PHYSICAL REVIEW X 9, 021015 (2019)

Understanding the Formation of PbSe Honeycomb Superstructures by Dynamics Simulations

Giuseppe Soligno* and Daniel Vanmaekelbergh Condensed Matter and Interfaces, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 1, Utrecht 3584 CC, Netherlands

(Received 28 October 2018; revised manuscript received 28 January 2019; published 23 April 2019)

Using a coarse-grained molecular dynamics model, we simulate the self-assembly of PbSe nanocrystals (NCs) adsorbed at a flat fluid-fluid interface. The model includes all key forces involved: NC-NC short- range facet-specific attractive and repulsive interactions, entropic effects, and forces due to the NC adsorption at fluid-fluid interfaces. Realistic values are used for the input parameters regulating the various forces. The interface-adsorption parameters are estimated using a recently introduced sharp- interface numerical method which includes capillary deformation effects. We find that the final structure in which the NCs self-assemble is drastically affected by the input values of the parameters of our coarse- grained model. In particular, by slightly tuning just a few parameters of the model, we can induce NC self- assembly into either silicene-honeycomb superstructures, where all NCs have a f111g facet parallel to the fluid-fluid interface plane, or square superstructures, where all NCs have a f100g facet parallel to the interface plane. Both of these nanostructures have been observed experimentally. However, it is still not clear their formation mechanism, and, in particular, which are the factors directing the NC self-assembly into one or another structure. In this work, we identify and quantify such factors, showing illustrative assembled-phase diagrams obtained from our simulations. In addition, with our model, we can study the self-assembly dynamics, simulating how the NCs’ structures evolve from few-NCs aggregates to gradually larger domains. For example, we observe linear chains, where all NCs have a f110g facet parallel to the interface plane as typical precursors of the square superstructure, and zigzag aggregates, where all NCs have a f111g facet parallel to the interface plane as typical precursors of the silicene-honeycomb superstructure. Both of these aggregates have also been observed experimentally. Finally, we show indications that our method can be applied to study defects of the obtained superstructures.

DOI: 10.1103/PhysRevX.9.021015 Subject Areas: Computational Physics, Materials Science, Nanophysics

I. INTRODUCTION superstructure: (i) synthesis of the NCs, i.e., the super- structure building blocks, from their atomic precursors Nanoparticle self-assembly is an emerging route with [48–60], and (ii) self-assembly of the NCs into ordered tremendous potential to build novel nanostructured materi- structures [61–66]. Typically, NCs are, after their synthesis, als [1,2]. In the past two decades, increasing interest has dispersed in solution and fully covered by organic molecules been devoted toward the formation of 3D [3–20] and 2D (ligands), chemically attached (chemisorbed) at the NC [21–32] nanogeometric materials for optoelectronic appli- surface, which largely screen NC-NC attractive interactions. cations [33–47]. These nanomaterials are formed by the The NC self-assembly can be induced, for example, by self-assembly of semiconductor nanocrystals (NCs) with solvent evaporation (i.e., by increasing the NC density). In the size of a few nanometers, and often with a roughly this case, a close-packed NC superstructure is obtained, spherical shape, into various types of superstructures. whose geometry is dictated mainly by entropic factors, i.e., Two main stages are required for the formation of a the shape and size of the NCs [67–71]. NC-NC attractive interactions can be activated (e.g., by ligand-exchange [72] *Corresponding author. or facet-specific ligand-detachment [73] reactions), inducing [email protected] NC self-assembly in different types of superstructures. A special class of NC superstructures, referred to as NC Published by the American Physical Society under the terms of superlattices, is formed when NC self-assembly is followed the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to by crystal-structure alignment and oriented attachment the author(s) and the published article’s title, journal citation, [74–83] of close-by NCs, resulting in atomically coherent and DOI. nanogeometric materials. The formation process of NC

2160-3308=19=9(2)=021015(32) 021015-1 Published by the American Physical Society GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019) superlattices can be confined in 2D when the NCs are superstructures obtained with our MD model present the adsorbed at a fluid-fluid interface [84]. The NCs self- same kind of defects observed in the experimental super- assemble in ordered monolayers at the fluid-fluid interface, lattices, indicating that our MD model can also be used to and subsequently perform oriented attachment [73,85–88], study the formation and stability of defects in these resulting in 2D atomically coherent nanogeometric materi- structures. als. In this last class of processes, in addition to NC entropy and NC-NC interactions, also the different interactions of the II. COARSE-GRAINED MD MODEL NCs with the different molecules of the two fluids forming First, we briefly illustrate our simulation model (more the interface (i.e., interface-adsorption effects) are involved details are provided in Appendix A). in the superlattice formation mechanism. First experimentally observed a few years ago [73,85,86,88], the PbSe silicene-honeycomb 2D super- A. Outline lattice is expected to exhibit a Dirac-type band structure, We model a PbSe NC using a polybead structure. The with the semiconductor band gap preserved [89–91]. beads forming the NC are disposed to reproduce a Hence, such a material would combine the properties of rhombicuboctahedron of size 6 nm, that is a typical semiconductors with those of graphene, making it very experimental shape [76,92]. We simulate the dynamics interesting for optoelectronic applications. This superlattice of the NCs using the position-Verlet algorithm [94] to is formed by PbSe NCs, approximately having a 5-nm size compute the bead’s motion. Bead-bead pair potentials are and a rhombicuboctahedron shape, disposed as in the used to reproduce NC-NC short-range facet-specific attrac- silicene lattice, i.e., in a periodically buckled honeycomb tive and repulsive interactions. The distance between beads lattice, and all oriented with a f111g facet parallel to the belonging to the same NC is kept fixed by constraint forces superlattice plane. The PbSe NCs, before the self-assembly, [95]. The solvent is treated implicitly by modeling the NC i.e., when still dispersed in solvent, are fully covered by Brownian motion. External potentials are applied to the NC oleic acid ligands, which are strongly attached at the f111g, beads to mimic the interface-adsorption forces experienced f110g facets and weakly bonded at the f100g facets [92]. by NCs at a fluid-fluid interface. The formation process of the PbSe silicene-honeycomb superlattice typically occurs at the solvent-air interface. B. NC-NC interactions During the process, ligands desorb from the NC f100g facets, allowing oriented attachment between PbSe NCs The polybead structure representing each NC is formed by opposite f100g facets [73]. In similar experiments by 144 beads (shell beads) disposed to represent the NC [73,85,87], self-assembly of NCs into square superlattices surface, see Fig. 1, plus seven beads (core beads) used, see was observed, with all NCs oriented with a f100g facet later, for the Brownian motion of the NCs. To model parallel to the superlattice plane, and with oriented attach- van der Waals and electrostatic-chemical atomic inter- ment between PbSe NCs by opposite f100g facets also actions between NCs at almost-contact distance, shell performed. Although the relevant forces involved in the beads of different NCs interact with each other by the formation of these superlattices are known [93], the precise Lennard-Jones pair potential, mechanisms are far from understood. In particular, it is not σ 12 σ 6 clear yet why the PbSe NCs form square superlattices in U ¼ 4ϵ − ; ð1Þ some cases and silicene-honeycomb superlattices in others. LJ r r In this work, we shed light on this, introducing a new coarse-grained molecular dynamics (MD) model to study truncated and shifted in r ¼ 2.4σ [see Eq. (A3)], where r is the self-assembly of NCs at fluid-fluid interfaces. By the center-to-center bead-bead distance, σ ¼ 1 nm, and ϵ presenting simulations with our MD model, we illustrate sets the interaction strength for the bead pair. PbSe NCs are, how the intricate interplay between interface-adsorption after their synthesis, dispersed in solution, and ligands, forces orienting and keeping the NCs at the interface and typically oleic acid molecules, are chemisorbed at the NC attractive forces only between f100g facets of close-by facets. At this stage, ligands largely screen NC-NC attrac- NCs drive the self-assembly of PbSe NCs into either square tive interactions and prevent NC assembly. In the experi- or silicene-honeycomb superstructures. As a confirmation ments of interest [73,85–88], the NC self-assembly is that our model correctly captures the key features of the activated by chemically inducing the detachment of ligands NC self-assembly, in our simulations we observe, in the from the PbSe NC f100g facets, i.e., where the bonding formation of both the silicene-honeycomb and square energy between ligand molecules and NC surface atoms is superstructures, the same few-NCs precursors observed lowest [73,92]. Hence, during the self-assembly, NCs experimentally, that is zigzag aggregates, where all NCs attract (and attach to) each other only by f100g facets, have a f111g facet parallel to the interface plane, and while NC-NC bonds by f110g and f111g facets are linear chains, where all NCs have a f110g facet parallel prevented by ligands. To model these NC-NC facet-specific to the interface plane, respectively. In addition, the NC repulsive steric interactions due to ligands, all pairs of shell

021015-2 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019)

due to the NC interactions with the molecules of the two fluids. Such a potential binds the NC at a certain height at the fluid-fluid interface and drives it to energetically favorable orientations. The specific potentials arising for PbSe NCs at a typical fluid-fluid interface (toluene-air) are predicted in the next section. To reproduce in our coarse- grained MD model a fluid-fluid interface, here identified by 0 FIG. 1. Left: Shell beads, here plotted as spheres with diameter the plane z ¼ (where a Cartesian coordinate system x, y, z 1 nm, reproducing the surface of a NC in our coarse-grained MD is introduced), external potentials are applied to the NC model. Each pair of shell beads of different NCs interacts by a beads. To mimic the force bonding a NC to the interface, Lennard-Jones pair potential [Eq. (1)], representing van der Waals the potential and other atomic interactions. In addition, each bead-bead pair 8 involving a red or orange colored bead interacts by a soft repulsive > E if z < −z potential [Eq. (2)], modeling oleic acid molecules chemisorbed at < s c d 110 111 U ≡ − 2 ≤ the NC f g, f g facets. Right: Representation with facets of zðzcÞ > E1 þ uzðzc z0Þ if jzcj zd ð3Þ the NC, with f100g, f110g, f111g facets in different colors. :> Ea if zc >zd beads of different NCs involving a red or orange colored is applied to the central core bead of each NC, with zc the bead, see Fig. 1, interact by the soft repulsive pair potential central core bead z coordinate, uz setting the strength of the force bonding the NC to the interface, z0 the minimum-U σ 6 z R position of the NC center of mass, zd the maximum UR ¼ ϵR − 1 ; ð2Þ r distance at which the NC feels the interface (so zd is, roughly, the NC size), and E , E , E1 constants represent- σ ϵ 2 10−22 s a truncated in r ¼ R [see Eq. (A5)], with R ¼ × J, ing the interface-adsorption energy E (see Sec. III)ofa σ 2 5 σ R ¼ . nm when both beads are red or orange, and R ¼ single NC, respectively, fully immersed in the fluid below 1 25 . nm otherwise. These values are order-of-magnitude z ¼ 0, fully immersed in the fluid above z ¼ 0, and estimations to reproduce typical soft repulsive potentials adsorbed at the interface in its minimum-E position and between ligand-capped NCs in solvent [96]. Since the NC orientation. To reproduce the forces due to the fluid-fluid 100 f g facets are ligand-free, NCs can attach to each other interface that drive an adsorbed NC in a specific orienta- 100 by opposite f g facets, performing oriented attachment. tion, we define [see Fig. 2(a)] φ ∈ 0; π as the polar angle E ½ The maximum bonding energy of two NCs attached to of a NC vertical axis (given by one of its six h100i 100 ∼10−19 each other by f g facets is expected to be J [97]. directions) with respect to the plane z ¼ 0, and ψ ∈ This value is due not only to NC-NC van der Waals ½0; 2πÞ as the NC internal Euler angle around its vertical interactions, but also (and mostly) to electrostatic-chemical axis. Force couples are applied to the NC to induce torques atomic interactions between surface atoms of the PbSe with, respectively, direction orthogonal to the φ, ψ rota- f100g opposite facets. Note that E is an input parameter tional plane, modulus jdUφ=dφj, jdUψ =dψj, and sign of our model and its exact value can be tuned by regulating U φ U ψ ϵ U depending on the sign of d φ=d , d ψ =d , where the value of in LJ [Eq. (1)] for the various bead pairs; see Appendix A. 2 UφðφÞ¼up½φ − φ0ðφÞ ξðzcÞ; ð4Þ

