, can we unify
hard sphere/shape models, explicit ligand models and all atom simulations?.
The Orbifold Topological Model (OTM) provides a positive answer to these three
questions. In this paper, I give a detailed review of OTM, describing the
concept of ligand vortices and how it leads to spontaneous valence and
nanoparticle “eigenshapes” while providing a prediction of the lattice
structure, without fitting parameters, which accounts for many body effects
not captured in (two-body) potentials. I present a thorough survey of
experiments and simulations and show that, to this date, they are in full
agreement with the OTM predictions. I will conclude with a discussion on
whether NC superlattices are equilibrium structures and some significant
challenges in structure prediction.

###### keywords:

Nanocrystal, Assembly, Nanoscience, Crystallization

[1]organization=Iowa State University and Ames Lab, addressline=Department of
Physics and Astronomy, postcode=50010, city=Ames, IA, country=USA

††footnotetext: Abbreviations:NC, Nanocrystal; HS, Hard Sphere/Hard Shape;
BNSL, Binary Nanocrystal Super Lattice; PF, Packing Fraction; Optimal Packing
Model, OPM; OTM, Orbital Topological Model; OCM, Optimal Cone Model;MOLT,
Molecular Theory; WHAM, Weighted Histogram Analysis Method; PMF, Potential of
Mean Force; fcc, face centered cubic; bcc, body centered cubic; bct, body
centered tetragonal.

###### Contents

  1. 1 Introduction
  2. 2 Packing and Hard Sphere/Shape Models
    1. 2.1 Summary of HS models
    2. 2.2 Mapping NC to HS
    3. 2.3 AlB2 is a paradigmatic HS phase
    4. 2.4 How well HS predictions compare to experiments
  3. 3 The Orbifold Topological Model (OTM)
    1. 3.1 A case for OTM
    2. 3.2 The formalism
    3. 3.3 OTM beyond quasi-spherical NCs
    4. 3.4 Experimental validation of OTM
    5. 3.5 Towards an OTM Free Energy
  4. 4 Numerical Simulations
    1. 4.1 HOODLT and technical simulation details
    2. 4.2 PMF for pairs and N-PMF for clusters
    3. 4.3 Superlattices
    4. 4.4 The polymer limit
    5. 4.5 NCs on a solid support
  5. 5 Dynamics of NC Assembly
    1. 5.1 Solvent evaporation models
    2. 5.2 Equilibrium structures from solvent evaporation
    3. 5.3 Are superlattices in thermal equilibrium?
  6. 6 Conclusions
  7. 7 Acknowledgements

##  1 Introduction

Nanocrystals (NCs), also referred to as nanoparticles, are conceptualized as
”Programmable Atom Equivalents” [1], i.e. “large” atoms with many degrees of
freedom: size, shape, chemical composition[2], and type and structure of
capping ligands[3]. Motivation to understand and predict NC assembly is driven
by both fundamental science and new applications as described in many previous
references, for example [4, 5, 6, 7, 8, 9].

Figure 1: Example of a NC: core consists of Au(yellow) and S(red). The ligand
is a simple hydrocarbon, schematically showing its model as hard spheres
(billiard balls). On the right is a schematic relation between different
assembly strategies in binary systems as discussed in the text. The bccAB6
phase is also labeled Cs6C60 in many references.

Many strategies have been developed to assemble NCs into materials. Most of
them, but not all as with liquid crystals[10] for example, consist in
functionalizing NC cores with appropriate ligands, as depicted in Fig. 1.
Ligands are necessary because the bare interactions are not specific enough
for a controlled assembly, so their role is to make NCs stable in a solution
and enable assembly into a material in a controlled and reproducible manner.
There are many strategies in which the assembled NC material is swollen, that
is, the solvent is part of the final structure. This is the case for DNA
assembly[11, 12, 13, 14], Nanocomposite Tectons[15], Interpolymer
complexation[16] and many others. In other cases, the end process in assembly
is a solvent free structure, that is, a dry system. This is the case in
solvent evaporation[17, 18, 19] or tuning ionic strength[20].

Binary systems, i.e consisting of two NC species, have very sophisticated
phase diagrams, so they provide an excellent paradigm in which to put model
predictions to a test. As an example, Fig. 1 shows a schematic representation
of the different phases parameterized by two coordinates: the strength of
hydrogen bonds and the quality of the solvent. In a good solvent
($\chi>\frac{1}{2}$) and weak hydrogen bonds, NCs are stable: NCs repeal each
other and remain in a liquid-like state. As the strength of the hydrogen bonds
increases, the ensuing attraction drives the assembly into several phases for
DNA assembly[1] and different ones for nanocomposite tectons[15], with the
differentiating factor being the flexibility of the ligand[21]. Yet, another
set of phases are obtained in poor solvent and no hydrogen bonds[22, 23, 24].
The example in Fig. 1 illustrates that different strategies are related and
may give rise to new opportunities, such as, for example, dry systems with
strong hydrogen bonds.

This paper focuses on the assembly of solvent-free structures. Herein, unless
otherwise stated, swollen systems will not be specifically discussed. Hard
sphere or shape (HS) models have played a central role in describing assembly
with significant predictive power, which I review in Sect. 2. There are other
models that explicitly take into account ligands. Particularly relevant for
this study are the Optimal Packing Model (OPM)[25] and the Optimal Cone Model
(OCM)[26]. Additionally, numerical simulations have been extensively applied
to understand NC assembly and are discussed in great detail in this paper.
Yet, simulations by themselves are not enough: At the end of the day, they
must be integrated and provide validation for an overarching model that
identifies the few relevant parameters that control the dynamics and
equilibrium of the system.

HS does provide an excellent model for the NC core; see Fig. 1, but it is far
from intuitively obvious why HS would provide a fair representation of the
entire NC, as the ligands appear to be soft and very compressible. There are
many other reasons to be skeptical about HS models. The internal energy of $N$
HS particles is given by the expression of the ideal gas and the free energy
is of the order of $k_{B}T$

$U=\frac{3}{2}Nk_{B}T\quad,\quad F\approx Nk_{B}T\quad\mbox{NC modeled as HS
systems}$ (1)

with a possible kinetic energy contribution in the internal energy for
rotation, which implies that the internal energy is always constant and phase
transitions are entropically driven, with cohesive energies of the order of
$k_{B}T$ per NC. Numerical simulations, see the discussion in Sect. 4, show
that Eqs. 1 are in complete disagreement for any real NC system. In fact,
typical internal energies and free energies for a solid phase are

$U\approx-1200Nk_{B}T\quad,\quad F\approx-500Nk_{B}T\quad\mbox{ real NC
systems}\ ,$ (2)

that is, almost three orders of magnitude larger than HS Eq. 1 (and with
opposite signs). In fact, the main contribution to the entropy comes from
ligand conformations, while translational (and rotational) contributions,
which are the only ones that the HS models account for, are almost negligible.
Given these obvious discrepancies, one might be tempted to abandon HS
altogether. Yet, HS models are extremely valuable, with many sophisticated
predictions verified in actual experiments. As happens in many areas of
physics, there are aspects that are independent of the details of the
microscopic (more properly, nanoscopic) interactions. Basically, HS models are
relevant because they capture the underlying universality of packing with
elegant and minimal simplicity[27]; that is, model-independent results that
optimize the available space.

Therefore, HS provides a correct first-order approximation to NC assembly
structure prediction. It would seem obvious that the next order would simply
consist of adding reasonable enthalpic terms. For example, Ref. [28] has
successfully characterized all possible binary NC structures at an interface
(2D) with the addition of relative interaction energy between NCs and the
solvent interface. DNA with double-stranded (rigid) ligands is another example
in which this approach is very successful[29]. However, as shown in more
detail below, see Fig. 3, when ligands are flexible (and their size is not
negligible compared to the core), experiments reveal a dependence of nearest-
neighbor lattice constants with the NC coordination, i.e. the number of
nearest-neighbors, which in turn points to the presence of many-body effects,
which may suggest the need to go beyond pair potentials, requiring a model
where ligands are described in a more explicit form.

Ligands are explicitly modeled in OXM models (X = P, C) and therefore provide
more detail than HS models. The OPM is very successful in describing the
lattice constants in single-component fcc (or bcc) lattices[25], while the OCM
accurately describes the equilibrium nearest-neighbor separation of two or
three isolated NCs[26]. However, the closest-neighbor NC separations fall in
between OPM and OCM in most other experimental situations[7, 30, 31]. OXM
models account for the ligand conformations, so at this point in the
discussion they may appear to be conceptually different and unrelated to the
HS models.

The previous discussion provides the motivation for this paper: How to bring
together HS models, the OXM (X=P,C) models and all atom simulations with the
goal of developing a general framework for NC prediction. In this paper, I
make the case that the Orbifold Topological Model (OTM)[32, 33], provides such
a unified model. I will show that at this point in time there is not a single
experiment in disagreement. Yet, there are still challenges and aspects that
need to be developed towards a completely predictive framework, which are
discussed throughout the paper.

The end result of a NC assembly experiment is usually a lattice of NCs, i.e. a
superlattice, see simulation snapshot in Fig. 19 or electron microscopy images
in Fig. 4. Ref. [34] provides a detailed account of the structures, relates
them to other soft- and hard-matter systems, and describes strategies to
expand their range. Superlattices are basically lattices of crystals (or
crystals of crystals), as the NC cores themselves are already crystals.
Predicting the superlattices that define equilibrium for given NCs is a
fundamental goal. A more modest problem, yet still very important, is given a
superlattice consisting of several NCs species, predict its structure: lattice
constants, packing fraction, ligand arrangements, etc. the OTM provides a
general solution to this problem., independent of the microscopic interactions
details. To address the more difficult problem of predicting what
superlattices define equilibrium, that is, what superlattices should be
observed in an actual experiment, a free energy is needed, which OTM does not,
in principle, provide. However, I will discuss exciting recent developments
that bring us closer to achieving this goal.

The flow of the paper will start by discussing HS models and their triumphs,
and, of course, will provide a detailed account of the failures that motivate
the OTM. Then, I introduce the OTM and provide its validation through
experiment and simulations while pointing out its most outstanding challenges.
Over the years, different papers have reviewed the role of ligands in NC
assembly and this is a very large field. Unavoidably, there are many papers
that I will for sure unjustly not cite for which I apologize, and different
subjects that are not covered here. I refer to Ref. [35, 36, 7, 37, 38, 39,
40] for reviews that cover overlapping and complementary topics.

Figure 2: Optimal PF phases are adapted from Ref. [41](in blue). The HS Phase
diagram is estimated from Ref. [42](in red, phases not present in PF). The
experimental data is estimated from the data gathered in Ref. [32] (Orange for
phases not present in neither HS Phase diagram or PF). variables $\gamma,x$
defined in Eq. 4. Naming of the phases is the same as in the orginal
references.

##  2 Packing and Hard Sphere/Shape Models

###  2.1 Summary of HS models

I describe a few aspects of HS that are of particular relevance for this
paper. I refer to Ref. [43] for a recent authoritative review of the state of
the art and many other aspects.

The Packing Fraction (PF) ($\eta$) is defined by

$\eta_{HS}=\frac{\mbox{volume occupied by particles}}{\mbox{total
volume}}\stackrel{{\scriptstyle s.c.\
spheres}}{{=}}\frac{\pi}{6}\frac{N}{V}d_{HS}^{3}\ ,$ (3)

where the sub-index HS is to emphasize that PF is computed within a HS model.
The rhs of the equation illustrates the case of a single component system
consisting of $N$ HS of diameter $d_{HS}$ within a volume $V$. In other cases,
for example, binary systems of spherical particles, there are two other
thermodynamic coordinates: $\gamma$(relative particle size) and
$x$(stoichiometry), defined as

$\gamma=\frac{d_{HS}^{B}}{d_{HS}^{A}}<1\quad,\quad
x=\frac{N^{B}}{N^{A}+N^{B}}$ (4)

where $d_{HS}^{i},N^{i}$ $i=A,B$ are the diameters and numbers of the large(A)
and small(B) particles.

There are two types of problems that are particularly relevant for NC
assembly:

  1. A

) Maximum PF ($\eta$) for given parameters, e.g ($\gamma,x$)

  2. B

) Phase diagram as a function of $\eta$(or pressure, which is the conjugate
thermodynamic variable) and other parameters, e.g ($\gamma,x$).