C. NC-solvent interactions 2 Uψ ðψÞ¼up½ψ − ψ 0ðψÞ ξðzcÞ: ð5Þ The seven core beads of each NC are placed one in the

NC center of mass and the other six at a distance rB from it The parameter up sets the orientation force’s strength, and, and in the six h100i directions. To model the NC Brownian for simplicity, is assumed constant (although, in principle, motion, the Verlet-type algorithm in Ref. [98] is applied, for it could depend on φ, ψ and be different in Uψ and Uφ). The each NC, to its central core bead and to three of the function ξðzcÞ, which is 1 for zc ¼ z0, 0 for jzcj >zd, and a remaining six core beads (nonaligned with the central core linear interpolation of these values for jzcj

021015-3 GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019)

see Sec. III. For the rotational symmetry of the NC shape in φ, ψ, different values of φ0, ψ 0 correspond to the same NC orientation. For example, there are eight different combi- nations of φ0, ψ 0 corresponding to the orientation of a NC with a specific f111g facet parallel to z ¼ 0 and pointing toward up, one combination for each f111g facet of the NC. In the simulations, the φ0, ψ 0 corresponding to the NC minimum-Uφ, minimum-Uψ orientation closest to the current NC orientation is chosen, respectively. Cubic and cantellated-rhombicuboctahedron particles, when adsorbed with a f111g facet parallel to the interface, generate hexapolar capillary deformations in the interface height profile inducing capillary interactions between the particles [99,102]. This effect is, however, negligible for the experiments considered here, as we show and discuss extensively in Appendix B3. Nonetheless, for complete- ness and future applications, a NC-NC pair capillary potential Uc is introduced in our coarse-grained MD model, see Appendix A4 (although it is irrelevant for the results presented in this work).

E. Thermodynamics considerations In our coarse-grained MD model, the Helmholtz free energy F of N NCs at a fluid-fluid interface is F U − S ¼ tot T N; ð6Þ

where T is the (constant) temperature, SN is the entropy of φ ψ FIG. 2. (a) Sketch of the configuration zc, , ofaPbSeNC the N NCs, and U ≡ U þ U is the total internal 0 tot kin pot with respect to the toluene-air interface plane z ¼ ,wherezc U φ ∈ 0 π energy of the system, with kin the total kinetic energy of is the z coordinate of the NC center of mass, ½ ; is the the NCs and polar angle of the NC vertical axis vφ (corresponding to one of the NC h100i directions), and ψ ∈ ½0; 2πÞ is the NC internal U ≡ U U U U U U pot LJ þ R þ z þ φ þ ψ þ c ð7Þ Euler angle around vφ (and is defined by the vectors vψ and w, see Appendix A3). (b)–(d) Interface-adsorption energy E the total potential energy of the NCs. In the definition of [Eqs. (8)–(10), (B1)] of a PbSe NC, modeled as a rhombicu- U U U pot [Eq. (7)], it is implicit that LJ, R are summed over all boctahedron of size 6 nm, at a toluene-air interface, with U U U U respect to φ, ψ, z , computed using the numerical approach in bead-bead pairs, z, φ, ψ over all NCs, and c over all c U S Refs. [99–101],withE ¼ 0 corresponding to the NC desorbed NC-NC pairs. The terms kin and T N are included in our from the interface and immersed in toluene. The surface model by the NC Brownian motion plus the bead-bead hard U U tensions, with toluene and air, assigned to the various NC interactions (that set the NC shape). The terms LJ, R U facets take into account that ligands (oleic acid molecules) are represent the contribution to pot due to the NC-NC 110 111 100 chemisorbed at the NC f g, f g facets, while f g interactions. The interactions of the NCs with the fluid facets are ligand-free (see text). (b) EðzcÞ for a NC oriented molecules (and the interactions of the fluid molecules with, respectively, a f100g and a f111g facet parallel to z ¼ 0. (c),(d) EðφÞ and Eðψ) plotted, respectively, for various values between themselves) are implicitly included in our model of ψ and φ, and minimized with respect to z .Fortherotational by implementing the Brownian motion of the NCs. The c U U U U symmetry of the NC shape in φ and ψ, E is shown only external potentials z, φ, ψ , c represent the effects due for φ ∈ ½0; π=2, ψ ∈ ½0; π=4. The black curves in (b)–(d) to the different interactions that NCs at fluid-fluid inter- U U show, for various values of uz, up, the potentials z, φ, faces have with the molecules of the two different fluids. Uψ [Eqs. (3)–(5), here opportunely shifted] used in our coarse- We call interface-adsorption energy the total energy E ≡ grained MD model to represent the interface-adsorption energy U þ Uφ þ Uψ þ U due to these “asymmetric” inter- E of a PbSe NC at the toluene-air interface. (e),(f) 3D view, z c actions that NCs, when adsorbed at fluid-fluid interfaces, close to the NC, of the toluene-air interface minimum-E shape (blue grid, with the toluene the fluid below), computed by the experience with the different molecules of the two different numerical method of Soligno et al. [99–101], for the meta- fluids forming the interface (in this definition of E,itis stable and minimum-E orientation of the NC, i.e., with a f100g implicit that Uz, Uφ, Uψ are summed over all NCs, Uc over and a f111g facet parallel to z ¼ 0, respectively. all NC-NC pairs). In Sec. III (and in Appendix B)we

021015-4 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019) discuss in more detail the thermodynamic origin of E, and as −PV, with P the fluid pressure and V the fluid volume. we predict E for the experimental systems of interest Hence, E can be written as (i.e., PbSe NCs at toluene-air interfaces). Since our − − simulations are performed at constant T, N, and system E ¼ Ui PaVa PtVt; ð9Þ volume, the NCs will evolve toward a configuration with with P , P the pressure of air and toluene, respectively, and lower F [Eq. (6)], until a stable minimum in F is reached. a t U corresponding to the remaining terms of the internal A reduction of F (i.e., an energy bonus) can be achieved by i energy U, due to the toluene-air, NC-toluene, and NC-air increasing S and/or by decreasing U . When the NCs N tot interfaces (i.e., the terms not included in the grand self-assemble, they are evolving toward more ordered potentials of toluene and air molecules). The interface- structures, so they are decreasing S (except at very N adsorption energy E, for a fixed position of the NC, is high packing fraction [103]), which corresponds to an minimum at equilibrium, since T, V, μ , μ are constant. As increment in F (i.e., an energy penalty). Hence, the NCs t a proven in Ref. [100], the toluene-air interface shape hðx; yÞ self-assemble only if U decreases (more than the incre- tot that minimizes E, hence the equilibrium hðx; yÞ, is the ment in −TS ). N solution of the Young-Laplace equation, with Young’slaw for the contact angle holding along the three-phase toluene- III. INTERFACE-ADSORPTION POTENTIALS air-NC contact lines. With respect to the NC, the potential FOR PbSe NCs E is an internal energy. In our coarse-grained MD model, E In this section, we compute the potential energy of an is represented by the potentials Uz, Uφ, Uψ , defined in isolated PbSe NC at the typical interface toluene-air with Eqs. (3)–(5). In the remainder of this section, we compute E respect to the NC configuration zc, φ, ψ at the interface for a single-adsorbed PbSe NC at the toluene-air interface, plane, see Fig. 2(a), where a Cartesian coordinate system x, and from it we estimate the parameters for the potentials y, z is introduced such that the flat toluene-air interface Uz, Uφ, Uψ . 0 coincides with the plane z ¼ , the toluene is in the Since the total volume V is fixed, −PaVa − PtVt in 0 0 semiplane z< and air in the semiplane z> . The Eq. (9) can be rewritten as −ΔPVt (plus a constant), where ultimate purpose of the calculations reported in this section ΔP ≡ Pt − Pa. In our model, we assume a flat toluene-air is to estimate the interface-adsorption parameters in our interface far away from the NC, implying that ΔP ¼ 0 – coarse-grained MD model [precisely, in Eqs. (3) (5)]. [101],soE is equal to the internal energy Ui due to the toluene-air, NC-toluene, and NC-air interfaces. Writing Ui A. Macroscopic sharp-interface model explicitly for a single PbSe NC at the toluene-air interface, We use a macroscopic model, treating toluene and air as E is [100] homogeneous fluids, the toluene-air interface as a possibly curved 2D surface with no thickness and equilibrium height X26 X26 γ γðaÞ ðaÞ γðtÞ ðtÞ profile z ¼ hðx; yÞ, with hðx; yÞ¼0 when no NC is E ¼ S þ k Wk þ k Wk ; ð10Þ 1 1 adsorbed, and the PbSe NC as a homogeneous solid with k¼ k¼ 100 rhombicuboctahedron shape (with two opposite f g γ 0 028 facets at distance 6 nm). Diffuse-interface effects, not with ¼ . N=m the toluene-air surface tension, S the ðaÞ ðtÞ included in this approach, become relevant only for particle toluene-air interface surface area, Wk and Wk the surface sizes below 3–4nm[104]. We assume constant the area of the portion of the NC kth facet (k ¼ 1; …; 26)in temperature T, the chemical potentials μ , μ of toluene γðaÞ γðtÞ t a contact with air and toluene, respectively, and k , k their and air molecules, respectively, and the total volume V ¼ respective surface tension. Vt þ Va þ VNC of the system, with Va, Vt the (non- constant) volumes of air and toluene, respectively, and B. PbSe NC with ligands on f111g, f110g facets VNC the (constant) volume of the NC. The convenient thermodynamic potential E for this toluene-air-NC system, To different facets of the NC we assign different surface henceforth called interface-adsorption energy, is tensions with air and toluene. To all the NC f111g and f110g facets, assumed covered by ligands (typically, oleic E ≡ U − TSt − TSa − Ntμt − Naμa; ð8Þ acid molecules), we assign surface tension 0.018 N=m S with air, corresponding to the hexane-air surface tension, with U the total internal energy of the system, and t, Nt, and zero with toluene, since ligands are soluble in toluene. S , N the entropy and total number of molecules of a a 111 110 θ ≡ γðaÞ− toluene and air, respectively. Gravity is not included in Hence, for the f g and f g facets, cos ½ k γðtÞ γ ≈ 0 64 θ ’ Eq. (8), since it is negligible for the systems of interest. k = . , where is Young s contact angle as defined With respect to the toluene and air molecules, E is a grand by Young’slaw[101]. To all the NC f100g facets, assumed potential (also called Landau potential). For extensiveness, instead ligands-free, we assign cos θ ¼ 0.3, to account for the grand potential of a homogeneous fluid can be written the energetically less-favorable interactions of these facets

021015-5 GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019) with toluene (compared to ligand-toluene interactions). In cases, see Sec. IV, the NCs prefer to orient with a f100g ðaÞ ðtÞ 0 Eq. (10), the values of S, W , W (k ¼ 1; …26) depend facet parallel to z ¼ , since the energy penalty in the free k k F U on the NC configuration z , φ, ψ at the interface, and on the energy [Eq. (6)] due to the orientational potentials φ, c U interface equilibrium shape h (which also depends on ψ is compensated by an energy bonus (greater in absolute z , φ, ψ). We use the numerical approach introduced by value) in the electrostatic-chemical bonds between NCs c 100 U Soligno et al. [99–101], where the equilibrium interface attached by f g facets, i.e., in LJ. height profile hðx; yÞ is computed with respect to zc, φ, ψ, and then E is obtained. More details on this method are C. PbSe NC fully covered by ligands reported in Appendix B1. Note that, as a first-order In Appendix B2, we show analogous calculations to φ ψ approximation, Eðzc; ; Þ could be computed assuming those presented in Fig. 2, but for cos θ ¼ 0.64 on all NC hðx; yÞ flat, i.e., neglecting capillary deformation effects. facets, corresponding to a NC fully covered by ligands. In However, as shown in Ref. [102], this can lead not only to this case, multiple metastable orientations of the NC with overestimations of the energy, but even to erroneous energy barriers ∼10−20 J are found, suggesting that, when equilibrium orientations of the NC. fully covered by ligands, NCs do adsorb at the interface, φ ψ In Figs. 2(d), we show Eðzc; ; Þ [Eq. (10)], shifted by a but, essentially, with random orientation. This is con- 0 constant such that E ¼ corresponds to a PbSe NC firmed by experimental observations [87], where a mono- desorbed from the interface and fully immersed in toluene layer of randomly oriented PbSe NCs was observed. Then, [see Eq. (B1)]. In Fig. 2(b), EðzcÞ is plotted for a few fixed in the same experiments [87], a closer-packed monolayer φ ψ orientations (i.e., fixed ; ) of the NC and compared with of NCs oriented with a f100g facet parallel to the interface U the potential z [Eq. (3)], used in our MD model to was observed at a later stage. Our interpretation is that approximately represent EðzcÞ, i.e., to induce the forces ligands were still chemisorbed at (mostly) all the PbSe that keep a NC at the interface plane. As shown, the NCs facets when the monolayer of randomly oriented NCs parameter uz, setting the strength of these forces, is was observed, while they were (mostly) detached from −20 expected to be ≃2 × 10 J for a NC oriented with a the NCs f100g facets when the NCs were found oriented. −20 f111g facet parallel to z ¼ 0 and ≃5 × 10 J for a NC As shown in this section, the expected equilibrium oriented with a f100g facet parallel to z ¼ 0. In our MD orientation, at the toluene-air interface, of isolated PbSe model, for simplicity, we keep uz constant with respect NCs covered by ligands only on their f110g, f111g facets to the NC orientation. In Figs. 2(c) and 2(d), the energy is with a f111g facet parallel to the interface plane. EðφÞ and EðψÞ, minimized over zc, is plotted for various However, short-range electrostatic-chemical attractive fixed ψ and φ, respectively, and compared with the interactions between ligands-free f100g facets of close- potentials Uφ, Uψ [Eqs. (4) and (5)], used in our MD by NCs can affect this orientation (see the results with our model to approximately represent Eðφ; ψÞ, i.e., to induce MD model in Sec. IV), forcing the NCs to have a f100g the forces that drive an adsorbed NC toward a certain facet parallel to the interface and resulting in the NCs orientation. As shown, the parameter up, setting the square superstructures observed in Ref. [87]. strength of these forces, is expected to be approximately between 2 × 10−19 J and 4 × 10−19. Interestingly, the D. Considerations on many-particle effects results with our MD model (see Sec. IV) show that the The free energy F [see Eq. (6), with E represented by NC self-assembly can be directed toward either a square or Uz þ Uφ þ Uψ ] can induce a nonminimum-E orientation of a silicene-honeycomb superstructure solely by tuning up the NCs, when many NCs are adsorbed at the interface, (with realistic values for all the other parameters of the because of short-range NC-NC attractive and repulsive MD model). In particular, the turning value for up is chemical interactions (represented by U þ U ); see −19 LJ R ≃2.5 × 10 (see Sec. IV), i.e., close to its expected value. Sec. IV. However, the interface-adsorption energy E is In the experiments, the self-assembly of PbSe NCs often expected to be negligibly affected, with respect to the results in coexistence between square and silicene-honey- position and orientation zc, φ, ψ of each NC, by multi- comb superstructures, corroborating our predictions. As particle effects, for the experimental conditions of interest; shown in Fig. 2(f), the NC orientation with minimum E is see Ref. [99]. That is, each adsorbed PbSe NC experiences 111 0 with a f g facet parallel to the interface plane z ¼ . The the same potential Eðzc; φ; ψÞ we computed for an isolated energy Eðφ; ψÞ indicates also a metastable orientation of PbSe NC. So, in our MD model, we are justified in 100 0 the NC with a f g facet parallel to z ¼ ; see Figs. 2(c) applying the potentials Uz, Uφ, Uψ , here estimated from −20 and 2(e). Since the energy barriers are small (∼10 J), in the interface-adsorption energy E of a single NC, inde- our MD simulations we neglect this metastable orientation, pendently to each interface-adsorbed NC. Capillary inter- defining the interface-adsorption potentials Uφ; Uψ to be actions between adsorbed NCs, due to the variations of the minimum only for the NC orientation with a f111g facet interface-adsorption energy E with respect to NC-NC parallel to z ¼ 0. Despite this, we show that in certain reciprocal distances and azimuthal orientations in the

021015-6 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019) interface plane, are also negligible for PbSe NCs adsorbed interface. The NCs’ initial configuration (t ¼ 0)isa at a toluene-air interface, as shown and discussed exten- hexagonal monolayer, with lattice spacing 11 nm, of sively in Appendix B3. randomly oriented NCs at the interface.

IV. RESULTS FROM OUR COARSE-GRAINED 2. Initial formation stage MD MODEL In both simulations (Figs. 3 and 4), the initially dispersed In this section, we present the results from our coarse- NCs, driven by the potentials Uφ, Uψ , Uz, orient almost grained MD model for the self-assembly of PbSe nano- immediately (by t ∼ 0.1 ns) with a f111g facet parallel to crystals (NCs) of size 6 nm at a fluid-fluid interface. the interface plane, that is the minimum-Uφ, minimum-Uψ

orientation, and keep their minimum-Uz height zc at the A. Self-assembly in square and silicene-honeycomb interface plane. At this stage, the NCs are too far apart to superstructures feel the attractive interactions between f100g facets, due to U In Figs. 3 and 4, we present two simulations, with our the potential LJ. Hence, the NCs minimize the total U MD model, of N ¼ 625 NCs, where the NCs self-assemble internal potential energy pot of the system, see Eq. (7), into, respectively, silicene-honeycomb superstructures by keeping the configuration zc, ψ, φ with minimum Uφ, (Fig. 3) and square superstructures (Fig. 4). Uψ , Uz (see the graphs of zc, ψ, φ relative to the first snapshot of each simulation, i.e., at t ¼ 0.01 μs). Then, at a 1. Simulation setup later stage (t ≳ 0.01 μs), because of Brownian motion, NCs A Cartesian coordinate system x, y, z is introduced start bumping into each other, feeling the attractive inter- 100 U such that the plane z ¼ 0 is the fluid-fluid interface. actions between f g facets, due to LJ.However,as Periodic boundary conditions are applied in the x and y long as the NCs keep the orientation with a f111g facet directions, with period 275 nm. Snapshots of the simu- parallel to the interface plane and the same height zc at the lations at different times t show, in Figs. 3 and 4, a top interface plane, they cannot attach by f100g facets, for view of the interface, with the NCs represented as in clear geometrical reasons. Note that NC-NC attachment by Fig. 1, and ligands (oleic acid molecules, chemisorbed at f111g or f110g facets would be geometrically possible the NC f110g, f111g facets and preventing NC-NC for NCs in this configuration, but it is prevented by attachment by these facets) sketched with black line ligands chemisorbed at these NC facets [which, in our segments. The graphs at the right of each snapshot show model, are represented by the soft repulsive potential U ; ψ φ R the distribution of zc, , for the NCs, with zc the NC see Eq. (2)]. height at the interface plane and ψ, φ the NC orientation; see Fig. 2(a). The parameter up, that sets the strength of 3. Formation of silicene-honeycomb superstructures the forces due to the interface-adsorption potentials Uφ, Uψ [Eqs. (4) and (5)], i.e., orienting the NCs with a f111g In Fig. 3, the NCs manage to attach to each other by −19 f100g facets by shifting their height z at the interface, half facet parallel to the interface plane, is up ¼ 5.5 × 10 J c −19 of the NCs moving slightly above the interface plane and in Fig. 3 and up ¼ 1.0 × 10 J in Fig. 4. As shown in Sec. III, the orientation with a f111g facet parallel to the half of them slightly below, that is, by partially desorbing interface plane minimizes the interface-adsorption energy from the interface (see the two peaks that appear in the zc E [Eqs. (8)–(10)] for PbSe NCs with f111g, f110g facets distribution for increasing time). This shift in zc increases the total potential U , corresponding to an energy penalty. covered by ligands at a toluene-air interface, and up is z expected to be between the two values considered in However, by attaching by f100g facets, the NCs decrease U U Figs. 3 and 4. All the other parameters of our MD model the total potential LJ more than the increment in z; U have the same values (listed in Appendix C) in the two hence, the total internal energy pot actually decreases. This simulations of Figs. 3 and 4. The total bonding energy is confirmed by the energy plots shown in the bottom right- U between two NCs attached by f100g facets is set to the hand panel of Fig. 3, where the black line is potðtÞ, the −19 U typical value E ≃ 1.0 × 10 J [97]. This value takes into light blue line is LJðtÞ summed over all bead-bead pairs, account not only NC-NC van der Waals interactions, but and the green line is UzðtÞ summed over all NCs. The NCs’ also (and mostly) electrostatic-chemical atomic inter- orientation at the interface remains with a f111g facet actions between PbSe f100g facets at (almost) contact parallel to the interface plane, i.e., the orientation minimiz- distance, expected since the f100g facets are ligand-free. ing Uφ; Uψ , as confirmed by the plot of UφðtÞþUψ ðtÞ, The parameter uz, setting the strength of the forces due to summed over all NCs, see fuchsia line in the energy graph, the interface-adsorption potential Uz [Eq. (3)], which are which remains essentially constant (see also the distribu- parallel to the z direction and keep the NCs at the tions of ψ, φ). As clearly shown also by the simulation −20 interface plane, is uz ¼ 2 × 10 J, i.e., close to the snapshots, thanks to this mechanism the NCs aggregate into values estimated in Sec. III for PbSe NCs at a toluene-air silicene-honeycomb superstructures, equivalent to those

021015-7 GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019)

FIG. 3. Simulation, with our coarse-grained MD model, of N ¼ 625 PbSe NCs at a fluid-fluid interface. Snapshots, at different times t, show a top view of the interface, with the black borders indicating the periodic boundary conditions. The NCs are represented as in Fig. 1. Ligands attached to the NC f110g, f111g facets (preventing NC-NC attachment by these facets) are sketched with black line segments. The graphs at the right of each snapshot show the number n of NCs with a certain value of (green) zc, (red) ψ, and (blue) φ, with zc the NC height at the interface plane and ψ, φ the NC orientation; see Fig. 2(a). Because of the interface-adsorption potentials Uz, Uφ, Uψ [Eqs. (3)– (5)], the NCs are initially in their minimum-Ezc, ψ, φ configuration (marked by vertical dotted lines in the zc, ψ, φ distribution graphs), corresponding to NCs all in same plane and with a f111g facet parallel to the interface. As long as the NCs are in this configuration, they are geometrically unable to attach by opposite f100g facets. The attractive forces between opposite f100g facets, due to the potential ULJ [Eq. (1)], drive the NCs out of plane—half of the NCs below, and half of them above, see zc distributions—inducing self-assembly in silicene-honeycomb superstructures. The parameter up, setting the strength of the interface-adsorption forces orienting the NCs with a −19 f111g facet parallel to the interface plane, i.e., due to Uφ, Uψ ,isup ¼ 5.5 × 10 J (for comparison, see Fig. 6, where an equivalent simulation is shown, but for a lower value of up). Inset (a) shows an enlargement of a zigzag aggregate of NCs oriented with a f111g facet parallel to the interface plane. Inset (b) shows an enlargement of a large portion of the silicene-honeycomb superstructure formed by the NCs. In particular, defects of this superstructure are highlighted: vacancies (fuchsia circle), lattice’s misalignment (see lattices highlighted in blue and in green), trilayer silicene-honeycomb (see bottom left-hand corner of the inset). The graph in the bottom right-hand panel U 100 U shows, with respect to t, the total potential (light blue) LJ, i.e., inducing attractions between f g facets, (green) z, i.e., keeping the NCs at the interface plane, (fuchsia) Uφ þ Uψ , i.e., orienting the NCs at the interface with a f111g facet parallel to the interface plane, and (black) the total internal potential energy Upot; see Eq. (7). An animation of this simulation is available in the Supplemental Material [105].