For single component spheres the solution to problem A is degenerate, as both
fcc and hcp have the highest PF $\eta_{HS}=\frac{\pi\sqrt{2}}{6}\approx
0.74048$. Solution for problem B) contains liquid and crystalline phases at a
transition $\eta_{HS}\approx 0.5$, glass phases for increasing PF. Near the
highest PF, fcc is favored over hcp[44, 45]. The case of binary systems of
spherical particles is illustrated in Fig. 2 for $\gamma>0.41$. The phases
with the highest PF [46, 41] are many and diverse. The phase diagram, obtained
by minimizing the free energy (or maximizing the entropy, as the internal
energy is given by the monovalent ideal gas formula, see Eq. 1), is (very)
schematically constructed from the data provided by Marjolein Dijkstra in Ref.
[42].

It would seem reasonable to expect that close to the maximum PF, the solutions
of problems A and B would be equivalent. In other words, the maximum PF phases
would have the lowest free energy. However, this expectation is not fulfilled.
This is clear for binary systems in Fig. 2, as only the AlB2 phase appears as
a solution of both problems A and B. Further evidence is provided from other
rigorous results with more complex shapes[47, 48, 49, 50, 51, 52].

The dichotomy between the minimum of free energy and the highest PF motivates
us to consider the assembly of HS in terms of “eigenshapes”. That is, the
system is described by an effective shape that completely packs space. This
shape is obviously different from the actual HS, and there are ways to
actually calculate it[53]. In the following, I show that it is possible to
generalize this idea when ligands are included, although in that case, the
eigenshape is not a property of the NC only and depends on the environment.

Figure 3: Electron microscopy image of a single component system (4.1 nm
Au-C18 NC) deposited on a substrate. From there, NCs with coordination 3,4,5,6
are identified and histogrammed from where the nearest neighbor separation is
obtained. It is apparent that the nearest neighbor distance increases with
coordination and that for coordination 6 the results are almost equivalent
with fcc. Also shown are the results of a similar experiment with fcc
(coordination 12). The PF of many different single component NCs is completely
consistent with the fcc HS PF. Results are from Ref. [54] (permission from D.
Talapin).

###  2.2 Mapping NC to HS

NC core shapes are usually polyhedra, although spherical cores do exist see
Fig. 5, but in some cases they may be approximated as spherical because the
flexibility of the ligand helps round off edges and corners. For a spherical
NC, the first issue is what is the correct effective HS diameter, which must
include a contribution from the ligands.

Early studies [22, 23, 7] defined the NC HS diameter as the lattice constant
of the resulting hexagonal single component lattice, deposited on a solid
substrate. Due to the interactions of the bottom NC ligands with the solid
substrate, and the unconstrained ligands at the top, this definition is not
optimal; this is apparent from Fig. 3, where the maximum of the histograms for
coordination 6 is broad and slightly below the one for fcc. In Ref.[55] we
showed that the Optimal Packing Model (OPM) [25] gives the best possible
definition. Therefore, the HS diameter $d_{\text{HS}}$ of a spherical NC with
a core radius $R$ and grafting density $\sigma$ is defined as

$\tau(\lambda,\xi)=(1+3\xi\lambda)^{1/3}$ (5)

with

$\tau\equiv\frac{d_{\text{HS}}}{2R}\quad,\quad\xi\equiv\frac{\sigma}{\sigma_{\text{max}}}\quad\mbox{and}\quad\lambda\equiv\frac{L}{R}$
(6)

where $\sigma_{\text{max}}=\frac{1}{A_{0}}$ is the maximum grafting density of
the core (or $A_{0}$ is the ligand footprint, which is $d^{2}$ in the language
of polymer theory[56]) and $L$ is the maximum (extended) ligand length
($R_{max}$ in the language of polymer theory) shown schematically in Fig. 8.
The parameter $\lambda$ is sometimes referred as the “softness”. In the
language of polymer physics[56], the OPM Eq. 5 defines the radius of a
spherical brush where the ligands are in a non-solvent condition. If the
spherical NC is divided as non-overlapping cones, see a cone shown in Fig. 8,
then the OPM results follows by assuming that the ligand entirely fills out
its own available cone.

A very important property that cements Eq. 5 as the correct definition of the
HS diameter is that the resulting PF is exactly the same as the corresponding
HS PF. This is proved as follows: The amount of space occupied by matter for a
single NC is (note that $A_{0}L$ is the volume occupied by one ligand)

$V_{matter}=V_{core}+V_{ligands}=\frac{\pi}{6}(2R)^{3}+4\pi R^{2}\sigma
A_{o}L=\frac{\pi}{6}(2R)^{3}\left(1+3\xi\lambda\right)=\frac{\pi}{6}(2R)^{3}\tau^{3}=\frac{\pi}{6}d_{HS}^{3},$
(7)

that is, the matter, i.e. the NC, occupies the same volume as a HS of diameter
$d_{HS}$[54, 33]. Therefore for a BNSL with $n_{A},n_{B}$ NCs in the Wigner
cell, the PF is

$\displaystyle\eta=\frac{V_{matter}}{V}=\frac{V_{A}+V_{B}}{V_{WS}}\stackrel{{\scriptstyle
superlattice}}{{=}}\frac{\pi}{6}\frac{n_{A}(d^{A}_{HS})^{3}+n_{B}(d^{A}_{HS})^{3}}{V_{WS}}\stackrel{{\scriptstyle
HS\ PF\ definition}}{{=}}\eta_{HS}(\gamma)\ ,$ (8)

where $V_{WS}$ is the volume of the Wigner-Seitz cell. This can easily be
generalized to any other configuration.

If one accepts the OPM as the correct definition of the HS, the structure of
the phase: lattice constant, nearest-neighbor separations, PF or ligand
conformations are quantitatively predicted, see Sect 2.3 for an example worked
out in full detail. The other conclusion that follows is that if the NCs pack
as HS, where the nearest neighbor distances are determined by
$d^{A}_{HS},d^{B}_{HS}$, there will be a fraction $1-\eta_{HS}$ of empty space
within the superlattice. It should be noted that in principle, it is possible
for NCs to be closer than the distances that follow from HS. The closest
approach that could be theoretically achieved is if ligands completely fill
any empty space, which would requite a NC diameter given by

$d_{OCM}=\eta^{1/3}_{HS}d_{HS}<d_{HS}\
\rightarrow\eta_{OCM}=1>\eta_{HS}\quad,$ (9)

which defines the Optimal Cone Model (OCM) diameter[26]. As I show below, the
actual effective NC diameter lies in between the HS (or OPM) and OCM values,,
see also the simulations in Fig. 21.

The generalization of Eq. 5 to NC cubes has been developed[57]. If the core
has edge length $l_{E}$, the HS edge $l_{HS}$ including the contribution of
the ligands is given as

$l_{HS}=l_{E}\left(1+6\xi\lambda\right)^{1/3}\ ,$ (10)

note the appearance of a factor $6$, where the parameters $\xi$ and $\lambda$
are defined in the same way as Eq. 4:

$\xi=\frac{\sigma}{\sigma_{Max}}\quad,\quad\lambda=\frac{L}{l_{E}}.$ (11)

The validity of OPM and OCM as the length of the ligand increases, reaching
the polymer limit, is discussed in SubSect. 4.4. Related models have been
developed in this limit: Ref. [58] is, in fact, a generalization of OCM for
spherical NCs functionalized with long polymers. The $n$ monomers of each
ligand are divided as $n=n_{dry}+n_{inter}$, so that $n_{dry}$ of them are
within a size $h_{dry}$ and are described by the OPM formula Eq. 5, and
$n_{dry}$ are shared among all nearest-neighbor NCs by an “interpenetrating”
layer $h_{inter}$, where each ligand has the ideal chain size $\propto
n^{1/2}_{inter}$ within. The model assumes an interpenetration layer of
dimension $h_{inter}$. With respect to the structure of the ligands, the model
is equivalent to OCM if $h_{inter}/h_{dry}<<1$ or, what is the same if the
vortices, see Fig. 18 are effectively two-dimensional. However, the lattice
constant (or nearest-neighbor distance) is entirely determined by space
filling constraints and therefore is the same as OCM Eq. 9. NC lattice
constants for short and moderately long ligands are consistent with OCM and
space-filling constraints for low coordination only. Simulations in Ref. [57]
have analyzed the overlap region for cubic NCs, showing a finite overlap
region so that $h_{inter}$ is finite, although the ligands are too short to
have a measurable effect.

###  2.3 AlB2 is a paradigmatic HS phase

It is illustrative to show that experimental measurements allow a very
detailed comparison with theoretical predictions. Fig. 4 shows electron
microscopy images for the AlB2 BNSL that were reported in Ref. [54]. As
discussed, the HS diameters given by Eq. 5 are

$\displaystyle d_{HS}^{A}=10.04\mbox{ nm}$ $\displaystyle 2R=7.0\mbox{ nm
},n=18$ $\displaystyle d_{HS}^{B}=5.75\mbox{ nm}$ $\displaystyle 2R=4.1\mbox{
nm },n=9\ ,$ (12)

here $n$ is the number of Carbons in the ligand chain. The maximum extension
$L$ of the ligands are given (in nm) either

$L=\left\\{\begin{array}[]{c c}0.12(n+1)&\text{from
Ref.~{}\cite[cite]{[\@@bibref{Number}{Boles2015}{}{}]}}\\\
0.128n+0.2&\text{from
Ref.~{}\cite[cite]{[\@@bibref{Number}{Travesset2017a}{}{}]}}\end{array}\right.\
.$ (13)

Figure 4: Electron Microscopy images of the AlB2 BNSL as reported in Ref. [54]
for the $xy$ plane(top) and the $zx$ plane bottom. The AlB2 lattice (for the
corresponding $\gamma=0.57$) is shown with spheres not at scale for proper
visualization. The measurement of NC distances is performed and histogrammed
(on the right) and in this way, the lattice constant $a$ and c-axis $c$ are
determined. (permission from D. Talapin)

The second formulas has a better theoretical justification and for ligands
like polystyrene it gives more accurate results, but for simple hydrocarbon
ligands both give completely consistent results. In this review I will follow
the first formula for hydrocarbon ligands as it seems to be more widely used
in experiments. The HS diameters have used $\xi=1$, consistently with the
grafting density reported in the experiments[54]. Therefore, this BNSL occurs
at a value of $\gamma$, see Eq. 4

$\gamma=\frac{d_{HS}^{B}}{d_{HS}^{A}}=0.57$ (14)

The Wickoff positions of AlB2 state that one of the large A NC(red) and a
small B NC(blue) NC in Fig. 4 are located at

Position A $\displaystyle:$ $\displaystyle(0,0,0)$ (15) Position B
$\displaystyle=$ $\displaystyle(0,\frac{a}{\sqrt{3}},\frac{c}{2})$

and also, as the figure illustrates, for this value of $\gamma$, the A and B
NCs are in contact (or “kissing”) so the distance between A and B is given by
$d^{AB}_{HS}=\frac{1}{2}\left(d_{HS}^{A}+d_{HS}^{B}\right)$. Then from Fig. 4
and Eq. 2.3 and 15, we have a pure theoretical prediction, because I have
assumed that the NCs are HS (through the OPM formula) and the geometrical
relations of the BNSL without fitting or measured experimental parameters. It
is

(Theory), HS through OPM $\displaystyle
c=2\sqrt{(d^{AB}_{HS})^{2}-\frac{(d_{HS}^{A})^{2}}{3}}$ $\displaystyle=$
$\displaystyle 10.7\mbox{ nm}\quad$ (16) $\displaystyle a=d_{HS}^{A}$
$\displaystyle=$ $\displaystyle 10.0\mbox{ nm}$ (Experiment), Electron
Microscopy, Fig. 4 $\displaystyle c$ $\displaystyle=$ $\displaystyle
10.6\mbox{ nm}\quad$ $\displaystyle a$ $\displaystyle=$ $\displaystyle
10.0\mbox{ nm}\ ,$

and I compare these results with the ones obtained from the experiment, the
orange and red dashed drawn lines from the histogram obtained from electron
microscopy in Fig. 4.

The agreement between theory and experiment is within an accuracy of less than
1$\%$. The experimental PF follows from the measured lattice constants and the
amount of matter, i.e. core and ligands within the BNSL unit cell. It gives

$\eta_{exp}=\frac{\mbox{Volume
Matter}}{\sqrt{3}a^{2}c/2}=\eta_{HS}(\gamma=0.57)=0.78\ ,$ (17)

where the last identity follows because the general result Eq. 8 applies in
this case. The important point here is that the available experimental data
allows precise and quantitative comparisons with theoretical models. One may
conclude from this example that HS provides a successful description to the
point that HS is all we need. Certainly, if HS models were always to work this
well, there is no need to work much further, perhaps adding an enthalpic term
as outlined in the introduction. Yet, the same methods applied to other BNSLs,
as shown, for example, in Fig. 6 exhibit clear discrepancies between HS and
experiments. What the results in Fig. 4 tell us is that these discrepancies
reflect real physical effects and motivate us to develop a theory that goes
beyond HS.