021015-8 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019)

FIG. 4. Simulation, with our coarse-grained MD model, of N ¼ 625 PbSe NCs at a fluid-fluid interface. Snapshots, at different times t, show a top view of the interface, with the black borders indicating the periodic boundary conditions. The NCs are represented as in Fig. 1. Ligands attached to the NC f110g, f111g facets (preventing NC-NC attachment by these facets) are sketched with black line segments. The graphs at the right of each snapshot show the number n of NCs with a certain value of (green) zc, (red) ψ, and (blue) φ, with zc the NC height at the interface plane and ψ, φ the NC orientation; see Fig. 2(a). Because of the interface-adsorption potentials Uz, Uφ, Uψ [Eqs. (3)–(5)], the NCs are initially in their minimum-Ezc, ψ, φ configuration (marked by vertical dotted lines in the zc, ψ, φ distribution graphs), corresponding to NCs all in the same plane and with a f111g facet parallel to the interface. As long as the NCs are in this configuration, they are geometrically unable to attach by opposite f100g facets. The attractive forces between opposite f100g facets, due to the potential ULJ [Eq. (1)], tilt the NCs toward non-minimum-E orientations—i.e., with a f100g facet parallel to the interface, see ψ, φ distributions—inducing self-assembly in square superstructures. The parameter up, setting the strength of the −19 interface-adsorption forces orienting the NCs with a f111g facet parallel to the interface plane, i.e., due to Uφ, Uψ ,isup ¼ 1.0 × 10 J (for comparison, see Fig. 5, where an equivalent simulation is shown, but for a higher value of up). Inset (a) shows an enlargement of a linear aggregate of NCs oriented with a f110g facet parallel to the interface plane. Insets (b) and (c) show an enlargement of a large portion of the square superstructure formed by the NCs. In particular, defects of this superstructure are highlighted: vacancies (fuchsia circle), lattice’s misalignment (see lattices highlighted in blue and in green). The graph in the bottom right-hand panel shows, with U 100 U respect to t, the total potential (light blue) LJ, i.e., inducing attractions between f g facets, (green) z, i.e., keeping the NCs at the interface plane, (fuchsia) Uφ þ Uψ , i.e., orienting the NCs at the interface with a f111g facet parallel to the interface plane, and (black) U the total internal potential energy pot; see Eq. (7). An animation of this simulation is available in the Supplemental Material [105].

021015-9 GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019) experimentally observed, i.e., where all the NCs have a B. Assembled-phase diagram 111 f g facet parallel to the interface plane and are disposed In Figs. 3 and 4 we showed that the NC self-assembly in a buckled honeycomb lattice. In inset (a) of Fig. 3,we can be drastically affected by solely tuning the parameter show an enlargement of a typical precursor of the silicene- up, regulating the strength of the forces orienting the NCs honeycomb superstructure, that is, a zigzag aggregate of 111 111 at the interface with a f g facet parallel to the interface. NCs with a f g facet parallel to the interface (also Reasonably, on the basis of the formation mechanism experimentally observed [73]). In inset (b) of Fig. 3, for silicene-honeycomb and square superstructures that we show an enlargement of a large portion of a silicene- emerged from the simulations of Figs. 3 and 4, the other honeycomb superstructure formed by the NCs, highlighting two key parameters that should affect the NC self- defects that have formed: vacancies (i.e., a NC missing assembly are E, setting the strength of the forces between from the lattice, see fuchsia circles), misalignment between f100g facets of different NCs, and u , setting the strength different domains of the superstructure (see the lattices z of the forces keeping all the NCs aligned at the same highlighted in light blue and green), and a trilayer silicene- plane at the interface. To confirm this reasoning, in Fig. 5 honeycomb (see the bottom left-hand corner of the inset, we report the assembled phase (square superstructure, where the presence of the extra NCs forming the third layer silicene-honeycomb superstructure, or disordered, i.e., no of the silicene-honeycomb lattice is indicated by the light self-assembly) obtained from a simulation, with our blue hexagonal lattice). All these defects have been coarse-grained MD model, of 64 NCs for 2 μs, with observed experimentally [88]. Their presence in our MD respect to the values used for E, u , u (using, as NC simulations is an additional confirmation that our model z p initial configuration, a hexagonal monolayer with lattice correctly captures the key features of the PbSe NC self- 0 assembly in silicene-honeycomb superstructures. spacing 12 nm at the interface plane z ¼ ). All the other parameters of our MD model are set to the same 4. Formation of square superstructures values used in the simulations of Figs. 3 and 4 (see Appendix C). Periodic boundary conditions are applied in Differently from the mechanism described for Fig. 3,in the x, y directions, with period 96 nm. Fig. 4 the NCs manage to attach to each other by 100 facets f g As shown in Fig. 5, the obtained NC assembled phase by changing their orientation at the interface, orienting with a nontrivially depends on E, u , u . A low u (corresponding f100g facet parallel to the interface plane (see the φ; ψ z p p to weak forces orienting the NCs with a 111 facet parallel distributions for increasing time). This change in NC f g to the interface plane) favors formation of square super- orientation increases the total potential Uφ þ Uψ , corre- structures and, vice versa, a high up (corresponding to sponding to an energy penalty. However, by attaching by 111 100 U strong forces orienting the NCs with a f g facet parallel f g facets, the NCs decrease the total potential LJ more than the increment in Uφ þ Uψ ; hence, the total internal U energy pot actually decreases. This is confirmed by the energy plots shown in the bottom right-hand panel of Fig. 4, U U where the black line is potðtÞ, the light blue line is LJðtÞ summed over all bead-bead pairs, and the fuchsia line is UφðtÞþUψ ðtÞ summed over all NCs. All the NCs remain at the same height at the interface plane, keeping the zc value with minimum Uz, as confirmed by the plot of UzðtÞ, summed over all NCs, see green line in the energy graph, which remains essentially constant (see also the distributions FIG. 5. Assembled-phase diagram for 64 PbSe NCs at a fluid- of zc). As clearly shown also by the simulation snapshots, fluid interface with respect to uz (regulating the strength of the thanks to this mechanism the NCs aggregate into square forces keeping the NCs in plane), up (regulating the strength of superstructures, equivalent to those experimentally ob- the forces orienting the NCs with a f111g facet parallel to the served, i.e., where all the NCs are aligned in the same plane interface plane), and E (total bonding energy between two NCs and have a f100g facet parallel to the interface plane. In inset attached by f100g facets). The symbols indicate (blue) disordered (a) of Fig. 4, we show an enlargement of a typical precursor of phase, (red) square superstructure, and (green) silicene-honeycomb the square superstructure, that is, a linear aggregate of NCs superstructure. Multiple symbols in a single cell indicate as- sembled-phase coexistence. The labels “Fig. 3” and “Fig._4” with a 110 facet parallel to the interface (also experimen- f g indicate the cases with the same parameters of the simulations tally observed [73,85]). In insets (b) and (c) of Fig. 4,we shown in Figs. 3 and 4, respectively. For the experiments of interest show an enlargement of a large portion of square super- [73,85–88], the parameters uz, up, E are expected to fall within the structure formed by the NCs, highlighting defects that have ranges considered here, hence close to the triple point where square, formed: vacancies (see fuchsia circles) and misalignment silicene-honeycomb, and disordered phase coexist. See also between different domains of the superstructure (see the Figs. 12–14, in Appendix C, where a snapshot of the NC final lattices highlighted in light blue and green). configuration and the energy plot are shown for each simulation.

021015-10 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019) to the interface) favors formation of silicene-honeycomb systematic studies—based on the model and results superstructures. However, silicene-honeycomb superstruc- presented in this paper—for optimizing the synthesis tures are formed only for low values of uz (corresponding procedures and will illustrate their findings in future to weak forces keeping the NCs at the interface plane, interdisciplinary works with simulations and experiments. i.e., preventing partial desorption of the NCs from the interface), while square superstructures are essentially V. UNDERSTANDING THE FORMATION unaffected by uz (for the range considered). Finally, tuning MECHANISM the NC-NC attachment energy E (i.e., the strength of the In this work, we uncovered the formation mechanism of NC-NC attractive forces) affects the turning values for uz square [73,85,87] and silicene-honeycomb superstructures and up at which different superstructures are observed: a higher NC-NC attraction allows formation of silicene- [85,86,88]; see Secs. IVA 3, IVA 4. Our simulations indicate that silicene-honeycomb self-assembly will occur only if, at honeycomb superstructures for higher values of uz,but on the other hand, a higher value of up is required to some point after the NCs adsorb at the interface, there are deep prevent formation of square superstructure. enough wells in the energy due to NC-NC attachment by f100g facets (i.e., E is low enough) and in the interface- C. Outlook for the experimental synthesis adsorption energy corresponding to the NC orientation with a f111g facet parallel to the interface plane (i.e., in the potential The range of values considered in Fig. 5 for E, uz, up Uφ þ Uψ ), to compensate for the partial desorbment of the represents the realistic intervals in which the experiments NCs from the interface (half of the NCs moving slightly above [73,85–88] are expect to fall, on the basis of the calcu- lations presented in Sec. III for the adsorption at toluene-air the interface plane, and the other half slightly below) which interfaces of PbSe NCs (regarding the values of u , u ), and causes an energy penalty, because the interface-adsorption z p U on the basis of typical bonding energies between oriented- energy due to z increases. Conversely, square self-assembly attached nanocrystals [97] (regarding the value of E). As will occur only if, at some point after the NCs adsorb at the shown by Fig. 5, within these ranges of values, all three interface, there are deep enough wells in the energy due to phases (disordered, silicene-honeycomb, square) are pos- NC-NC attachment by f100g facets (i.e., E is low enough) sible for the experiments. That is, the experiments are very and in the interface-adsorption energy keeping the NCs at the close to the three-phase coexistence point, and slight same height at the interface plane (i.e., in the potential Uz), to variations in the experimental conditions can drastically compensate for the change in orientation of the NCs at affect the PbSe NC self-assembly, resulting in a different the interface (from a f111g to a f100g facet parallel to the assembled structure. This explains the difficulties encoun- interface plane) which causes an energy penalty, because tered in the experiments in controlling and reproducing the the interface-adsorption energy due to Uφ þ Uψ increases. NC self-assembly. Finally, no NC-NC attachment by f100g facets, hence no In this work, we uncovered the formation mechanism of self-assembly, is expected if, at any stage after the NCs have these superstructures, paving the way toward improved synthesis routes. For example, on the basis of our results, adsorbed, the energy well in the NC-NC attachment energy by 100 one deduces that the NC self-assembly rate in silicene- f g facets is not deep enough to compensate for either a honeycomb superstructures can be optimized by increasing reorientation of the NCs (required to form the square super- the parameters E and/or up in the experiments, and/or by structure) or a partial desorbment of the NCs from the decreasing uz. The parameter uz can be decreased, e.g., by interface (required to form the silicene-honeycomb super- lowering the solvent-air surface tension, or by using smaller structure), both giving an energy penalty, due to an increase in U U U NCs. The parameter up can be regulated by playing with φ þ ψ and z, respectively. the solvent-air surface tension and with the size, surface The most essential results emerging from our work and properties, and shape of the synthesized NCs—even relevant for understanding the formation mechanism of slightly different degrees of truncation in the NC shape PbSe square and silicene-honeycomb superstructures are — can considerably affect up where NC shape and surface schematized in the next few concise points. properties can also affect the equilibrium orientation of (1) PbSe NCs (size ∼6 nm) fully capped by oleic acid isolated NCs at the interface. The sharp-interface method of molecules adsorb at a toluene-air interface with Soligno et al. [99–101], here applied in Sec. III, is well energy bonding ≃1.3 × 10−19 J and, essentially, suited to predict how these NC properties affect uz, up. The with random orientation; see Appendix B2. If many parameter E can be regulated, e.g., by tuning the NC size or of such NCs adsorb at the interface, a hexagonal by using a different compound. Clearly, the experimental monolayer (with randomly oriented NCs) is ex- conditions affecting one parameter in the desired manner pected in the close-packed limit, due to the approx- might affect another parameter in the opposite way, so imately hard-sphere interactions between NCs (since care should be used in regulating them. We will carry on NC-NC attractions are largely screened by ligands).