Figure 5: Spherical NCs are possible to synthesize. These images are from Ref.
[59]. On the right 6 nm spherical NCs from the Zhou lab. Provided by Shan Zhou
and printed with permission.

###  2.4 How well HS predictions compare to experiments

The definition of the HS diameter in SubSect 2.2 combined with the detailed
phase diagrams (or packing fractions) from SubSect 2.1 allow rigorously
establishing the range of validity of the HS to predict the assembly of NC by
solvent evaporation. Although perfectly spherical NCs are available, see Fig.
5, most NCs are polyhedral in nature, yet ligands help round off edges and are
well described as spherical, see Subsect. 3.3 and Sect. 3.5 for a more
detailed discussion. Therefore, failures of HS models will highlight missing
physics and help develop accurate models that minimally extend HS.

2D single-component spherical NCs, see Fig. 3, show that the nearest-neighbor
separation strongly depends on coordination: The maximum of histograms, which
are identified as the nearest-neighbor distance, increases with coordination,
demonstrating a clear breakdown of the HS description. The simulations to be
presented in Sect. 4 confirm these results; see Fig. 23 and the subsequent
discussion. Additional confirmation results are provided in Ref. [31] and
include polystyrene ligands, which shows that the breakdown of the HS model is
not specific to a particular ligand or NC.

In 3D single component spherical NCs an fcc phase should be expected. For
short ligands $\lambda<\lambda_{c}\approx 0.7$, experimental results are in
agreement[17, 60]. However, for longer ligands $\lambda>\lambda_{c}$, bcc is
found to be more stable. This is explained by the argument put forward in Ref.
[25], where next to nearest distances are reached for sufficiently long
ligands, making the actual number of nearest neighbors from 12(fcc) to 14(bcc)
(8 nearest neighbors plus 6 next to nearest neighbors), the condition that
ligands are long enough to reach next to nearest neighbors leads to the
equation

$\frac{2}{\sqrt{3}}d_{HS}=2R+2L\rightarrow\frac{2}{\sqrt{3}}\tau(\lambda_{c},\xi)=1+\lambda_{c}$
(18)

with solution $\lambda_{c}\approx 0.7$ if $\xi=1$, see Eq. 5 for definitions
of the different parameters. At a quantitative level, the fcc lattice
constants are accurately predicted by the HS formula Eq. 5. However, bcc
requires a small correction, departing from HS to slightly compressible
potentials, so that its PF becomes the same as fcc, that is,

$d_{bcc}=\left(\frac{\eta_{HS}(bcc)}{\eta_{HS}(fcc)}\right)^{1/3}d_{HS}=\left(\frac{3\sqrt{3}}{4\sqrt{2}}\right)^{1/3}d_{HS}=0.972081\cdot
d_{HS}$ (19)

as confirmed from both experiment[61, 62, 63] and simulations[64](see Fig. 19
and Fig. 22 and discussion). In summary, whenever single components assemble
into NCs with short ligands $\lambda<\lambda_{c}$ HS predictions match both
qualitatively (the fcc phase) and quantitatively (lattice constants) HS
predictions. For longer ligands $\lambda>\lambda_{c}$ HS fail: the stable
phase is bcc and with slightly smaller nearest-neighbor distance given by Eq.
19. Still, regardless of what is the equilibrium state, Fig. 3 shows that when
NC coordination is 12, the PF takes on the HS fcc value.

I should mention that body-centered tetragonal (bct)[60, 25, 65], hcp[66, 65]
and A15[67] lattices have been reported with single component NCs. It is
possible to continuously transform a bcc to fcc through an intermediate bct,
so at this point the evidence[63] is that such bct structures are not
equilibrium phases but metastable intermediates that remain stable only for
certain solvent contents, a point elaborated further when discussing MOLT in
SubSect. 3.5. The appearance of the A15 phase is attributed to polydispersity:
NCs with different diameters spontaneously phase separate into a binary
system, that is, crystal fractionalization[68, 69], since the A15 lattice is
basically a single component $\gamma=1$ MgZn2[70]. Here I mention another
possibility where the 4 coordinated NC within the A15 may develop the four-
valence eigenshape in Fig. 11, a point that can experimentally verified by
analysis of lattice constants. Whether such an MgZn2 structure with single-
component NCs would be stable remains to be proved with detailed calculations.

Figure 6: PF computed from experiment[54] compared with HS and with OTM[33,
32]. The case of AuCu is discussed in the text.

For NC mixtures consisting of two species, that is,  binary NC systems, the
predictions depend on both $\gamma$ and $x$, see the definitions in Eq. 4. I
have already discussed a BNSL, see Fig. 4, where AlB2 is successfully
described by HS. Yet, this is the exception rather than the rule. At the
qualitative level, Fig. 2 shows that some of the phases present in the HS
phase diagram: NaCl, AlB2, NaZn13, MgZn2(Laves) and bccAB6(for HS exists at a
lower value $\gamma$ and is not shown in the figure) are also found in
experiments. There are, however, several missing phases: CaCu5, CaB6, CsCl and
Li3Bi.

(a) AlB2 BNSL

(b) MgZn2 BNSL

Figure 7: Calculation of maximum HS PF as a function of $\gamma$ and
experiments where the BNSL is found. The dashed lines are the fcc and bcc PF,
respectively. The experiments are from Ref. [23, 71, 72, 73, 54, 74, 75](AlB2)
and Ref. [54, 23, 72, 74, 73](MgZn2). There is some uncertainty in the
determination of $\gamma$ from the experiments, as the grafting density
(needed in Eq. 5) was not available, so $\xi=1$ was assumed in those cases.

At a quantitative level, Fig. 6 compares the measured PF obtained in the
experimental results[54] with the predictions of HS. Agreement is obtained for
AlB2 only and one of the NaZn13 BNSLs. For the other cases, actual PF
fractions predicted from HS are underestimated and in some cases, like MgZn2
or Li3Bi, the disagreement is quite large. Furthermore, the actual critical
values of $\gamma$ where the phases are stable are also in disagreement,
although experimentally there is significant uncertainty.

Regardless of actual phase diagrams or optimal PF, what has made HS models so
appealing is the correlation between the maximum PF and the values of $\gamma$
(see Eq. 4) where the BNSL is experimentally realized. The calculation of the
MgZn2 BNSL HS PF is as follows. When $\gamma<<1$, the most favorable packing
consists of the A particles in contact, each with coordination (or kissing
number 4) and the B-particles filling the interstitials. As the diameter of B
particles increases ($\gamma$ becomes larger), there is a critical value
$\gamma_{c}=\sqrt{\frac{2}{3}}$, where the B particles in the pyrochloric
structure, see Fig. 10, kiss each other, and then, A-A contacts cannot hold
any longer and are replaced by the B-B contacts. The resulting PF is given by

$\eta_{HS}(\gamma)=\left\\{\begin{array}[]{cc}\frac{\sqrt{3}\pi}{16}(1+2\gamma^{3})&\gamma\leq\gamma_{c}\\\
\frac{\sqrt{2}\pi}{24}(\frac{1}{\gamma^{3}}+2)&\gamma\geq\gamma_{c}\end{array}\right.$
(20)

Note that for $\gamma=0$, the above expression reduces to
$\frac{\sqrt{3}}{16}\pi\approx 0.34087$. which is the PF of the diamond
lattice. Eq. 20 is plotted in Fig. 7, where the maximum PF corresponds to the
cusp at $\gamma=\gamma_{c}$.

Similarly to Eq. 20, the PF for all other BNSLs are calculated. Fig. 7
illustrates the two representative examples that I mostly use in this paper:
AlB2 and MgZn2. For AlB2 all experiments occur around the PF maximum, and the
values of $\eta$ are high. Fig. 6 shows that the PF (and lattice constants)
for AlB2 are consistent with the HS predictions, making stronger the case
already made from Fig. 4 that NC pack as HS in this BNSL. MgZn2 still shows a
correlation between the maximum PF and $\gamma$ where the experiments are
found. However, there are quite a few cases in which the resulting PF is too
low $\eta<0.65$, so these BNSL cannot be stable because there is too much free
space available. Another conspicuous aspect is that experiments are only found
on the left of the PF maximum, a point that is quite important and will be
elaborated further in the following. Therefore, we must conclude that HS
models describe AlB2 well but break down when applied to MgZn2. Similar
considerations follow for all other reported BNSLs[32].

Edge length (nm) | Lattice constant (nm)
---|---
| Experiment | Theory (Eq. 10)
9.8(9) | 11.1 | 11.2
9.3(8) | 10.8 | 10.7
8.8(7) | 10.3 | 10.2
5.6(4) | 7.2 | 7.0
Table 1: Comparison between experimental results and theoretical prediction
Eq. 10 (with $\sigma_{Max}=2.7$ chains/nm2), see discussion in Ref. [57] for
complete details. The experimental grafting density is $\sigma=0.81$
chains/nm2.

In addition to spherical-like NCs, there are other systems in which
experiments can be compared against HS predictions. In the context of
Perovskite NCs, BNSL with cubic NCs have been thoroughly investigated[76, 77,
78, 57]. Single-component NC cubes are packed as simple cubic superlattices
(except for small NCs whose nuances are discussed in Ref. [79]), in agreement
with HS models[80, 81] with lattice constants predicted from Eq. 10[57], see
Table 1. However, analysis of BNSLs that include both cubes and spheres show
clear breakdowns from HS[76, 57]. A similar situation takes place for
truncated tetrahedral NCs[82, 83], where experiments report a monoclinic and
two diamond phases. The HS phase diagram of truncated tetrahedra[84] contains
many phases including diamond, but does not include the monoclinic phase.

The conclusions of the extensive comparison between HS and experiment are that
there are examples where NCs behave as HS, like fcc or the AlB2 BNSL, but
often the HS description breaks down. It should therefore become apparent that
ligands often cannot be coarse-grained into a hard shape because conformations
play a fundamental role. Yet, despite its obvious shortcomings, HS do provide
a simple and very valuable starting point that will be generalized into a
complete framework for structure prediction.

Figure 8: Example of OPM vs OTM: Within OPM each ligands fills out its own
cone, as shown with the shaded volume. The only way to get NCs closer is for
ligands to splay and creating a vortex. Note that once ligands splay, they
occupy space belonging to other cone ligands.

##  3 The Orbifold Topological Model (OTM)

###  3.1 A case for OTM

I have introduced two models, the OPM see Eq. 5 and the OCM see Eq. 9. Both
define an effective NC diameter. As discussed in Sect. 2.4, the OPM defines
the HS diameter of a NC and leave a significant amount of “empty” space; see
also Eq. 8, and the OCM defines a situation where the NCs are packed in such a
way that the ligands fill out the entire available space. When analyzing
experimental results, there are situations where OPM or OCM work, but neither
of them provides a complete solution. Basically, what is needed is a new model
that reproduces the OPM and OCM in the limits when they are correct.

Before dwelling into how OTM provides a general solution that reduces to OPM
and OTM in the appropriate limits, it is worth describing the effect of
potentials that go beyond the HS discussed in Sect. 2.4. For example, [85, 86]
(also [87, 88, 89]) and [90] have discussed softening the potential from HS to
an inverse repulsive power law. We recently provided a detailed determination
of the phase diagram of binary particles that interact with Lennard-Jones
potentials[91]. These systems have rich phase diagrams, but when applied to NC
prediction do not provide any real improvement over the simpler HS models and,
furthermore, require defining additional parameters that are not obvious how
to relate to the actual physical properties of the ligands. In summary, going
beyond HS by simply modifying a two-body potential may provide an ad-hoc
solution for a particular case but leads to models that have less
predictability. As I elaborate further, this is hardly surprising, as ligand
deformations preclude a description of NC interactions in terms of two-body
potentials.

Figure 9: Two neighboring vortices must contain a neutral line where ligands
are perpendicular and escape to the third dimension. On the surface of a
sphere, the neutral lines define regions surrounding each vortex.

###  3.2 The formalism

I deal with spherical NCs first. For a given arrangement, I first assume that
the entire system consists of HS, with diameter given by OPM Eq. 5, which pack
optimally. That means that a given BNSL will have a PF exactly equal to
$\eta_{HS}(\gamma)$, see Fig. 7 and Eq. 4, Eq. 8. Then, the coordination of a
given NC within the nanostructure is defined as all of the NCs that are in
actual contact. Mathematicians define the coordination of a particle as the
“kissing number”. It is important to emphasize that the coordination is a
rigorous and unambiguous quantity for any arrangement because the NCs are
modeled as HS. Thus, the coordination is the number of nearest neighbors for a
HS arrangement where the particles touch each other, and in the same way the
number of next-to-nearest and higher-order neighbors are rigorously defined.

The main idea in OTM is that at low coordination, large ligand deformations,
consisting of splayed ligands as shown in Fig. 8 and Fig. 9, the vortices, are
possible. The Orbifold Topological Model is then derived from two
principles[32, 33]:

  1. 1.

For a given lattice, the minimum free energy occurs at the lattice constant
with maximum HS PF.

  2. 2.

If the coordination is six or more, then NCs can only pack as HS, no large
ligand deformations are allowed.

The first assumption is justified by the notion that assembly into a dry state
is driven by attractive interactions, and therefore the lowest free energy
will optimize the number of ligand monomers at a given unit volume, consistent
with existing constraints.

The second assumption reflects that in order to break the HS description and
get NCs closer than their respective HS separation, ligands must splay as
shown in Fig. 8. These splayed configurations were dubbed vortices in the
original papers. Vortices do interact repulsively within a given NC; This is
illustrated in Fig. 9 where, mapping the ligands into spins, they must twist
for an energy cost, similarly as domain walls in ferromagnets. Therefore, it
becomes energetically costly for a given NC to have a large number of
vortices. Also, because ligands within vortices need to occupy neighboring
cones, see Fig. 8, volume constraints make a large number of vortices very
challenging to realize. The conclusion is that if $N_{v}$ defines the number
of vortices for a given NC, configurations with $N_{v}\geq 6$ are forbidden.
There is nothing “magical” about the impossibility of $N_{v}\geq 6$ vortices:
As we have seen already from experiments in Fig. 3 and will see with numerical
simulations, the nearest neighbor distance increases with coordination and a
putative NC configuration with $N_{v}\geq 6$ would have a nearest neighbor
distance that is very close to $N_{v}=0$, as it is demonstrated experimentally
in Fig. 3. Also, we have already seen a similar situation where the bcc
structure develops a small correction, see Eq. 19 for 8 nearest neighbors. I
will use a notation as in Fig. 8:

$\displaystyle d^{i}_{HS}$ $\displaystyle=$ HS diameter of NC i
$\displaystyle\bar{d}^{i}$ $\displaystyle=$ $\displaystyle\mbox{ OTM diameter
of NC i}\ .$ (21)

$d^{i}_{HS}$ is defined from Eq. 5. The formula for $\bar{d}^{i}$ is more
complex and examples will be derived in the following.

For those familiar with the theory of topological defects[92], it is useful to
think of a given NC as a skyrmion: Eeach ligand chain grafted to the NC deﬁnes
a vector whose origin is at the NC and whose end is at the center of the
terminal ligand group. Thus, the ligands define a mapping $S^{2}\rightarrow
S^{2}$ which enables to define a Skyrmion. Because of the way the ligands are
attached, this Skyrmion has always a topological charge $q_{S}=-1$. Still, by
focusing on the projection of the ligand vector to the sphere surface, see
Fig. 8, the order parameter defines a vortex. 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, the neutral lines
see Fig. 9, and as a result, there are no topological constraints on the total
number of vortices $N_{v}$. Ligands are ﬂexible, so the concept of Skyrmion
topological charge $q_{S}$ is only approximate, so it seems more appropriate
to denote it as a soft Skyrmion.

Figure 10: OTM applied to the MgZn2 BNSL, which consists of two
interpenetrating lattices, here shown at $\gamma=\gamma_{c}$. The large A
NCs(red) for a diamond lattice, while the smaller B ones(blue) a pyrochloric
lattice. For $\gamma<\gamma_{c}$ the large NCs are in contact and have
coordination 4, while for $\gamma>\gamma_{c}$ the smaller are in contact and
have coordination 6. According to OTM, only for $\gamma<\gamma_{c}$ vortices,
see Fig. 8, are allowed.

As a concrete example, I discuss the application of OTM to MgZn2. This BNSL is
described by space group P63/mmc (194) with a unit cell with $n_{A}=4$ in
Wyckoﬀ positions 4f and $n_{B}=8$ in positions 2a and 6h; see Fig. 10. The
maximum of $\eta_{HS}(\gamma)$, see Fig. 7b), is reached for
$\gamma=\gamma_{c}=\left(\frac{2}{3}\right)^{1/2}\approx 0.8165$. At
$\gamma<\gamma_{c}$ each A-NC is in contact with four other A-NCs while for
$\gamma>\gamma_{c}$ each B-NC with 6-B-NCs.

For $\gamma<\gamma_{c}$ each NC has coordination 4 so $N_{v}=4$ is possible.
Let us obtain the formula for $\bar{d}^{A}$ (see Eq. 3.2 for the notation
definition). We can define the following quantities

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{d^{B}_{HS}}{d^{A}_{HS}}\quad\mbox{ Same as
Eq.~{}\ref{Eq:hs:def_gamma}}$ $\displaystyle\bar{\gamma}$ $\displaystyle=$
$\displaystyle\frac{d^{B}_{HS}}{\bar{d}^{A}}$ (22)

Following assumption I, it should be that

$\displaystyle\bar{\gamma}=\gamma_{c}=\frac{d^{B}_{HS}}{\bar{d}^{A}}\rightarrow\bar{d}^{A}=\frac{d^{B}_{HS}}{\gamma_{c}}=\frac{\gamma}{\gamma_{c}}d^{A}_{HS}<d^{A}_{HS}\
.$ (23)

The condition $\bar{\gamma}=\gamma_{c}$ cannot be hold indefinitely. At
$\gamma=\gamma^{c}$ defined as

$\gamma^{c}=\frac{\sqrt{12}\gamma_{c}}{2\sqrt{11}-\sqrt{12}\gamma_{c}}=\frac{\sqrt{2}}{\sqrt{11}-\sqrt{2}}\approx
0.7434$ (24)

B-NCs, which are HS, come into contact with A-NCs and the condition
$\bar{\gamma}=\gamma_{c}$ cannot be held anymore, so B-particles need to be
displaced. Therefore, it is

$\bar{d}^{A}=\left(\frac{3}{11}\right)^{1/2}(1+\gamma)d^{A}_{HS}\quad\mbox{
for }\gamma<\gamma^{c}\ .$ (25)

The PF becomes

$\displaystyle\eta(\gamma)\equiv\eta_{OTM}(\gamma)=\left\\{\begin{array}[]{cc}\eta_{HS}(\gamma)&\gamma\geq\gamma_{c}\\\
\left(\frac{1/\gamma^{3}+2}{1/\gamma_{c}^{3}+2}\right)\eta_{HS}(\gamma_{c})&\gamma^{c}\geq\gamma<\gamma_{c}\\\
\pi\frac{(\sqrt{11})^{3}}{48}\frac{1+2\gamma^{3}}{(1+\gamma)^{3}}&\gamma<\gamma^{c}\end{array}\right.$
(29)

where the explicit expression for $\eta_{HS}(\gamma)$ is given in Eq. 20.
Therefore, the MgZn2 BNSL has an OTM branch only for $\gamma<\gamma_{c}$. For
$\gamma>\gamma_{c}$ NCs pack as HS.

It is really important to stress that OTM predictions like Eq. 23,24,25 and
Eq. 29 do not require additional or fitting parameters and are independent of
the microscopic potentials defining the interaction details. Derivations for
all other BNSLs follow the same steps and are provided in Ref. [32, 33]. AlB2,
see Fig. 7, for example, does not have any OTM branch, which explains why NCs
pack as HS, as it has already been extensively shown.

Figure 11: Simulations from Ref. [93], discussed in Sect. 4, demonstrating the
NC spontaneous valence depending on the environment, as predicted from OTM. A
NC with coordination 2 is described with a cylindrical “eigenshape”, a NC with
coordination 4 with a tetrahedron and coordination 12 is a sphere. Nearest
neighbor NCs are shown with transparencies for proper visualization. Other
eigenshapes are described in Ref. [32].

Fig. 10 shows that for $\gamma<\gamma_{c}$ the NC within an MgZn2 BNSL
effectively becomes a tetrahedron. That is, the NC shape after assembly, the
“eigenshape”, is different than the “bare” spherical shape. This ability to
spontaneously develop a valence is determined by the environment, i.e. the NC
coordination, is a general result illustrated in Fig. 11, where different
valences are shown with the help of simulation results that will be presented
later on. It should be emphasized that even for a given coordination valences
are not unique: there are other valences for coordination 4, for example, see
Ref. [32].

Figure 12: The parameter $S$ defined in Eq. 30 and illustration of the
different limits. In the planar brush limit, each face defines a planar brush.
In the spherical limit, the polyhedral nature of the NC core becomes
irrelevant. The limit $S\sim 1$ defines a nontrivial situation where ligands
lead to complex eigenshapes.

###  3.3 OTM beyond quasi-spherical NCs

Although spherical NCs have been made[59], see Fig. 5, the NC cores in
assembly experiments have the shape of a polyhedra consisting of vertices,
edges, and faces. The faces have the shape of a polygon. Generally, not all
polygons that define the faces have the same size or shape, but I will
consider $H$ the characteristic size of the largest polygon. The relevant
parameter is

$S\equiv\frac{\sigma}{\sigma_{Max}}\frac{L}{H}=\xi\frac{L}{H}\ .$ (30)

Then there are three situations, depicted in Fig. 12

$S\quad\left\\{\begin{array}[]{cc}>>1&\mbox{ Spherical limit}\\\ \sim 1&\mbox{
OTM}\\\ <<1&\mbox{ Planar limit}\end{array}\right.$ (31)

In the spherical limit, shape becomes irrelevant and the core is effectively a
sphere, see Fig. 12, so the OTM for spherical NCs as described previously
apply. In the planar limit, each of the NC faces consist of planar
brushes[56], therefore the NC may be coarse grained as a HS with the same
shape as the core, with an additional enthalpic interaction. If the ligand is
long enough to be considered a polymer, a rigorous method to quantitatively
describe the shape is provided from the methods in Ref. [94]. Therefore,
$S\sim 1$ represents the case in which quantitative solutions are more
difficult to obtain. It also happens to be the most common situation in NC
assembly where $H$ is in the nanometer scale and the ligand $L$ size is
typically of the same order.

The new variable that appears when considering shapes other than spheres is
the relative orientation among NCs. Detailed formulas for other shapes beyond
spheres or cubes are not yet available, but the cases of cubes in the relevant
limit $S\sim 1$, which I discuss following Ref. [57] is general and includes
all the elements necessary to generalize to any other shape. If the cubes
interact through faces, then vortices are not possible, since the cones, see
Fig. 8, become cylinders as there is no curvature, so there is no free volume
for neighboring chains to occupy and the NCs are effective HS cubes, see Fig.
13. This is the reason why the HS prediction works so accurately for single NC
cubes assembled into a simple cubic lattice; see Table 1. Therefore,
nontrivial OTM solutions will occur when NCs interact through vertices and/or
edges. The general cases, including cubes with sphere interactions are
described in Ref. [57]. I discuss here only the vertex-vertex case, in which
the characteristic edge length $\bar{l}$ and center-center separation
$\bar{d}$ are given by

$\bar{l}=l_{core}\rightarrow\bar{d}=\sqrt{3}\bar{l}=\sqrt{3}l_{core}<\sqrt{3}l_{HS}\
,$ (32)

where it has been assumed that the NC core vertices are in contact. This is
illustrated by Fig. 13 from the simulations in Ref. [57]. This example serves
to illustrate how the HS breaks down for shapes other than spherical, and I
refer to [57] for other cube orientations and also for systems of spheres and
cubes.

Figure 13: Example of OTM for cubes for the case of face-face, where cubes are
described as HS and corner-corner, were the cube vertices of the core are in
contact and the HS description breaks down. The ligands are drawn at a reduced
scale for visualization. Other cases are discussed in Ref. [57]. Figure 14:
Compilation of almost 200 experiments reporting BNSLs. The data is from the
same references as in Fig. 7 plus Ref. [30], which is drawn with diamond cyan
symbols. The AuCu BNSL is special as it corresponds to a situation with ligand
loss, discussed in the text.