021015-11 GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019)

(2) PbSe NCs (size ∼6 nm) partially capped by oleic PbSe NCs in square and silicene-honeycomb superstruc- acid molecules (i.e., ligands cover only their f110g, tures, experimentally observed [73,85–88]. We show that f111g facets) at a toluene-air interface. the final structure is determined by an intricate interplay (a) When isolated, they are bonded to the interface between interface-adsorption forces orienting and keeping by an energy ≃2.3 × 10−19 J, and are oriented the NCs in plane at the fluid-fluid interface where with a f111g facet parallel to the interface plane; the self-assembly occurs and short-range electrostatic- see Sec, III B. chemical attractive forces between specific facets of the (b) When close enough to each other, their behavior NCs. Hence, in the formation of these 2D superstructures, a is determined by the interplay between NC-NC fundamental role is played by the NC interactions with the electrostatic-chemical attractive forces (between fluid-fluid interface, in contrast with 3D NC superstructures NC ligand-free f100g facets) and the interface- where, typically, the self-assembly is driven solely by adsorption forces keeping the NCs in plane at the NC-NC entropic and (possibly) attractive interactions. interface and orienting the NCs with a f111g facet The interface-adsorption forces experienced by the NCs, parallel to the interface plane. Depending onwhich for typical experimental conditions, are estimated using a forces prevail, the following scenarios occur. macroscopic sharp-interface model; see Fig. 2 (Sec. III). (i) Interface-adsorption forces keeping the NCs in Ligand molecules chemisorbed on the NC surface (weakly plane at the interface and orienting them with on the f100g facets, and strongly on the f110g, f111g a f111g facet parallel to the interface plane facets) also play a fundamental role. Indeed, during the prevail over attractive forces between NC formation process of the PbSe superstructures, ligands are 100 f100g facets. Therefore, NCs cannot attach expected to detach from the NC f g facets, allowing 100 by opposite f100g facets (because of clear NC-NC attractions only by opposite f g facets, while geometrical reasons). A monolayer is formed NC-NC attachment by f110g, f111g facets is prevented at the interface—with all NCs oriented with a by the ligands still chemisorbed on these facets. The f111g facet parallel to the interface plane— basic mechanism of formation for square and silicene- disordered at low NC densities and with honeycomb superstructures, arising from the different hexagonal geometry at high packing fractions. interplay between the various forces in the system, is (ii) Attractive forces between NC f100g facets illustrated by the simulations shown in Figs. 3 and 4 and interface-adsorption forces orienting (Sec. IVA). Then, by tuning the parameters of our coarse- the NCs with a f111g facet parallel to the grained MD model around their expected values in the interface plane prevail over interface- experiments, we show that the experimental conditions adsorption forces keeping the NCs in plane are very close to the three-phase point where square at the interface. The NCs partially desorb from superstructure, silicene-honeycomb superstructure, and the interface plane (half of the NCs slightly disordered-monolayer phase coexist; see Fig. 5 (Sec. IV B). moving up and half of them slightly down) to This justifies the difficulties in controlling and reproducing allow attachment by opposite f100g facets the self-assembly in the experiments, since slight changes while keeping a f111g facetparalleltothe in the experimental conditions can result in a completely interface plane. Therefore, NCs self-assemble different behavior of the NCs. The most important results in silicene-honeycomb superstructures. presented in this work and relevant for understanding (iii) Interface-adsorption forces keeping the NCs the formation mechanism of PbSe square and silicene- in plane at the interface and attractive forces honeycomb superstructures are summarized in Sec. V.Our between NC f100g facets prevail over inter- results pave the way toward an improved control over the face-adsorption forces orienting the NCs with nanostructure synthesis; see Sec. IV C. As a confirmation that our coarse-grained MD model a f111g facet parallel to the interface plane. The NCs orient with a f100g rather than correctly captures the key features of the dynamics of the f111g facet parallel to the interface plane (the NC self-assembly, we show that the formation dynamics of former orientation is energetically less favor- the silicene-honeycomb and square superstructures seems able for an isolated NC) to allow attachment to match the experiments; see Figs. 3 and 4. At the early by opposite f100g facets while remaining all formation stages of the silicene-honeycomb and square in plane at the interface. Therefore, NCs self- superstructures, we recognize their typical precursors, i.e., assemble in square superstructures. zigzag and linear aggregates, respectively (both experi- mentally observed). In addition, the superstructures formed by the NCs present the same kinds of defects observed VI. CONCLUSIONS experimentally. This suggests that our coarse-grained MD In conclusion, introducing a new coarse-grained MD model can also be used to study the formation of these model for the self-assembly of NCs at fluid-fluid interfaces defects and their stability with respect to, e.g., the temper- (see Sec. II), we study the self-assembly mechanism of ature or other parameters.

021015-12 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019)

We remark that our coarse-grained MD model, here (iii) Compute applied to PbSe NCs, can easily be adapted to study the Δt2 self-assembly at fluid-fluid interfaces of NCs with other Δ 0 Δ riðt þ tÞ¼riðt þ tÞþgiðtÞ ; ðA2Þ shape and interactions. The interface-adsorption potentials, mi here defined to keep the NCs in the same plane and orient for i ¼ 1; …;N N . them with a f111g facet parallel to the interface, can easily a p Δt 10−12 be modified to reproduce different preferred orientations The time step used in our simulations is ¼ s. The force F t is the total force—applied at the time t to of the NCs, possibly also including metastable orientations. ið Þ the ith bead—due to the various potentials acting on the A graphic-user-interface software implementing our beads, that is—see later definitions—the bead-bead pair coarse-grained MD model and usable to simulate the NC U U U U U potentials LJ, R, and the external potentials z, φ, ψ , self-assembly in customizable user-defined conditions will U be made (freely) available to the community soon [106]. c used to reproduce the forces on the NCs due to the fluid- Our coarse-grained MD approach is computationally much fluid interface. The N beads of each NC are given by seven core beads, cheaper than fully atomistic or lattice Boltzmann methods, a used to simulate the NC Brownian motion (see Appendix A2), and simulations with ∼1000 NCs at a fluid-fluid interface plus N − 7 144 shell beads, used to reproduce the NC can easily be performed on a single computer. a ¼ surface. Shell beads of different NCs interact with each other U by the Lennard-Jones pair potential LJ. Given the generic ith ACKNOWLEDGMENTS and jth shell beads (i; j ¼ 1; …;NaNp, i ≠ j), the potential U The authors acknowledge financial support by the Dutch LJ between this pair of beads is defined as NWO Physics, FOM program nr. 152 "Engineering Dirac 4ϵ½ðσÞ12 − ðσÞ6þc if r ≤ 2.4σ physics in semiconductor superlattices", and the ERC U ðrÞ ≡ r r ðA3Þ LJ 0 2 4σ Advanced Grant No. 692691 "FIRST STEP". if r> . ;

where r ≡ jri − rjj is the distance between the beads, σ is set to 1 nm in our simulations, and c is a constant such that APPENDIX A: NC POLYBEAD MODEL U 2 4σ 0 LJð . Þ¼ . The forces f ij and f ji,appliedontheith and U Here we provide more technical details on our coarse- jth bead, respectively, and due to the pair potential LJ grained molecular dynamics model to simulate the self- between these two beads, are zero if r>2.4σ,and assembly of nanocrystals at fluid-fluid interfaces. dU ðrÞ 24ϵ σ 12 σ 6 f ¼ − LJ ¼ 2 − r ; ij dr r2 r r ij 1. Bead dynamics and interactions ij f ¼ −f ; ðA4Þ Each NC is represented by a polybead structure formed ji ij by N 151 beads. Each ith bead, with i 1; …;N N , a ¼ ¼ a p otherwise, with rij ≡ ri − rj. In addition, some pairs of shell N and p the number of NCs, is formally a pointlike particle, beads interact also by the soft repulsive pair potential UR, with mass mi and position at the time t given by riðtÞ≡ defined as ½xiðtÞ;yiðtÞ;ziðtÞ, where a Cartesian coordinate system x, σ ϵ ½ð RÞ6 − 1 if r ≤ σ y, z is introduced. The dynamics of the beads is computed U ≡ R r R ’ RðrÞ ðA5Þ by the position Verlet s algorithm [94]. Constraints forces, 0 if r>σR; to keep beads belonging to the same NC at constant ϵ 2 10−22 σ reciprocal distances, are calculated and applied using the where we set R ¼ × J in our simulations (and R is U method of Ciccotti and Ryckaert [95]. Hence, the basic defined later). If R is active for the ith and jth bead pair, then algorithm to compute the position of the beads at the time the forces f ij and f ji,appliedontheith and jth bead, U t þ Δt, given their positions at the time t and at the time respectively, and due to the pair potential R between these σ t − Δt, is the following. two beads, are zero if r> R,and (i) Compute the unconstrained bead positions, U 6ϵ σ 6 − d RðrÞ R R f ij ¼ ¼ 2 rij; ðA6Þ 2 dr r r Δt ij 0 Δ 2 − − Δ riðt þ tÞ¼ riðtÞ riðt tÞþFiðtÞ ; ðA1Þ mi f ji ¼ −f ij; ðA7Þ 1 … U U for i ¼ ; ;NaNp. otherwise. In Fig. 6, we show a plot of LJ and R for (ii) Compute the desired constraint force giðtÞ to be illustrative parameters. 0 Δ applied on each ith bead, given riðt þ tÞ, for Periodic boundary conditions can be applied in our i ¼ 1; …;NaNp, as illustrated in Ref. [95]. model; that is, all the beads in the system are periodically

021015-13 GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019)

FIG. 6. Blue curve: Lennard-Jones pair potential ULJ [Eq. (A3)], in units of ϵ and with respect to r=σ, where r is the bead-bead distance. Black curve: Soft repulsive pair potential UR [Eq. (A5)], FIG. 7. 3D views of the polybead structure, in different in units of ϵ and with respect to r=σ, for σR ¼ 2.5σ and orientations, representing a NC in our model, showing (a) the ϵR ¼ 0.08ϵ. Red curve: Sum of ULJ and UR, for the same correspondent representation with facets of the NC (with the parameters used before. f100g, f110g, f111g facets in orange, dark orange, and red, respectively), (b) only the shell beads of the NC, (c) both the shell beads and the core beads of the NC, (d) only the core beads of the repeated in the x, y, and z directions with the chosen period. NC. Beads are plotted as spheres with diameter 1 nm. The shell If the minimum image criterium is satisfied, that is, the beads (colored in blue, light blue, black, orange, and red) are cutoff radius of each bead-bead pair potential is smaller positioned to model, approximately, the surface of a 6-nm-sized than half of the periodic box side, then all bead-bead pairs rhombicuboctahedron, with shell beads of different NCs inter- with a given bead can be computed considering only the acting with each other by Lennard-Jones and soft repulsive pair closest copy of every other bead (since all other pairs are potentials; see Eqs. (A3) and (A5). Each of the 24 black colored beads is placed with the center in one of the 24 vertexes of a certainly at a distance greater than the cutoff radius). slightly cantellated rhombicuboctahedron with side 2.5 nm for In Fig. 7, we illustrate the disposition of the Na beads used the f100g square facets, and distance 5 nm between two opposite to reproduce a NC. The 144 shell beads are used to reproduce f100g facets. The position of the center of each blue, light blue, the surface of a NC with rhombicuboctahedron shape and and red colored bead is obtained by a linear interpolation of the size 6 nm. The seven core beads are placed one in the NC centers of two black beads belonging to the same facet. The center of mass and the other six at distance rB from it and in position of the center of each orange colored bead is obtained by a the six h100i directions of the NC. For each NC, the mass of linear interpolation of the centers of the two closest blue colored the central core bead is 2.114 × 10−22 kg, the mass of each beads. The seven core beads, exploited for the Brownian motion of the remaining six core beads is 10−25 kg, and the mass of of the NC and here colored in green or gray, are placed one in the 2 10−24 NC center of mass and the other six at distance rB from it (in these each of the 144 shell beads is × kg, corresponding to 3 100 5 10−22 plots, rB ¼ nm) and in the six h i directions of the NC, a total mass × kg for the NC. Of course, by using a respectively. Note that, since the core beads do not interact by different number of shell beads and/or placing them in bead-bead pair potentials, their position does not affect the shape different positions, one can easily tune in our model the NC of the NC even if they are placed outside the NC surface formed shape and the accuracy in its representation. by the shell beads. The dynamics of the four green colored core Shell beads of different NCs interact by the Lennard-Jones beads follows the Verlet-type Brownian algorithm in Eq. (A8), U pair potential LJ [Eq. (A3)]. To pairs of beads colored with while the dynamics of the three gray colored core beads, like for light blue, see Fig. 7, we assign the interaction parameter all the shell beads, follows the standard position-Verlet algorithm in Eq. (A1). ϵ ¼ ϵ2. To pairs of a light blue colored bead and a non-light- blue colored bead, we assign the interaction parameter ϵ ¼ðϵ1 þ ϵ2Þ=2. Finally, to all the other shell-bead pairs, σR ¼ 2.5 nm, while for pairs where one bead is red or orange we assign the interaction parameter ϵ ¼ ϵ1. In our simulations, σ 1 25 −21 colored and the other is not, we assign R ¼ . nm. ϵ1 ¼ 10 J, and ϵ2 is tuned to regulate the desired energy bond E between two NCs attached by opposite f100g facets. −21 −19 2. NC Brownian motion We set ϵ2 ¼ 10 JtoobtainE ≈−0.7 × 10 J, ϵ2 ¼ 2.5 × −21 −19 −21 10 JtoobtainE ≈−1.0 × 10 J, and ϵ2 ¼ 4.0 × 10 J The core beads do not interact by pair potentials, but they to obtain E ≈−1.3 × 10−19 J. are exploited for inducing the Brownian motion of the NCs. In addition, in our simulations, all shell-bead pairs For this, the Verlet-type algorithm of Ref. [98] is applied to involving a red or orange colored bead in Fig. 7 interact the central core bead and to three of the remaining six core also by the soft repulsive pair potential UR [Eq. (A5)]. For beads, nonaligned with the central core bead; see the four pairs where both beads are red or orange colored, we assign green colored beads in Fig. 7. That is, the unconstrained

021015-14 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019)

0 position ri of these four beads, with i the index referring to the bead, is not computed as described in step (i) of the algorithm in [Eq. (A1)], but as follows:

b Δt2 0 Δ 2 − − Δ i riðt þ tÞ¼ biriðtÞ airiðt tÞþ FiðtÞ mi biΔt þ ½F iðtÞþF iðt þ ΔtÞ; ðA8Þ 2mi where λΔt −1 FIG. 8. Using our coarse-grained MD model (see text), we ≡ 1 consider here a single NC in bulk (i.e., no fluid-fluid interface is bi þ 2 ; ðA9Þ mi introduced) and measure the NC average translational and rota- tional diffusion with respect to the simulation time t, for rB ¼ λΔt 3.3 nm (see Fig. 7) and λ ¼ 3 × 10−11 kg s−1 in Eq. (A8). In (a), a ≡ 1 − b ; A10 i i ð Þ the red line shows the squared displacement of the NC center of 2mi mass with respect to t, averaged over 1000 simulations. The blue line is the analytical prediction [Eq. (A14)] for a sphere of and F i ≡ ðF i1; F i2; F i3Þ, with diameter 6 nm immersed in toluene at room temperature. In (b), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 the red line shows hφ ðtÞi, that is, φ [with φ the polar angle of F ω t Δt ≡ ξ ω t 2λk TΔt; i ð þ Þ i ð Þ B ðA11Þ the NC vertical axis in the plane z ¼ 0, see Fig. 2(a)], with respect to t and averaged over 1000 simulations, setting φ ¼ 0 as initial ω 1 2 for ¼ , 2, 3, where kB is the Boltzmann constant, T is configuration. After a time long enough, hφ ðtÞi becomes the temperature expressed in kelvin (in our simulations we constant, since, as expected, hφðtÞi (light blue line) goes to 2 set T to room temperature), and ξiω, for any given time, is π=2. Before this regime takes place, hφ ðtÞi is expected to be an independent random number obtained from a Gaussian linear in t, as predicted by Eq. (A15). The black line is the distribution with zero mean and unitary variance. The analytical prediction [Eq. (A15)] for a sphere of diameter 6 nm values of the (friction) parameter λ in Eq. (A8) and of immersed in toluene at room temperature. The inset is an 2 r (i.e., the distance of the side core beads from the central enlargement of the plot in the initial linear regime of hφ ðtÞi, B ≲ 0 05 μ core bead) are tuned to induce the expected translational i.e., for t . s in our case. and rotational average diffusion of a NC immersed in a typical solvent at room temperature. Assuming that a NC −11 −1 λ ¼ 3 × 10 kg s in Eq. (A8), and we place the side approximately behaves like a sphere of diameter D, its core beads at a distance r ¼ 3.3 nm from the central core translational and rotational diffusion coefficients are [107] B bead. Using these values (heuristically found), the obtained k T translational and rotational diffusion of the NC fairly D ¼ B ; ðA12Þ matches, see Fig. 8, the diffusion predicted by Eqs. (A14) t 3πηD and (A15), at room temperature, for D ¼ 6 nm, and for η ¼ 5 6 10−4 −1 −1 k T . × kg s m , that is, the viscosity of toluene at D B ; r ¼ πη 3 ðA13Þ room temperature. Note that if rB is such that the core beads D are placed outside the NC surface formed by the shell beads, the actual shape of the NC is not affected, since the core beads respectively, where η is the solvent viscosity. The mean do not interact by bead-bead pair potentials. The remaining squared displacement (in 3D) of a NC is expected to fulfill three core beads (gray colored beads in Fig. 7) follow the the Stokes-Einstein equation [107], i.e., non-Brownian dynamics in Eq. (A1), and are introduced in 2 the model to ensure a distribution of the mass with cubic hjrðtÞ − rð0Þj i¼6Dtt; ðA14Þ symmetry with respect to the NC central core bead. with rðtÞ the NC center-of-mass position at the time t. Thus far we have assumed the NCs as immersed in a The mean squared angular displacement (in 3D) of a NC homogeneous fluid. In the next sections, a fluid-fluid is expected to fulfill the Stokes-Einstein-Debye relation interface is introduced in our MD model, by defining [107], i.e., external potentials for the NC beads that mimic the forces, due to the interface, experienced by adsorbed NCs. 2 hjφðtÞ − φð0Þj i¼4Drt; ðA15Þ Regarding the NC Brownian motion, we assume that a NC has approximately the same rotational and translational with φ the angular displacement of a given direction diffusion coefficients when at the fluid-fluid interface. of the NC after a time t. In our simulations, we set Hence, we keep the same procedure and parameters

021015-15 GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019) illustrated in this section to induce the NC Brownian The potential Uz represents the interface-adsorption forces motion when the NCs are at a fluid-fluid interface. The binding a NC to the interface, i.e., keeping the NC close to interface effects on the NC dynamics are already taken into the interface, with z0 the NC Uz height at the interface and account by introducing the external potentials on the beads. uz setting the force strength. The parameter zd represents Finally, note that hydrodynamics interactions between NCs the maximum distance at which the NC feels the interface are not included in our approach, since these are expected (so zd is roughly the NC size). For simplicity, we assumed to be negligible in the experiments of interest. that Uz depends only on the NC center-of-mass distance from the interface, and we used a simple parabolic equation 3. Single-NC interface-adsorption potentials to represent this dependence. More complex and accurate The remaining fundamental ingredient to correctly expressions, which, e.g., are not symmetric with respect to reproduce in our coarse-grained MD model the self- zc ¼ z0 and/or also include the NC orientation, can easily assembly of PbSe NCs in quasi-2D superstructures is the be introduced in our model. fluid-fluid interface where the NCs are adsorbed (typically, To the side core beads of the NC we apply the external the interface between the solvent where the NCs are potentials inducing the interface-adsorption forces driving initially dispersed and air). We apply external potentials the NC toward a specific orientation at the interface. During to the NC beads to mimic the forces due to the fluid-fluid each time step of a simulation with our MD model, first the orientation of the NC with respect to the interface plane interface. 0 z 0 (i.e., z ¼ ) is computed. Then, given the NC orientation, We assume that the plane ¼ is the fluid-fluid inter- U U face, where a 3D Cartesian coordinate system x, y, z is the interface-adsorption potentials φ and ψ are applied introduced, with the semiplane z<0 being one fluid and to the various side core beads of the NC. To define the the semiplane z>0 being the other fluid. For simplicity, orientation of a NC, with respect to the interface plane, we φ ∈ 0 π ψ ∈ 0 2π we consider here a single NC. When more NCs are use the angles ½ ; and ½ ; Þ, i.e., the polar adsorbed at the interface, the same definitions can be angle between the NC vertical axis and the interface plane, applied independently for each NC (Ref. [99] shows and the internal Euler angle of the NC around its vertical that, for the experiments of interest, single-NC interface- axis, respectively; see Fig. 2(a). The vertical axis of the NC adsorption properties are negligibly affected by multi- is defined as the vector vφ going from the central core bead to the side core bead with position ðx2;y2;z2Þ; that is, particle effects). Given the Na beads forming the NC, labeled by i ¼ 1; …;Na and having position ri ¼ vφ ≡ ðx2;y2;z2Þ − ðx1;y1;z1Þ: ðA19Þ ðxi;yi;ziÞ, we indicate by i ¼ 1 the core bead in the NC center of mass (central core bead), and by i ¼ 2; …; 7 the φ remaining six core beads of the NC (side core beads). Then, the angle of the NC is The i ¼ 2 and i ¼ 5 side core beads are aligned with the vφ · ð0; 0; 1Þ central core bead, and analogously the i ¼ 3 and i ¼ 6, and φ ≡ arccos : ðA20Þ the i ¼ 4 and i ¼ 7. The three lines passing through these jvφj three pairs of side core beads are orthogonal to each other, see Fig. 7, and represent the six h100i directions of the NC. Given the side core bead with position in ðx3;y3;z3Þ, which is not aligned to ðx1;y1;z1Þ and ðx2;y2;z2Þ, we define All the other NC beads, i.e., with i ¼ 8; …;Na, are the shell beads. We define external potentials only for the i ¼ 1, ≡ − i ¼ 2, i ¼ 3, i ¼ 5, and i ¼ 6 beads. The forces exerted on vψ ðx3;y3;z3Þ ðx1;y1;z1Þ; ðA21Þ these beads, thanks to the constrained dynamics, are ≡ transmitted to the whole NC polybead structure. and w ðwx;wy;wzÞ as the vector orthogonal to the NC To the central core bead of the NC, i.e., with position vertical axis vφ and belonging to the plane identified by the vectors (0,0,1) and vφ. Therefore, w · vφ 0 and ðx1;y1;z1Þ, we apply the potential Uz defined in Eq. (3). ¼ 0 0 1 0 1 The force f ¼ðfx;fy;fzÞ experienced by the NC central w · ½ð ; ; Þ × vφ¼ . Adding the condition jwj¼ ,we core bead because of Uz [Eq. (3)]is obtain

∂U qvφffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixvφz z wx ¼ ∓ ; fx ¼ − ¼ 0; ðA16Þ 2 2 ∂x1 jvφj vφx þ vφy vφ vφ ∓ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy z ∂Uz wy ¼ ; fy ¼ − ¼ 0; ðA17Þ 2 2 ∂y1 jvφj vφx þ vφy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ∂U −2u ðz − z0Þ if jz j ≤ z vφx þ vφy − z z c c d fz ¼ ∂ ¼ ðA18Þ wz ¼ ; ðA22Þ z1 0 if jzcj >zd: jvφj

021015-16 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019)