As a final note, it would be interesting from a fundamental aspect at least,
to functionalize spherical NCs, see Fig. 5 and provide a detailed comparison
of assembled structures with shapes such as truncated octahedra, which are
well described within the spherical approximation, and in this way establish
if there are subtle effects driven from NC shape.

###  3.4 Experimental validation of OTM

At this point in time, there is no experiment or simulation that contradicts a
single OTM prediction. In this subsection, I will emphasize those experiments
that enable quantitative verification of OTM and how they fix the HS failures
described in SubSect. 2.4. Validation of OTM has also been accomplished
through simulations, to be discussed in great detail in Sect. 4 and also in
Sect. 5.

As a first example, I discuss the results presented in Fig. 6. In all cases
except for AuCu, the OTM predictions are in excellent agreement with the
experimental results by Boles and Talapin [54](see SubSect 2.3 for a
discussion on the high level of accuracy available from experiments).
Particularly notable is the agreement for Li3Bi, as the HS prediction is off
by almost 100%.

First, I discuss the MgZn2 BNSL as it has been extensively used in previous
discussions, see Eq. 23-29 and Fig. 10. For $\gamma\leq\gamma_{c}$ there is an
OTM branch calculated in Eq. 23 and therein, which is shown in Fig. 14. The
putative low-PF experiments that were of concern following the discussion in
Fig. 7 no longer exist, see the orange arrow in Fig. 14, and have now become
PF larger than the one in fcc. More importantly, the predicted lattice
constants (and PF) agree with those from the experiment, as shown in Fig. 6.
The observation that all experiments in Fig. 6 sit on the left of the maximum
PF, also consistent with the newer data from Ref. [30], clearly demonstrates
that the larger A NCs do form vortices and that A-NCs within the BNSL are well
represented by the tetrahedral eigenshape described in Fig. 11.

All BNSLs shown in Fig. 14 include almost 200 experiments, where I emphasize
Ref. [30] (in cyan diamonds), as this is the most systematic and complete data
available to this date. In some cases: NaCl, AlB2 there are no OTM branches
and the NCs are described as HS. The few experiments corresponding to low PFs
correspond to old data where the grafting density was not reported, so this is
the most likely explanation. In all those other BNSLs, the OTM branches fix
the problem of low PFs and OTM correctly predicts the lattice constants (or
PF) whenever they are available, as shown in Fig. 6. The Li3Bi BNSL is
interesting as it corresponds to a situation where the breakdown of the HS is
maximal. The latest data from Ref. [30] included in Fig. 14, which also
provides detailed measurements of lattice constants, present a very strong
validation for OTM. Applications to the DDQC/AT quasicrystal phase are in
qualitative agreement but have not been elaborated.

The failure of OTM to predict AuCu in Fig. 6 was discussed in the original
references[32, 33], where it became apparent that almost all the values of
$\gamma$ for which they were reported correspond to a low HS PF for which
there is no OTM branch. All AuCu BNSLs were recognized to have PbS (or PbSe)
cores, whose ligands detach due to weak binding[95]. The reason why OTM does
not work is that this BNSL corresponds to an NC in which some facets have lost
all their ligands and the actual cores are in physical contact. The assembly
of partially grafted NCs has very significant interest[96, 65, 97] as direct
contact among the cores allows charge transport throughout the material, so
extensions of OTM to these important cases will need to be developed. In
either case, it remains an outstanding success that OTM could identify AuCu as
inconsistent with a BNSL where NCs are fully grafted, as shown in Fig. 14,
before the systematic experiments of Ref. [95] had been reported.

Figure 15: Experiments of 8.6 and 5.3 nm NC cubes functionalized with DDAB
(Didodecyldimethylammonium bromodide) with spherical NCs functionalized with
oleylamine lead to five BNSL[78]. The cases of ABO3 and NaCl are shown in the
left, together with the curves describing HS packing for spheres and cubes
(s$|$c) and binary spheres(s$|$s). The orange arrows highlight two ABO3
experiments whose HS PF is below 50$\%$. OTM fixes it and predicts the correct
lattice constant[76]. The figure on the right shows the OTM PF for all five
BNSL reported.

Perovskite NCs[98, 99, 100] take the form of a cube, which for single
component assemble into simple cubic superlattices[101] (except for small
cubes[79]) with lattice constants and PF well described by HS as demonstrated
in Table 1. Binary systems of cubes and spherical NCs form a very rich phase
diagram[76, 77, 78]. Each NC in NaCl has coordination six and NC cubes
interact through their faces, therefore, there are no OTM branches and this
superlattice is entirely described by HS. In ABO3, however, the NC cubes
interact through their vertices, and, as it follows from the discussion in
Fig. 13 this leads to the breakdown of HS descriptions and the emergence of
OTM branches. Fig. 15 highlights two experiments (red dashed arrows) whose PF
within HS would be less than 50 $\%$ and the inclusion of OTM increases the PF
above the fcc and also brings the lattice constants in agreement with the
experiment. This example illustrates how important the relative NC orientation
is in determining whether HS descriptions break down. Fig. 15 shows the OTM PF
for all five phases encountered (including only 2 NC sizes) and how all have
large PF ($>0.7$). Refer to [78, 57] for more detailed quantitative analysis.

Figure 16: Four representative polycatenar ligands used to assemble BNSLs.
$l_{eff}^{DH}$ is the effective size (in nm) of the ligand shell used in the
paper Ref. [102]. Notations follow the same convention as the original paper.

An interesting more complex example is provided by polycatenar ligands[102],
shown in Fig. 16. They consist of a thiol or carboxylic group hydrocarbon
chain linked to the NC core, with an intermediate linker with two or three
hydrocarbon chains linked through an ether group. The HS diameter of a NC in
the original reference was obtained from

$d_{HS}=d_{core}+l_{eff}^{HS}\ ,$ (33)

where $l_{eff}^{HS}$ is given in Fig. 16 for the four more relevant ligands.
Single component systems assemble into bcc and fcc or hcp consistently with
previous discussions.

BNSL phases are also reported. The most notable aspects relevant for this
review are the presence of AuCu3 (see discussion after Table 2), which is
obtained for three different combinations: 6.5nm 20p-Au+2.4nm 20d-CdSe
($\gamma=0.48$), 6.5nm 20p-Au + 5.5nm Oleate-CdSe ($\gamma=0.44$), 5.5nm
20b-PbSe + 2.4nm 20d-CdSe($\gamma=0.55$). There is also an ubiquitous presence
of AuCu for many values of $\gamma$ where there is no high PF. In light of our
previous discussion, the appearance of this phase is surprising, so newer
experiments will be necessary to clarify this case.

For MgZn2, BNSLs were reported for 5.3nm Oleate-CdSe + 2.4
20d-CdSe($\gamma=0.71$), 3.5nm 20i-CdSe + 2.4 20b-CdSe ($\gamma=0.88$), 3.5nm
20b-PbSe+2.7nm 20b-CdSe($\gamma=0.89$). These last two values of $\gamma$
would imply vortices for small B-NCs, which disagree with OTM. I believe this
can be explained because $\gamma$ is overestimated by using a fixed ligand
shell $l_{eff}^{HS}$ in Eq. 33, neglecting the curvature effect in the OPM
formula, which for small NCs is quite important. A rigorous calculation is
beyond the scope here, but a rough estimate can be obtained if we just count
the number of actual chemical bonds if the upper ligands were just a single
chain and assume an effective maximum grafting density $\xi=1$. This gives
$n=29,31$ for 20b,20i ligands, then using the OPM formula Eq. 5 for the
equivalent hydrocarbon chain, one obtains $d^{HS}=6.8$ nm (3.5nm 20b-PbSe) and
$d^{HS}=5.6$ nm (2.7nm 20b-CdSe), for a $\gamma=0.82$, which is in the
expected range according to previous discussion.

In summary, an extensive comparison with single- and binary-NC systems,
including spherical and cubic shapes, provides overwhelming experimental
verification for OTM. Additional stringent validation can be provided from
simulations, as I discuss in Sect. 4.

Figure 17: MOLT results from the papers by Missoni and Tagliazucchi[103, 104].
On the left, the case of a spherical NC were only bcc and fcc exist as
equilibrium phases. On the right, with truncated octahedra NCs bct becomes
stable at higher solvent content. The quantity $\Phi_{sv}$ is the solvent
volume fraction defined in Eq. 34 and has a somewhat different meaning as in
the rest of the paper, see discussion in the text. The grafting density is
$\sigma=4$ ligands per nm2. $D_{c}$ is the actual diameter of the truncated
octahedra, where different sizes vary the hexagonal edge, see the original
references for additional details.

###  3.5 Towards an OTM Free Energy

OTM provides a detailed prediction of a given BNSL structure but cannot
establish which lattice defines the equilibrium state because it lacks an
explicit free energy. Such a putative free energy must include ligand
conformations. The Molecular Theory (MOLT) developed in Ref. [103, 104],
inspired from previous models in lipid bilayers and other soft systems[105,
106, 107, 108] is able to incorporate ligand conformations in a minimal and
computationally affordable way.

The advantage of MOLT compared with simulations is fast turnaround and no
statistical fluctuations, i.e. within approximations made, free energies can,
in principle, be calculated to any needed precision. This also enables
calculations for processes consisting of a succession of equilibrium states,
i.e. quasistatic. A limitation for MOLT is that it requires NC positions and
orientations, so going beyond single-component systems requires specifying
many degrees of freedom and actual calculations require some prescription for
the NC positions and orientation. OTM does provide positions and orientations
so MOLT and OTM together enable, in principle, a first-principle determination
for any NC configuration.

The original MOLT papers[103, 104] assumed that each point in the space is
occupied by solvent or NC, so no voids are allowed in the configurations.
Therefore, the volume fraction of solvent is

$\Phi_{sv}=1-\eta$ (34)

where the PF $\eta$ is the volume fraction occupied by the NC. Here, however,
the volume is interpreted as defining a molar volume, rather an actual volume
of the corresponding moiety, thus imposing an incompressibility condition so
that $\eta$ can take values as large as one, as it is not the exact same
quantity as defined in Eq. 8. Applied to single-component systems, MOLT
predicts that the equilibrium structure for a NC with a spherical core is
either bcc or fcc only, see Fig. 17. However, the difference in free energy

$f_{bcc}-f_{fcc}\approx-0.5k_{B}T\ .$ (35)

at the lowest solvent volume fraction $\Phi_{sv}=0.1$ studied. Fig. 20 shows
that the actual values of the free energy per NC in the dry limit for both
$f_{bcc},f_{fcc}$ are of the order of 500$k_{B}T$. Therefore, calculating Eq.
35 from numerical simulations requires free energies obtained with a precision
of less than 0.1$\%$, which is extremely challenging and more than justifies
the efforts to develop methods complementary and alternative to simulations.

When the core is modeled with a more realistic shape, a truncated octahedra,
fcc and bcc are still the only equilibrium phases at low solvent fraction
$\Phi_{sv}\lesssim 0.1$, but the bct phase becomes stable for larger solvent
volume fractions, as illustrated in Fig. 17. These calculations establish that
bct is an intermediate during solvent evaporation and not an equilibrium phase
in the dry limit, as discussed in SubSect. 2.4. Ref. [63] has experimentally
shown the presence of bct intermediates. These examples illustrate the power
of MOLT, not just to describe equilibrium, but to characterize intermediates
during solvent evaporation, a topic that I elaborate on further in Sect. 5.

An exciting new development is the generalization of MOLT in the grand
canonical ensemble, enabling the solvent vapor pressure to be monitored as an
external parameter, much like an experiment. In this way, voids are
dynamically generated if they minimize the free energy and the dry limit can
be rigorously reached. Predictions for BNSL should become available in the
very near future.