1 ∂Uφ vφ vφ where we choose the positive sign for wz (and so the − ξ φ − φ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix z ψ fx ¼ ¼ ðzcÞð 0 Þup ; negative sign for wx and wy). Given the angle between w 2 ∂x2 2 2 2 jvφj vφx þ vφy and vψ , that is, 1 ∂U φ qvφffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiyvφz fy ¼ − ¼ ξðzcÞðφ0 − φÞup ; 2 ∂y2 2 2 2 vψ · w jvφj vφx þ vφy ψ ≡ arccos ; ðA23Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jvψ j 2 2 1 ∂U vφx þ vφy − φ −ξ φ − φ fz ¼ ¼ ðzcÞð 0 Þup 2 ; ðA27Þ 2 ∂z2 jvφj the angle ψ of the NC is defined as

while the force due to the potential Uφ=2 applied to the side x5;y5;z5 −f ψ if vφ · ðw × vψ Þ ≤ 0 core bead in ð Þ is . Therefore, we are applying a ψ ¼ ðA24Þ force couple, i.e., a torque, to the NC which, in total, 2π − ψ otherwise: induces a rotation of the NC around its center of mass and in the φ rotational plane. The total potential energy stored by the NC because of its orientation φ at the interface is Note that, if vφ is parallel to (0,0,1), then w,andsoψ,is UφðφÞ [Eq (4)]. Note that the forces induced by Uφ not defined. This is not a problem, since, for φ ¼ 0,the [Eq. (A27)] are singular in φ 0 and φ π, i.e., when NC has a f100g facet parallel to the interface plane ¼ ¼ vφ vφ 0. Indeed, in these cases, the plane of rotation z ¼ 0, so its orientation with respect to the interface x ¼ y ¼ φ plane is the same for any ψ;seeFig.2(a).Oncethe of the angle is undefined, since vφ is aligned to z, so the values of φ and ψ for the NC are computed at a given plane passing through vφ and z is undefined. Therefore, time step, the external potentials Uφ and Uψ are applied these cases need to be treated separately. However, since we U φ φ 0 to the NC side core beads to induce rotations (by consider a potential φð Þ which is maximum in ¼ and applying force couples) in the rotational planes of φ φ ¼ π, we simply set the forces to zero. and ψ. This drives the NC toward orientations with To both the side core beads with position ðx3;y3;z3Þ and minimum Uφ and minimum Uψ . ðx6;y6;z6Þ, i.e., aligned with the central core bead but not To both the side core beads with position ðx2;y2;z2Þ and along the direction of the NC vertical axis vφ, we apply the ðx5;y5;z5Þ, i.e., aligned with the central core bead along the rotational potential, direction of NC vertical axis vφ, we apply the rotational potential, Uψ ψ u ð Þ ≡ p ψ − ψ 2ξ 2 2 ð 0Þ ðzcÞ; ðA28Þ U φ φð Þ up 2 ≡ ðφ − φ0Þ ξðzcÞ; ðA25Þ 2 2 with up setting the force strength, ψ 0 the minimum-Uψ value of ψ for the NC, and ξðzcÞ defined in Eq. (A26) and representing the dependence of the interface-adsorption φ U with up setting the force strength, 0 the minimum- φ forces due to Uψ on the NC distance from the interface. value of φ for the NC, and ξ z defined as ð cÞ Therefore, given the vectors vψ ¼ðvψx;vψy;vψzÞ and w ≡ ðwx;wy;wzÞ defined in Eqs. (A21) and (A22), the force 8 f ¼ðfx;fy;fzÞ due to the potential Uψ =2 applied to the > 0 if jzcj >zd < side core bead in ðx3;y3;z3Þ is zcþzd − ξðz Þ ≡ if zd z0þzd :> z −z c d if z0

021015-17 GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019) where the sign is plus if ψ < π and minus otherwise. The with minimum-Uφ and minimum-Uψ considered in our force due to the potential Uψ =2 applied to the side core bead simulations, this problem does not occur. in ðx6;y6;z6Þ is −f. Therefore, we are applying a force couple, i.e., a torque, to the NC which, in total, induces a 4. NC-NC pair capillary potential rotation of the NC around its center of mass and in the ψ rotational plane. The total potential energy stored by the NC As predicted in Refs. [99,102], cubes and cantellated because of its orientation ψ at the interface is Uψ ðψÞ [Eq. (5)]. rhombicuboctahedrons, when adsorbed at a fluid-fluid 111 Note that, if φ ¼ 0 or φ ¼ π, the angle ψ and the vector w are interface with a f g facet parallel to the interface plane, generate hexapolar capillary deformations in the interface not defined. In this case, the forces induced by Uψ are set to height profile that induce capillary interactions between zero, since there cannot be any potential in ψ when φ ¼ 0 or the particles. We show in Appendix B3 that this effect is φ ¼ π (as the NC orientation is the same for any ψ when negligible for the self-assembly of PbSe NCs in the φ ¼ 0 or φ ¼ π). Note also that the forces induced by Uψ experiments of interest. Nonetheless, for completeness [Eq. (A27)] are singular in ψ ¼ 0 and ψ ¼ π, i.e., when and future applications, we introduce in our coarse-grained w ¼vψ , because the rotational plane of the angle ψ, i.e., the MD model the NC-NC pair potential Uc, with the form of plane passing through w and vψ , is undefined. Therefore, the expected interaction energy between two hexapolar these cases need to be treated separately. However, since we U ψ ψ 0 capillary deformations [108], to include NC-NC capillary consider a potential ψ ð Þ which is maximum in ¼ , interactions, at least in an approximate form (i.e., neglect- ψ π ψ 2π ¼ , and ¼ , we simply set the forces to zero. ing multiparticle effects and considering only the case of In our simulations, the orientation of a NC with mini- NCs with a f111g facet parallel to the interface). mum UφðφÞ and minimum Uψ ðψÞ is with a f111g facet To introduce the NC-NC capillary pair potential Uc,we parallel to the interface plane. This is implemented by assume here that only two NCs, each with Na beads, are at setting the interface. If more NCs are present, the same definitions are applied to each pair of NCs. The index i 1; …; 2N 0 3π φ ∈ 0 π 2 ¼ a . if ½ ; = is used to label the 2N beads in the system, each φ0 ¼ a 0.7π if φ ∈ ðπ=2; π; with position r ¼ðx ;y;zÞ, with the first N beads 8 i i i i a 0 25π ψ ∈ 0 π 2 (i ¼ 1; …;N ) belonging to the first NC and the remaining > . if ½ ; = Þ a <> N beads (i ¼ N þ 1; …; 2N ) belonging to the second 0.75π if ψ ∈ ½π=2; πÞ a a a ψ 0 ¼ ðA30Þ NC. The position of the center of mass, i.e., of the central > 1.25π if ψ ∈ ½π; 3π=2Þ :> core bead, of the first and second NC is, respectively, r1 ¼ x1;y1;z1 and r 1 x 1;y 1;z 1 . We intro- 1.75π if ψ ∈ ½3π=2; 2πÞ: ð Þ Naþ ¼ð Naþ Naþ Naþ Þ duce φp, ψ p, αp, with p ¼ 1, 2, where φp and ψ p are the For simplicity, we assumed up to be the same parameter angles φ and ψ, as defined in Eqs. (A20) and (A24),of in UφðφÞ [Eq. (A25)] and Uψ ðψÞ [Eq. (A28)], and we the first and second NC, respectively, and α1, α2 are the assumed UφðφÞ and Uψ ðψÞ independent from ψ and φ, azimuthal orientations in the plane z ¼ 0, and measured respectively, and used a simple parabolic equation to from the direction joining the NC centers of mass, of the represent them. More complex and accurate expressions, vertical axis of the first and second NC, respectively. Given e.g., including a dependence from both φ and ψ, can in the vertical axis of the first NC, i.e., principle be introduced in the method. We remark that the ≡ − forces induced by Uφ and Uψ are approximations meant to vφ1 ðx2;y2;z2Þ ðx1;y1;z1Þ; ðA31Þ capture the key features of the forces experienced by a NC at a fluid-fluid interface. Note that, although φ ∈ ½0; π and the vertical axis of the second NC, i.e., ψ ∈ ½0; 2πÞ provide a one-to-one map to all the possible ≡ − orientations of a NC with respect to the interface plane vφ2 ðx2þNa ;y2þNa ;z2þNa Þ ðx1þNa ;y1þNa ;z1þNa Þ; 0 φ ψ z ¼ , see Fig. 2(a), the rotational planes of and are not ðA32Þ necessarily the same if we consider certain symmetric orientations of the NC (consider, e.g., a NC with a f100g and the vector v defining the line joining the centers of facet parallel to the interface plane). This slight incon- D mass of the two NCs, i.e., sistency occurs because we are using only two planes of rotation (defined by φ and ψ) to change the orientation of a v ≡ ðx1 − x1;y1 − y1;z1 − z1Þ; ðA33Þ 3D object. To avoid this, one should introduce a third plane D þNa þNa þNa of rotation, orthogonal to the rotational planes of φ and ψ, ≡ 0 and then define a rotational potential also in this plane. we consider their projections vφ1xy ðvφ1x;vφ1y; Þ, Anyway, for the orientations of a NC with a f111g facet vφ2xy ≡ ðvφ2x;vφ2y; 0Þ, vDxy ≡ ðvDx;vDy; 0Þ on the z ¼ 0 parallel to the interface plane, which is the NC orientation plane, i.e.,

021015-18 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019) 8 − − 0 0 φ π 2 φ π 2 vφ1 ¼ðx2 x1;y2 y1; Þ; ðA34Þ > if 1 > = and 2 > = xy <> 0 if φ1 < π=2 and φ2 < π=2 α φ1;φ2 ¼ ðA44Þ vφ ¼ðx2 − x1 ;y2 − y1 ; 0Þ; ðA35Þ > π=3 if φ1 > π=2 and φ2 < π=2 2xy þNa þNa þNa þNa :> π=3 if φ1 < π=2 and φ2 > π=2; − − 0 vDxy ¼ðx1þN x1;y1þN y1; Þ: ðA36Þ a a and is introduced because, when φ ¼ 0.7π, all the signs of the capillary deformations of the hexapolar deformation Given α1 and α2, defined as induced by the NC are inverted from to the case φ ¼ 0.3π (i.e., rises and depressions in the interface height profile are vφ1 · v xy Dxy α1 ≡ arccos ; ðA37Þ inverted). The factor H is defined as jvφ1xyjjvDxyj H z1; φ1; ψ 1;z 1; φ2; ψ 2 ð Naþ Þ vφ2 · vDxy xy ≡ H Δ − z1 − z0 H Δ − z 1 − z0 α2 ≡ arccos − ; ðA38Þ ð z j jÞ ð z j Naþ jÞ jvφ jjv j 2xy Dxy ð1Þ ð2Þ × HðΔψ − jψ 1 − ψ 0 jÞHðΔψ − jψ 2 − ψ 0 jÞ α α ð1Þ ð2Þ we define 1 and 2 as × HðΔφ − jφ1 − φ0 jÞHðΔφ − jφ2 − φ0 jÞ; ðA45Þ α ≥ 0 1 if z · ðvDxy × vφ1xyÞ where HðxÞ is the Heaviside step function, z0 is the α1 ¼ ðA39Þ 1 1 ð Þ ð Þ 2π − α1 otherwise; minimum-Uz height of the NC center of mass, φ0 , ψ 0 are the minimum-Uφ, minimum-Uψ values closest to the α ≥ 0 current orientation φ1; ψ 1 of the first NC [see Eq. (A30)], and 2 if z · ðvDxy × vφ2xyÞ α2 ¼ ðA40Þ ð2Þ ð2Þ analogously φ0 , ψ 0 for the second NC. In our simulations, 2π − α2 otherwise: we set Δz ¼ 1.5 nm, Δφ ¼ Δψ ¼ 0.05π. These values are The total energy U of the two NCs, each inducing a estimations to take into account that a NC, when slightly c 111 hexapolar capillary deformation in the interface, is theo- deviating from the orientation with a f g facet parallel to 0 retically expected to be of the form [108] z ¼ , still induces a hexapolar capillary deformation in the interface height profile. Capillary deformations that may arise for NCs in other orientations, i.e., far from the uc U ðD; α1; α2Þ¼−H cosð3ΔαÞ; ðA41Þ orientation with a f111g facet parallel to z ¼ 0, are neglected c D6 since these orientations are nonstable for the experimental systems of interest, as shown in Sec. III. where The capillary forces induced by the pair potential U α α qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cðD; 1; 2Þ [Eq. (A41)] on the two NCs have a trans- ≡ − 2 − 2 lational component acting on the distance between the two D ðx1 xN þ1Þ þðy1 yN þ1Þ ðA42Þ a a NCs and a rotational component acting on the angles α1 and α2 of the two NCs. The translational component of the force 0 is the distance between the NC centers of mass in the z ¼ is obtained by applying to both the i ¼ 1 and i ¼ Na þ 1 plane, beads, i.e., the central core beads of the two NCs, the potential UcðDÞ [Eq. (A41)]. The force f ¼ðfx;fy;fzÞ 1 U Δα ≡ α − α α applied to the i ¼ bead and due to cðDÞ is 1 2 þ φ1;φ2 ; ðA43Þ