##  4 Numerical Simulations

###  4.1 HOODLT and technical simulation details

HOODLT is a software designed to perform calculations with NCs. It follows a
design using the best software practices available [109, 110, 111], allows
full reproducibility of results, simplifies enormously running new simulations
and drastically streamlines the complexities of NC calculations. Free energies
are calculated either approximately by Dynamic Lattice Theory[112, 113, 114,
115], or exactly using different algorithms[88, 64, 116]. HOODLT includes more
than 100 NC cores, as described in Refs. [117, 70]), ligands (hydrocarbons,
polystyrene, polyethylene glycol, etc.), and solvent (water, toluene,
hydrazine, etc.). There are methods to build initial configurations, run
simulations through HOOMD-Blue[118], algorithms (identify vortices, motifs,
etc.) or data analysis, for example, through WHAM[119] and Bridge
Sampling[120]. The crystal structure database implements all analytical OTM
formulas[121] such as Eq. 23-Eq. 29. HOODLT is freely available in bitbucket

https://bitbucket.org/trvsst/hoodlt/src/master

see Ref.[122]. The force field more commonly used in our simulations is
described in great detail in our previous papers [123, 64, 116, 93, 124, 125].
Atoms within the NC are modeled with rigid body dynamics[126, 127]. HOODLT[87]
has been used to implement the force field, prepare simulations, create
initial configurations, and analyze the data. Almost all simulations have been
run at a nominal temperature of $T_{m}=387$ K. This temperature is
significantly higher than the one most commonly used in experiments, but the
main goal of the simulations is to verify the OTM and higher temperatures
enable better thermalization. In regards of the parameter $\xi$, see Eq. 5, it
depends on the maximum grafting density $\sigma_{Max}=\frac{1}{A_{0}}$ (or the
ligand footprint). There are methods to calculate this quantity from
simulations[128, 129] and for very small NCs $<2$ nm where the core curvature
is large, $\sigma_{Max}$ may depend on NC size[55]. Herein I concentrate on
the simulation results. The simulation technical details are extensively
discussed in the original papers.

Figure 18: Calculation of the PMF and N-PMF for a tetrahedra with
Au1289(SC12)258, which are NC consisting of 1289 Au atoms for a 4.0 nm
diameter, with 258 dodecanethiol ligands. The N-PMF is a tetrahedra with a NC
at the center, where the edge coordinate and the NC-NC distance are
quasistatically decreased. The many body effects $\Delta^{MB}$ defined in Eq.
38, are obtained by subtracting from the actual free energy the two body free
energy: the sum of pairs of potentials of mean force.

The calculations are done by subjecting the NCs with harmonic springs and
obtaining different observables as a function of NC separation. There are
different ways, as discussed in the original references[64, 116] to eliminate
the possible bias introduced by the springs. The free energy of a superlattice
is obtained by integrating the pressure, where the equilibrium lattice
constant is obtained from basic thermodynamic identities

$P=-\frac{\partial{F}}{\partial V}\implies\left(\frac{\partial{F}}{\partial
V}\right)_{V=N_{WC}V_{WC}(a)}=0\ ,$ (36)

so the lattice constant is defined at zero pressure. Here $N_{WC},V_{WC}(a)$
is the number and volume of Wigner-Seitz cells, where the latter quantity
explicitly depends on the actual value of the lattice constant $a$.

###  4.2 PMF for pairs and N-PMF for clusters

The PMF, which here means the free energy of two NCs, is obtained by two
methods: 1) Force Integration of the total forces in the NCs and 2) Weighted
Histogram Analysis Method (WHAM)[119], with both methods providing equivalent
results. Fig. 18 shows a concrete example of a 4 nm core functionalized with
dodecanethiol at close to maximal grafting density, from which it is possible
to establish the following points:

  * •

At equilibrium (minimum of the PMF) HS breaks down: NC separation is
approximately 10 Å closer than predicted by the HS models, which is defined by
OPM Eq. 5.

  * •

The equilibrium separation is defined by OCM, with some nuances, as described
in Ref. [123].

  * •

The PMF minimum is very deep and on the order of -100$k_{B}T$.

  * •

The internal energy is 3-4 times higher (in magnitude) than the free energy.

Analysis of many other cases shows that these statements are general. The last
statement establishes that the free energy is the result of a significant
entropy-energy cancelation[130]. The PMF minimum ($F_{min}$) defines the
cohesive bonding energy between two NCs. Certainly, $100k_{B}Ts$ is a large
value, which often ignites many passionate discussions about whether these
systems are trapped in metastable states and can actually reach thermal
equilibrium. This point will be discussed further below in Sect 5. Here I just
emphasize that this large cohesive energy has been consistently obtained in
all PMF calculations using different force fields and NC shapes and ligands.
Also, cohesive energies grow with NC core size and the length of the ligand.
The Vlugt group, for example, summarized their results by the following
formula [26]

$F_{min}\approx-1.15(n+1)^{2}k_{B}T\ ,$ (37)

where $n$ is the number of carbon atoms within the alkyl group. For
dodecanethiol (n=12) ligands it is $F_{min}\approx-194k_{B}T$. Other estimates
are entirely consistent with these calculations and can be read in Ref. [131,
132, 133, 134, 123, 135, 136, 137, 138].

The $N-$ body PMF (N-PMF) has been obtained for many different configurations,
such as $P_{n}$ (NCs in a regular polygon with $n$ vertices surrounding a NC
at its center), tetrahedra shown in Fig. 18 and many others, see Ref. [116].
The N-PMF is obtained by holding the NCs with springs and computing the
reversible work as the NCs are brought together, as described in Fig. 18. Many
body effects quantify the difference between the actual free energy and the
one obtained from assuming two-body interactions only

$\Delta^{MB}(T,R)=F_{N}(T,R)-\sum_{i=0}^{N}\sum_{i>j}^{N}F_{2}(T,R_{ij})\ ,$
(38)

where $F_{N}(T,R)$ is the free energy (N-PMF) of the system and
$F_{2}(T,R_{ij})$ is the free energy (PMF) for two NCs separated by distance
$R_{ij}$,

Results for N-PMF are summarized in Fig. 21 and follow the general conclusions

  * •

For coordination less than six, the HS description breaks down, consistent
with the OTM assumptions.

  * •

Eigenshapes are developed for equilibrium configurations, such as shown in
Fig. 11.

  * •

Equilibrium distances increase with coordination: as an example, the N-PMF for
a tetrahedra in Fig. 18 clearly shows that the equilibrium, pointed as
OTM(dashed blue line) is further away from the minimum of the PMF(dashed red
line) thus indicating that with coordination 4 NC are separated further than
coordination 2.

  * •

There are significant many body effects (marked in green in Fig. 18), defined
by Eq. 38.

  * •

The cohesive free energy is of the order of hundreds of $k_{B}T$s.

These PMFs and N-PMFs provide a rigorous validation of the assumptions that
lead to OTM, as discussed in SubSect 3.2. Fig. 21 summarizes the results of
the simulations, clearly showing how the nearest-neighbor distance increases
with coordination and approximately saturates for 6, where the results are
basically consistent with the HS description, defined by the OPM, see Eq. 5.
Fig. 11 shows snapshots of representative simulations at the minimum of the
N-PMF, with the vortices explicitly drawn for coordination 2 and 4, thus
illustrating how NCs spontaneously generate a valence and an effective
eigenshape, as discussed in more detail above. At coordination 12 the NCs
become isotropic with the corresponding eigenshape of a sphere, in agreement
with OTM.

Figure 19: Free energy calculation of a bcc and fcc superlattice with
Au1289(SC12)258. The many body effects are obtained by subtracting from the
actual free energy the two body free energy: the sum of pairs of potentials of
mean force, calculated in Fig. 18. The lattice constant shown with blue dashed
line is the minimum of the free energy, see Eq. 36. OPM and $l_{f}$-OPM are
shown in black and grey dashed lines.

###  4.3 Superlattices

Simulations of superlattices with ligands containing between 5 and 20 carbons
have been reported in [64, 139] within all-atom and united models. Typical
single-component and binary systems are shown in Fig. 19 and Fig. 20, where
the free energy (in blue, bottom) is obtained by integrating the pressure (in
blue, top) as a function of the lattice constant. The equilibrium lattice
constant is obtained at zero pressure, i.e. the minimum of the free energy, as
described in Eq. 36.

Figure 20: Top right: Comparison of predicted lattice constant for fcc and
AlB2 for many different superlattices, showing excellent agreement with HS.
Bottom left shows the same calculation as in Fig. 19 for MgZn2 with the
minimum correctly predicted from OTM and in disagreement with HS. The right
side shows a detailed analysis of simulation snapshots at the free energy
minimum, with actual nearest neighbor separations predicted from OTM and
eigenshapes in agreement with Fig. 11.

OTM predicts that a fcc lattice consists of packing HS, and indeed, Fig. 19
shows that the lattice constant is accurately predicted by OPM. Fig. 20
summarizes the results of many fcc simulations, and all of them align very
accurately with the OPM result. For bcc, the OPM is not quite accurate, but
this is expected, as the lattice constant or nearest-neighbor distance is
given by Eq. 19, marked as $l_{f}$-OPM in Fig. 19.

In a fcc, the free energy as a function of lattice constant is very well
represented by a sum of two body potentials all the way to the equilibrium
result. That is, many body effects are very small. For the bcc, there are
significant many body effects, see Eq. 38, near the free energy minimum. The
difference in free energy between fcc and bcc is too small, and simulations
were not able to establish which lattice describes equilibrium and confirm the
derivation (and experimental results) in Eq. 18, see the discussion in Ref.
[64]. The MOLT[103, 104] mean field model briefly discussed in SubSect. 3.5 is
able to discriminate between bcc/fcc with excellent quantitative agreement
with Eq. 18, thus providing an important validation of its predictive power.

Coarse-grained models have been developed for these problems[140].
Particularly relevant are potentials derived from machine learning[141]. This
is an important area for further development, as all atom or united models
have obvious limitations that hamper reaching large spatial or temporal
scales. An interesting result from Ref. [141] is the negligible effect of many
body effects for fcc, in agreement with the results in Fig. 19.

Simulations for the MgZn2 BNSL at $\gamma=0.69$ are shown in Fig. 20. The
minimum (blue dashed line) is clearly off from the OPM result (grey dashed
line), thus illustrating the breakdown of the HS description. The OTM result,
see Eq. 25, predicts the free energy minimum, thus providing yet another
cross-check on its validity. Snapshots of the NC configurations at the minimum
demonstrate that the large A-NCs develop a four-valence with tetrahedral
eigenshapes, identical to the one obtained in Fig. 11 by direct simulations
with tetrahedral clusters, made possible by the presence of vortices. Explicit
comparisons show that the OPM sizes overlap and that OTM accurately defines NC
contact (or kissing). Fig. 20 includes a control simulation of a single
component with the same A-NCs in fcc where the sizes are described by OPM,
nicely demonstrating the OTM prediction where the same NC develops a different
valence and eigenshape depending on the environment (the coordination).

Simulations for the AlB2 BNSL are also shown in Fig. 20. All of them are in
excellent agreement with OPM, showing that this BNSL consists of packing HS.
The equilibrium snapshots for both A-A and A-B contacts are in excellent
agreement with the OPM. Thus, the simulations confirm that AlB2 is accurately
described by HS. This analysis has been carried out for all other BNSLs
reported in the literature and refer to [139] for a detailed discussion. The
most important point is that they fully validate OTM predictions.

Figure 21: Right: Summary of all simulations with clusters. Here $P_{n},n=2-6$
are planar clusters with the shape of regular polygons with $n$ vertices.
Three polyhedra (tetra=tretrahedra, octa=octahedra and ico=icoshaedra) are
also considered. The lattice constant is in between OPM (HS) and OCM, in
agreement with OTM and experimental results. Left: summary of BNSL stability,
Eq. 39 for all BNSL simulated in Ref. [139] as a function of $\gamma$, see
definition in Eq. 4.

A general $A_{p}B_{q}$ BNSL, with $p$ A-NCs and $q$ B-NCs within the Wigner-
Seitz cell has a free energy $F(A_{p}B_{q})$ normalized by the total number,
both $A,B$, of NCs. The quantity $\Delta F$

$\Delta F=F(A_{p}B_{q})-\frac{1}{p+q}\left(pF(A)+qF(B)\right)\ ,$ (39)

where $F(A/B)$ is the free energy of the corresponding single component
system, determines whether the BNSL is stable against phase separation (at
zero pressure) into two single component superlattices.

Fig. 21 shows $\Delta F$ as a function of $\gamma$, see Eq. 4, for all BNSLs
studied in Ref. [139]. The main conclusions are summarized as:

  * •

There are three BNSL: Fe4C, cub-AB13 and AuCu3 that are not stable against
phase separation for any of the $\gamma$ values for which they were simulated.

  * •

NaCl is the only BNSL that is clearly stable, at least for $\gamma<0.5$.

  * •

Stability for all other BNSLs could not be established within the accuracy of
the simulation.

Regarding the three BNSL that should separate into a single component, Fe4C
and cub-AB13 were identified in the early days in Ref. [23] and Ref. [142] but
have not been reported, at least to my knowledge, in more recent experiments.
It might be necessary to reanalyze the experimental data to clarify these
BNSLs. AuCu3, on the other hand, seec Fig. 14, has been reported for many
different cores and ligands, as shown in the five examples in Table 2: PbSe,
PbS, Ag and Fe3O4 and they are in the correct range of $\gamma$. The situation
of ligand loss, as is the case for AuCu, does not seem plausible. Although the
calculations in Fig. 21 do not apply, the AuCu3 is also present with
polycatenar ligands, as described above. Further work will be required to
clarify why free energy calculations predict that AuCu3 is unstable against
phase separation.

Reference | NCs (AuCu3 BNSL)
---|---
| core | ligand | $d_{HS}$ | $\gamma$
[71] | PbSe(7.7), Pd(3.4) | Oleic, C12 | 10.8, 5.3 | 0.49
[73] | Ag(9.6),Ag(4.0) | Oleylamine, C12 | 12.9, 6.0 | 0.46
[24] | Fe3O4(13.4), Au(3.8) | PS (5.3, 3.0)kDa | 20.0, 9.8∗ | 0.49
[143] | Fe3O4(10.0), Au(5.0) | Oleic, C12 | 15.9, 8.6 | 0.54
[30] | PbS(9.3),Au(5.1) | C18,C12 | 12.6, 7.2 | 0.57
Table 2: Five experimental examples reporting the AuCu3 BNSL. In parenthesis
is the size of the NC core in nm. PS: Polystyrene. The $d_{HS}$ value $\gamma$
is calculated from OPM in Eq. 4, except for the one marked with $*$, which is
quoted from the reference.

The difficulty in confirming whether BNSL define thermodynamic equilibrium
shows that many BNSL are marginally stable at best, raising the possibility
that subleading effects, like NC-core interactions, dipoles, and other
effects, as discussed in Ref. [139] may tilt the balance one way or another.
It emphasizes why it is so relevant to develop alternative calculations,
either with mean-field models like MOLT or from coarse-grained simulations,
specially with machine learning potentials with different environments
included in the training set.

Figure 22: Single component and BNSL results from Ref. [144] and Ref. [145].
If polymer ligands are sufficiently large they engulf each other cores, as
shown in the figure. Examples of vortices for a 3.8 nm core: for short
polystyrene ligands the results show nice vortices completely equivalent to
the ones with hydrocarbons. For longer ligands, the vortices are not as well
defined but still in agreement with OTM. The BNSLs are well described by OTM,
with a similar accuracy as in Fig. 19 and Fig. 20.

###  4.4 The polymer limit

BNSLs prior to 2015 were obtained with ligands that are relatively short, at
the very least short enough not to be considered polymers. This situation
changed with the results provided in Ref. [24], where Au NCs functionalized
with polystyrene with molecular weights in the range of 1.1-20 kDa were
reported to assemble into the same BNSLs already obtained with short ligands.
The relevance of this development is that expands the range of NC assembly
while connecting it with polymer theory and the abundant literature on polymer
brushes[56], which describes all solvent conditions. Since the focus of this
paper is on dry systems, only the case of poor solvent, actually non-solvent
in almost all cases, is relevant for this discussion. The excellent review in
Ref. [146] provides the general results for all solvent conditions.

Simulations of NCs with long ligands were carried out in Ref. [144] and Ref.
[145] using the coarse-grained model for polystyrene developed in Ref. [147].
There are, indeed, new effects when considering polymers. An example is given
in Fig. 22, where for two NCs, long enough ligands engulf each other NC cores:
The ligands from the back of the NC extend and surround the core of the other
NC. As the NCs become closer, the minimum of the PMF occurs at a separation
much closer to that predicted by OPM, so the HS description breaks down. It is
possible to define vortices and analyze how they evolve as the polymer size
increases. Fig. 22 shows 3.8 nm cores with increasing polystyrene monomer
length at the minimum of the PMF: short polystyrene display the same vortices
as found with short hydrocarbon chains, thus demonstrating the universality of
OTM predictions. For longer polymers, the vortices do not have any obvious
geometric structure but, as is elaborated further below, they remain
conceptually useful.

Superlattice free energies and lattice constants are determined by the same
methods discussed in SubSec 4.3. Many body effects are quantified by Eq. 38.
However, for long ligands, because of the engulfing effect already mentioned,
the PF includes ligand conformations that are not possible with short ligands,
so many body effects appear to be very large for both fcc and bcc in Fig. 22.
Still, the minimum for fcc is well described by OPM for fcc even for a 11.5
kDa polystyrene ligand and longer, see original reference [144]. The bcc
minimum is lower than the OPM, again in agreement with expectations; see Eq.
19. Therefore, the evidence from simulations is that the OTM still applies for
moderately long ligands.

Extensive simulations of BNSL functionalized with polystyrene have been
reported in Ref. [144] for all BNSL experimentally described in Ref. [24]
(simulated with the same core and ligand)and include explicit calculations of
lattice constants and free energies. Fig. 22 summarizes the results with two
plots: The left shows nearest-neighbor separations for all NCs within BNSL
with coordination six or larger, while the right shows those with coordination
less than six. The left plot follows the OPM result very accurately while in
the left the NCs are closer than OPM and in good agreement with the OTM
predictions, which are drawn with empty symbols. Snapshots at equilibrium are
shown for the MgZn2 BNSL and demonstrate the expected overlap in the OPM size,
with vortices explicitly drawn, illustrating once again, the universality of
OTM predictions.

The question on whether OTM applies to moderately long ligands is answered
positively by simulations. Still, additional work will be needed to establish
the limit of very long ligands, such as discussed in Ref. [58]; It seems
reasonable to expect that a NC functionalized with a long polymer ligand is
equivalent to a diblock, where one of the components is much shorter than the
other, in the strong segregation limit[148, 149] so that the minority
component forms a spherical micelle. In that case, the short concave polymer
component plays the role of the NC and the longer convex polymer plays the
role of the ligand. How to leverage the large body of work in block copolymer
melts into NC assembly is discussed further in conclusions.

Figure 23: (Left:) Example of a NC on a substrate, explictily showing the role
of the relative orientation. Example of a calculation with different bonds,
labeled as $(m,n)$ where m,n is the coordination of each NC. (Middle) N-PMF
calculated on a supported substrate (S) and Free Floating (FF). Illustration
of vortices as induced by the substrate. Right: Summary of all results and
example of how nearest neighbor depends on concentration.

###  4.5 NCs on a solid support

Experiments of NCs on substrates, like the ones in Fig. 3, enable the test of
important properties and effects and have played a very important role in the
development of OTM. The presence of a solid support on ligand conformations
and eigenshapes needs to be quantified, which provides the motivation for the
simulations carried out in Ref. [150]. Fig. 23(Left) shows the setup, where
NCs are investigated in different arrangements that include core orientations.
Of particular interest are nearest-neighbor distances of the form $(m,n)$
where $m\geq n$ are the coordination of the two NCs involved. A configuration
with $(6,n)$ $n=3,4,5,6$ is shown in the figure; see also Fig. 3.

The main physical intuition is that the substrate is akin to an NC with a very
large diameter and should induce a vortex on the NC ligands next to it. This
intuition is correct, as it is explicitly shown in Fig. 23(center). Therefore,
the effect of the substrate is to add one unit to the coordination or kissing
number. The N-PMF shown in Fig. 23(center) compares a $(6,3)$ configuration
without a substrate, i.e. “free floating” (FF) and with (S). The free energy
is reduced on the latter and there is a small but measurable shift in the
minimum, where NCs are further apart because of the increased coordination.

Fig. 23(Right) at the bottom shows a histogram of $(6,n)$ bonds within the
simulation for n = 3,4,5,6 illustrating that the NCs are further away as the
coordination increases, in agreement with OTM and the experimental results in
Fig. 3. Fig. 23(Right) summarizes all the results where as the coordination is
increased, the OPM result is attained.

Figure 24: Top: Illustration of the evaporation process: a droplet containing
$N$ NCs is quasistatically evaporated until the liquid phase completely
disappears, leaving behind an equilibrium NC cluster. Middle: Plot of NC
radius as a function of $\rho_{avg}$, defined in Eq. 40. Example of
equilibrium clusters for $N=13,28$ and summary of nearest-neighbor distance
for NCs within the interior of a cluster. Bottom: Vortices as a function of
solvent concentration and a Voronoi cell for one of the interior NC at $N=28$.

##  5 Dynamics of NC Assembly

###  5.1 Solvent evaporation models

Solvent evaporation for point(structureless) particles has been investigated
using implicit [151, 152] and explicit[153, 154, 155, 156] models. These
results are of direct relevance to actual solvent evaporation experiments as
there are many aspects that are independent of the microscopic details of the
particles. There are also effects related to the ligands, which require more
detailed models.

In order to reveal specific effects caused by the ligands, we have carried out
simulations at the united atom level with both toluene[93] and octane[124] as
solvent. Due to space and time limitations, these simulations included only a
relatively small number of NCs $N<30$. The average density is defined as

$\rho_{avg}=M_{sol}\frac{N_{sol}}{V}\ ,$ (40)

where $M_{sol}$ is the molecular weight of the solvent and $N_{sol}$ the
number of solvent molecules within a simulation box of volume $V$.

The main process is described in Fig. 24. Initially, the system consists of a
solvent droplet with $N$ NCs inside, coexisting with the vapor. Because this
is a good solvent for the ligands, the solution is stable and NCs repeal each
other within the solvent. The simulation box is quasi-statically expanded
until the liquid droplet is fully evaporated and an equilibrium with $N$ NC
cluster remains. These clusters undergo a crystallization process reminiscent
of the entropic crystallization in HS as the volume fraction (or PF)
increases. Analytical formulas describing the droplet radius as a function of
$\rho_{avg}$, defined in Eq. 40, have been derived[93] and are in excellent
agreement with the simulation results, as shown in Fig. 24.

These simulations allow for a rigorous determination of the relaxation times
$\tau_{R}$ involved in the evaporation process. For a sufficient amount of
solvent (more than 50% volume fraction, similar to HS), the relaxation times
are consistent with the Maxwell evaporation model[157], which assumes that
evaporation is a diffusion-limited process of the surrounding vapor phase.
However, Maxwell theory breaks down at lower solvent content, where relaxation
times increase in a way that is dependent on the total number of NCs within
the droplet, as shown in Fig. 25.

###  5.2 Equilibrium structures from solvent evaporation

The process described in Fig. 24 has been carried out for droplets containing
up to $N=30$ NCs. Examples of the final, fully evaporated structures are shown
for two cases, $N=13,28$ with the other cases $N<30$ available in the original
reference[124]. For $N=13$ the structure consists of a central NC surrounded
by 12 other NCs sitting at the vertices of the icosahedron. For $N=28$, the
structure cannot be identified as corresponding to any known polyhedra. It
contains 3 NCs inside the cluster and 25 outside. The Voronoi cell for each of
these three interior NCs consists of polyhedra that approximate an
icosahedron[124], as shown in Fig. 24 where the Voronoi cell contains 7
pentagons of the 12 that form an icosahedral cell.

The nearest-neighbor distance of interior NCs within the $N$-clusters $N>12$
are shown in Fig. 24. Within error bars, their radius is described by HS,
marked with a dashed line, as expected for NCs with coordination larger than
six. As pointed out in the original reference[124], the nearest-neighbor NC
distances of the clusters obtained after solvent evaporation are in perfect
agreement with the minimum of the N-PMF in completely dry systems, thus
proving that at least for small NC systems, solvent evaporation reproducibly
leads to the minimum of the free energy and therefore equilibrium, i.e. there
is no evidence for metastability.

The evaporation simulations enable to visualize how vortices are formed. This
is illustrated at the bottom of Fig. 24 as the solvent content is reduced,
quantified by $\rho_{avg}$ defined in Eq. 40. The radius of the droplet in
Fig. 24 as a function of $\rho_{avg}$ shows that the appearance of vortices is
sudden and occurs in the last stages of the drying process where the solvent
content is minimal, around $\rho_{avg}\sim 1$. Work is in progress to describe
similar processes in superlattices and to characterize the intermediates to
the final equilibrium structure.

Figure 25: Relaxation times $\tau_{R}$ for different $N$ clusters as a funcion
of PF. A clear transition is apparent at about 0.5 PF. The lattice-gas
evaporation model defined in Ref. [158, 159] provides a description of the
long time scales.

###  5.3 Are superlattices in thermal equilibrium?

The large interaction free energy, of the order of (at least) $100k_{B}T$ per
NC has prompted many spirited debates on whether dry superlattices are true
equilibrium or just metastable. A common argument is that if we had a system
of particles whose interaction potential is attractive and of the order of
$100k_{B}T$ or more, then such a system would agglomerate or quench and never
reach equilibrium. This argument, if true, would apply to the evaporation
simulations in Fig. 24, but as I have shown, N-clusters are reproducible and
minimize the free energy. Basically, the transition from repulsive NCs when
solvent is abundant to 100$k_{B}T$ attraction when solvent is absent occurs
slowly enough so that NC attraction is gradual. This is implied from Fig. 25,
where at PF approximating the HS crystallization of 50$\%$, relaxation times
increase and there is a weak NC attraction, very far from the 100 $k_{B}T$ of
the dry final state.

The follow-up argument is then that if a metastable structure is reached at
some point for sufficiently high PF then the large cohesive energies will
prevent the system from reaching the actual equilibrium state. In other words,
the experimentally observed superlattices reflect a metastable state with a
low activation energy rather than the true equilibrium. The example of bct as
an intermediate has been discussed in detail in SubSect. 3.5. Experimentally,
intermediate states have been characterized[61, 160, 63, 161, 65, 162], so
this is a real possibility. However, current evidence does not favor
metastability, at least for most of the cases that have been investigated in
detail; In addition to evaporation on a substrate, microfluidics[163] or
emulsification[143, 164, 165] for example, all lead to the same reproducible
structures, despite (or because) the intermediates; see Ref. [162]. I will add
that this argument is not specific to NC assembly: it can also be applied to
basically any crystallization process: simple carbon, for example, whose
cohesive free energy (in diamond or graphene) is 100 times or more larger than
$k_{B}T$ at room temperature.

It is certainly the case that at some point in the evaporation process the
cohesive energy becomes too large for the system to experience a phase
transition to another crystal structure. However, the main point is that this
statement being true does not imply that the system will be in a metastable
state. At that solvent content where the cohesive free energy exceeds many
$k_{B}T$ s, it may already have reached a partially swollen equilibrium
structure, isostructural to the final dry equilibrium, or the intermediate may
have an intrinsic instability at a certain solvent content and undergo a
solid-solid transition. This is a difficult problem and will require further
studies, such as simulations and mean-field calculations with MOLT[103, 104]
shown in in Fig. 17 for example. Still, I insist that this is a general
problem shared by many crystallization processes and it is not specific to NC
assembly.

I now make an argument showing that the free energy cohesive energies do not
imply irreversible or quenched long time dynamics. The lattice gas model
developed in Ref. [158, 159] has been shown to accurately model long time and
spatial scales, i.e. coarsening, in solvent evaporation processes. It is
schematically shown in Fig. 24: it approximates the space as a lattice
occupied by liquid or gas solvent (blue or gray), nanoparticle (red), and it
is possible to include an interface (green) although this is not necessary for
the argument here. The Hamiltonian has the form

$H=-\varepsilon_{l}\sum_{\langle
i,j\rangle}l_{i}l_{j}-\varepsilon_{n}\sum_{\langle
i,j\rangle}n_{i}n_{j}-\varepsilon_{n,l}\sum_{\langle
i,j\rangle}n_{i}l_{j}-\mu\sum_{i}l_{i}$ (41)

where $l_{i}=0$ states that there is gas and $l_{i}=1$ liquid at site $i$. The
sites occupied by NCs are labeled $n_{i}=1$ or $n_{i}=0$ otherwise. The
quantity $\varepsilon_{l}$ represents the free energy of attraction between
the liquid solvent and $\mu$ is the liquid chemical potential. The coexistence
gas-liquid is defined by the condition that the magnetic field term in the
Ising model equivalent to Eq. 41 is zero, which leads to
$\mu=-\frac{z}{2}\varepsilon_{l}$ (with z=4 for 2d square, z=6 for 3d cubic,
etc..). Therefore, simulations are carried in the gas phase, that is
$\mu<-\frac{z}{2}$ so that the solvent does evaporate. The parameter
$\varepsilon_{np}$ defines the NC-liquid interaction.

The parameter $\varepsilon_{nn}$ defines the NC-NC interaction in the dry
state and can therefore be related to the bonding free energy. Typical
simulations use $\varepsilon_{l}=4k_{B}T$ and
$\varepsilon_{n}=2\varepsilon_{l}=8k_{B}T$[158] or $\varepsilon_{n}=4k_{B}T$
[159] . Let us compute the putative free energy of a fcc superlattice. If I
assume that $\xi=1$ nm, see Fig. 25, then a 5nm NCs will include 25 of these
cubes per face (as opposed to the $2\times 2$ in Fig. 25). Therefore the
cohesive energy will be

$F_{cohesive}\approx\frac{12}{2}\times 25\times\varepsilon_{n}=6\times
25\times(4-8)k_{B}T=600-1200k_{B}T\ ,$ (42)

where the factor of 12 is the coordination for an fcc lattice. This result is
entirely consistent with the fcc cohesive free energies in Fig. 19 although
the numbers are somewhat dependent on the actual value of the coarse grained
linear length $\xi$ in nm.

The relevance of this discussion and the lattice-gas model in Eq. 41 is that a
large cohesive energy such as in Eq. 42 does not imply that the system leads
to irreversible dynamics, as the lattice-gas model does not lead to quenched
dynamics and random agglomeration.

Figure 26: Following the analogy in the introduction with billiard balls, OTM
establishes that at low coordination (basically less than six) NCs are more
like deflated soccer balls, where vortices naturally define valence and
eigenshapes, see Fig. 11. For large coordination, NCs are HS with diameter
defined by the OPM formula Eq. 5.

##  6 Conclusions

This article shows that the Orbifold Topological Model (OTM)[32, 33] provides
a unification of HS and OXM (X=P,C) models as well as numerical simulations,
into a general framework, summarized in Fig. 26. OTM predictions are
independent of the details of the NC interaction (or the force field in
numerical simulations) and have been extensively and thoroughly verified
throughout the paper; see, for example, Sect. 3 and Sect. 4, without a single
experiment or simulation in disagreement. The most significant limitation is
the lack of an explicit free energy, which is obviously needed to predict
equilibrium structures. Recent developments with MOLT[103, 104], see SubSect
3.5, provide a promising direction toward fixing this deficiency.

The paper shows that the many successes of HS models reflect the underlying
universality in packing problems, with its precise connection through the OPM
model[25] as discussed in Sect. 2, see also Ref. [55]. Yet, the failures of HS
show that NC systems are far more complex than HS models. For example, an fcc
phase in HS is possible because it maximizes the entropy, yet in real NC
systems, stabilization of any superlattice, fcc or otherwise, involves a
precise balance between enthalpy and entropy, see the discussion in Eq. 1, Eq.
2 and Sect. 4, with the former almost always larger in magnitude. Generally,
NC assembly is not entropically driven (and the largest contribution to the
entropy comes from ligand conformations). This point is one of many warning
examples against overusing HS results (and language) in interpreting
experimental results.

The OTM is a completely general model that applies to any (dry) NC system with
arbitrary ligands. The examples discussed in this paper have extensively used
single-component and binary superlattices in two- and three-dimensional
environments, mostly with spherical and cubic NCs. There are other cases where
more detailed comparisons with experiments will soon become available, such as
in quasicrystals[166], three-component superlattices[167, 168, 76], partially
grafted NCs[96, 95, 97], unbound ligands[169], the beautiful Moire patterns
recently reported[31] or NCs driven by different stimuli[38], among many other
examples. Furthermore, systematic studies on the role of NC shape[170], see
Fig. 12, should help clarify why the approximation of spherical NCs works so
well, even for big, i.e of the order of $8-10$ nm or larger, NCs[135].

The role of electrostatic interactions is one of the main topics left out.
There are many examples of solvent evaporation with water that contains
different salts as a solvent. For example, tetrahedra assembly[171, 83] The
final structure after solvent evaporation is free of water, but salt remains,
which will have a measurable effect on the lattice constant and will modify
the OPM formula Eq. 5. Additionally, the long-range effect of electrostatic
interactions may lead to additional nuances, as I [172, 173] and many others,
see for example, Ref. [174, 175, 176] have discussed before. Electrostatic
driven assembly strategies are a very important topic[177, 172], as at this
point in time it is the only successful strategy that leads to the assembly of
NC with electrically conducting ligands[178].

Another aspect mentioned but not elaborated on is the role of icosahedral
motifs, closely related to Frank-Kasper phases[179, 180]. The simulations
presented in Sect. 5 reveal a strong bias of small NC clusters towards
icosahedral order. Systematic investigations with much larger clusters
including thousands of NCs display icosahedral-order motifs before attaining
their bulk, presumably equilibrium, superlattice[181, 182, 183]. There is also
clear evidence that in binary systems icosahedral order clusters have lower
activation barriers during superlattice nucleation[30], which has significant
implications for glass states[184, 185]. Ref. [121] showed that all BNSL
experimentally reported so far can be regarded as generalizations of Frank-
Kasper phases, i.e quasi Frank-Kasper phases. Ref. [186] has used icosahedral
order to characterize the pitch in chiral NC structures. Methods have been
developed [187] to inverse design Frank-Kasper phases. In single-component
systems the main obstacle, however, is that HS Frank-Kasper phases have a PF
that is too low to become generally stable. This discord between icosahedral
order and packing is fixed in binary systems particularly around
$\gamma\approx 0.8$[121]: all Laves phases, for example, are Frank-Kasper
phases. The ubiquitous presence of icosahedral order and Frank-Kasper phases
in NC assembly is an important topic that will definitely require a detailed
review on its own.

The discussion in this paper has been restricted to equilibrium and dynamics
in dry systems, but, as illustrated in Fig. 1, there are many BNSL phases:
AlB2, bccAB6 (also labeled Cs6C60) that appear in both dry and DNA
assembly[1]. When ligands are flexible, such as single-stranded DNA[188] or
nanocomposite tectons[189] there are fewer phases shared with dry systems. Of
course, OTM does not strictly apply in swollen systems, as the ligand
conformations are different, but Fig. 1 illustrates that there are important
relations and perhaps a more universal description of the equilibrium
superlattices.

There is a rich and beautiful literature on polymer melts and brushes[56].
Subsect. 4.4 has presented results showing that OTM also holds for longer
ligands that reach the polymer limit. However, in my opinion, there is still a
significant disconnect between the methods and language in NC assembly and the
many beautiful developments available from polymer physics. NC assembly
provides a strategy for homogeneous NC dispersions, a problem of great
interest within the polymer community[190, 191]. Mean-field results, such as
block copolymer melts in the strong segregation limit[148, 192, 149, 193, 194,
94], or other models like Ref. [195, 58] should describe NC assembly for very
long ligands, but how exactly this specific connection and the existence of
voids (related to the way incompressibility conditions are implemented) within
the phase, as discussed in SubSect 3.5 applies, has yet to be made. Finally,
many aspects of Frank-Kasper phases in polymer melts[196, 197, 198, 199, 200]
expose general geometric principles common to all soft-matter systems that
should be relevant for NC assembly where icosahedral order is very present.

NC assembly has experienced extraordinary progress over the past two decades
and the future looks even brighter. These are exciting times for both theory
and experiment. The combined sophistication of theory and the ever-expanding
power of numerical simulations anticipate a future where new NC materials will
be led by computation. I hope that one of the contributions of this paper is
to help bridge that gap.

##  7 Acknowledgements

First and foremost, I would like to thank all students in my group who have
contributed to this work. This paper would not be possible without them. I
also would like to thank many researchers for educating me in this fascinating
field, especially: M. Boles, C. Calero, Q. Chen, I. Cherniukh, M. Engel, O.
Gang, M. Kovalenko, R. Macfarlane, S. Mallapragada, M. Olvera de la Cruz, E.
Rabani, M. Tagliazucchi, D. Talapin, D. Vaknin, W. Wang, X. Ye, S. Zhou and
Gerard Wong for inviting me to prepare this paper and to Shan Zhou and Dmitri
Talapin for permission to use their experimental figures. Finally, a big
thanks goes to all the participants in the KITP Santa Barbara workshop and
conference for the amazing environment that motivated this paper. The main
funding for this research is from the U.S. Department of Energy (U.S. DOE),
Office of Basic Energy Sciences, Division of Materials Sciences and
Engineering. Iowa State University operates Ames National Laboratory for the
U.S. DOE under Contract DE-AC02-07CH11358. Research is also supported in part
by the National Science Foundation under Grant No. NSF PHY-1748958 to the
Kavli Institute for Theoretical Physics. This work used EXPANSE, BRIDGES2 and
DELTA through allocation MCB140071 from the Advanced Cyberinfrastructure
Coordination Ecosystem: Services Support (ACCESS) program[201], which is
supported by National Science Foundation grants 2138259, 2138286, 2138307,
2137603, and 2138296.

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