∂U x1 − x 1 − c −6H 3Δα Naþ where the factor cosð3ΔαÞ takes into account the alignment fx ¼ ¼ uc cosð Þ 8 ; ∂x1 D of the hexapoles (such that the NC-NC force due to Uc is ∂U y1 − y 1 repulsive when a capillary rise of the hexapole induced by a − c −6H 3Δα Naþ fy ¼ ∂ ¼ uc cosð Þ 8 ; NC overlaps with a capillary depression of the hexapole y1 D induced by the other NC and attractive when two capillary ∂Uc fz ¼ − ¼ 0: ðA46Þ rises or two capillary depressions overlap), uc is a param- ∂z1 eter setting the interaction strength, and the factor H is used to switch on and off, during the simulation, the pair Then, the force applied to the i ¼ Na þ 1 bead and due to Uc capillary interaction, depending on whether both NCs are in is −f. To implement the rotational component of the force the orientation inducing a hexapolar capillary deformation due to Uc, we select two shell beads, for each NC, aligned α or not. The phase φ1;φ2 is with the NC center of mass (so that we can apply a force

021015-19 GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019)

≡ ≡ 14 4 couple to the NC) and in the same x, y plane, so that the D rUc . nm between the centers of mass of the two induced rotation occurs in the rotational planes of the angles interacting NCs. Since this value is definitely negligible α1 and α2, i.e., in a plane parallel to z ¼ 0. Given our compared to the energy scales involved in our MD model, definition of the shell bead positions for a PbSe NC, see we use rUc as a cutoff radius for the NC-NC pair interaction Fig. 7, we can always find two shell beads that fulfill these potential Uc [Eq. (A41)]; i.e., we neglect the pair capillary properties (within 0.02 nm of precision) when the NC has a interactions of NC pairs at a distance greater than rUc . f111g facet exactly parallel to the interface. When a NC is When periodic boundary conditions are applied, if rUc is slightly tilted from this orientation, we select the shell bead less than half of the periodic box side, then, for each NC, pair that fulfills these properties with the best approximation. only the NC-NC pairs with the closest copy of every other We now assume the two selected shell beads with such NC need to be considered. properties are the i ¼ 8 and i ¼ 30 beads for the first NC, and In all the simulations with our coarse-grained MD model the i N 8 and i N 30 beads for the second NC. ¼ a þ ¼ a þ presented in this paper, the pair potential Uc is used, setting Their position is, respectively, ðx8;y8;z1Þ, ðx30;y30;z1Þ, −16 6 uc ¼ 4 × 10 Jnm . Its effect on the NCs’ dynamics is, ðxNaþ8;yNaþ8;zNaþ1Þ, ðxNaþ30;yNaþ30;zNaþ1Þ.Tothe however, negligible. During each simulation, the total 8 0 i ¼ bead we apply a force f ¼ðfx;fy; Þ orthogonal to potential U (i.e., summed over all NC-NC pairs) is ≡ − − 0 ≡ 0 30 c a ðx8 x1;y8 y1; Þ ðax;ay; Þ, and to the i ¼ essentially always constant, compared to the other poten- − U U U U bead we apply a force f, such that the total modulus of tials z, φ, ψ (summed over all NCs) and LJ (summed τ the torque α1 induced on the first NC by this force couple is over all bead-bead pairs). Using values of uc a few times greater, and removing short-range attractive forces, we find ∂U 3Hu U ’ τ c c 3Δα instead that c affects the NCs dynamics, driving the NCs α ¼ ¼ 6 sinð Þ : ðA47Þ ∂α1 D toward the hexagonal-lattice assemblies predicted in Refs. [99,102]. However, for conciseness, we leave this This implies that the force f is given by out of this work. τ a τ ∓ α y α ax APPENDIX B: NC INTERFACE-ADSORPTION fx ¼ 2 ;fy ¼ 2 : ðA48Þ 2 jaj 2 jaj ENERGY CALCULATIONS

Note that, since the i ¼ 8 and i ¼ 30 beads are aligned with Here we report more details on the calculations presented the center of mass of the NC in the same x, y plane, the in Sec. III for the adsorption of PbSe NCs at a toluene-air direction of the torque induced by f and −f on the NC is interface and show additional results. orthogonal to the plane z ¼ 0. So, the NC rotation occurs, as 1. Sharp-interface method desired, in the rotational plane of α1. The signs for fy and ∓ τ for fx in Eq. (A48), determining the sign of the torque α1 , First, we report additional details on the numerical – depend on the sign of ∂Uc=∂α1: is þ (and ∓ is −)if method introduced by Soligno et al. [99 101] and used −∂Uc=∂α1 is positive, and is − (and ∓ is þ) otherwise. for the interface-adsorption calculations presented in Analogously, for the second NC, we apply to the i ¼ Na þ 8 Sec. III and in Appendixes B2 and B3. 0 ≡ − We consider a macroscopic model where the interface bead a force f ¼ðfx;fy; Þ orthogonal to b ðxNaþ8 − 0 ≡ 0 between two homogeneous fluids, here assumed toluene xNaþ1;yNaþ8 yNaþ1; Þ ðbx;by; Þ, and to the i ¼ Na þ 30 bead we apply a force −f, such that the total modulus of and air, is treated as a 2D possibly curved surface, which we the torque τα induced on the second NC by this force couple represent by a 2D triangular grid of points. A PbSe NC 2 adsorbed at this interface is modeled as a homogeneous is also τα [Eq. (A47)]. Hence, the force f is given by solid with rhombicuboctahedron shape (with distance 6 nm between two opposite f100g facets), and its surface is τα by τα b f ¼ ∓ ;f¼ x : ðA49Þ represented by a 2D closed triangular grid of points. We x 2 b 2 y 2 b 2 j j j j numerically calculate the energy E [Eq. (10)] of the NC- toluene-air system with respect to the NC position and The signs for f and ∓ for f in Eq. (A49), determining y x orientation at the interface. First, the equilibrium shape of the sign of the torque τα , depend on the sign of ∂U =∂α2: 2 c the toluene-air interface for the given NC position and − ∓ −∂U ∂α is (and is þ)if c= 2 is positive, and is þ orientation is computed, then E is obtained. As a first-order ∓ − ∂U α (and is ) otherwise. Note that, since c=d 2 ¼ approximation, one could assume the interface height −∂U ∂α τ τ τ c= 1, it follows that α1 ¼ α2 ; hence if α1 acts to profile always flat and unperturbed by the presence of a α τ α increase 1, then α2 acts to decrease 2, and vice versa. particle at the interface, that is the approach first introduced −16 6 Using the typical value uc ¼ 5 × 10 Jnm (see by Pieranski [109]. For example, numerical methods ap- Appendix B3), the maximum value for jUcj [Eq. (A41)] plying this approximation are used in Refs. [26,110–116]. −23 with respect to α1, α2 is ≃−5 × 10 J at a distance However, neglecting capillary deformations can lead to

021015-20 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019) erroneous predictions even for an isolated adsorbed par- index of the facet where the three-phase contact line is ticle; see Ref. [102]. considered). The position of the three-phase contact line, For convenience, the interface-adsorption energy E for a given NC configuration φ, ψ, zc, is automatically [Eq. (10)] is shifted by a constant and rewritten in the found by minimizing E with respect to the positions of the equivalent form air-toluene interface grid points; i.e., it is not imposed a priori. X26 The initial shape of the toluene-air interface grid, i.e., γ − ðaÞ θ E ¼ S A þ Wk cos k ; ðB1Þ before starting a simulated annealing simulation to mini- k¼1 mize E, is the plane z ¼ 0. To simulate a toluene-air interface which is flat far away from the NC, the toluene- where S and A are the surface areas of the toluene-air air-NC system is enclosed by a wall orthogonal to the plane interface with and without NC, respectively (so A is a z 0 and with Young’s contact angle π=2, placed far constant), the index k goes over the 26 facets of the NC, and ¼ ðaÞ enough from the NC to avoid finite-size effects. During the Wk is the surface area of the portion of the NC kth facet in simulated annealing, we can either allow volume exchange contact with air. To obtain Eq. (B1) from Eq. (10), we used between air and toluene or not. So, the equilibrium shape of ’ θ that Young s contact angle k for the kth facet is the toluene-air interface is found, respectively, for the θ γðaÞ − γðtÞ γ γ cos k ¼ð k k Þ= , with the toluene-air surface minimum-E volume of toluene and air (such that their γðaÞ γðtÞ sum is constant) or with the constraint of fixed toluene tension and k , k the surface tension of the kth facet with air and toluene, respectively, and that the total area and air volumes (defined by the initial interface shape). In Fig. 2(b), where the energy Eðz Þ is shown for some WðaÞ WðtÞ k WðtÞ c k þ k of the th facet is a constant (with k the fixed orientations φ, ψ of the NC, the interface shape surface area of the portion of the NC kth facet in contact minimizing E is found imposing fixed volumes for toluene with toluene). In the definition of E in Eq. (B1), the level E φ E ψ 0 and air. In Figs. 2(c) and 2(d), where the energy ð Þ, ð Þ E ¼ corresponds to the NC desorbed from the toluene-air is shown for some fixed values of ψ; φ, respectively, the interface and immersed in toluene, since in this case S ¼ A toluene-air interface shape minimizing E is computed ðaÞ 0 1 … 26 and Wk ¼ for k ¼ ; ; . allowing volume exchange between toluene and air; hence, First, the minimum-E shape of the toluene-air interface the obtained energy Eðφ; ψÞ is minimized over zc, i.e., the grid is computed, given as input a fixed position of the NC NC height at the interface plane. surface, defined, see Fig. 2(a), by the position and ori- Note that gravity is not taken into account in E entation zc, φ, ψ of the NC with respect to the toluene-air [Eqs. (8)–(10), (B1)], since it is negligible for nanoparticles interface plane (corresponding to z ¼ 0, where a Cartesian at fluid-fluid interfaces, corresponding to assuming a coordinate system x, y, z is introduced). For this calcu- capillary length l → ∞. Other applications and more lation, we use a simulated annealing algorithm (i.e., a details on this numerical method, here illustrated, can be Monte Carlo approach) to calculate the position with found in Refs. [99–102,117–120]. minimum E [Eq. (B1)] of the points of the toluene-air interface grid. As shown in Ref. [100], the obtained 2. Interface-adsorption energy of PbSe NCs interface shape is the solution of both the Young- Laplace equation and Young’slaw[101], that is, the In Sec. III, we presented results for the interface- equilibrium shape. After the interface minimum-E shape adsorption energy E of an isolated PbSe NC at a tol- uene-air interface, with respect to the NC position and is obtained, S and WðaÞ (for k ¼ 1; …; 26) are computed k orientation zc, φ, ψ [see Fig. 2(a)] at the interface plane from the position of the interface grid, and, finally, E is z ¼ 0. Ligand molecules (typically, oleic acid) were obtained from their values; see Eq. (B1). The toluene-air assumed chemisorbed at the NC f111g and f110g facets, interface area S is obtained by summing the areas of all the while the NC f100g facets were assumed ligands-free. To triangles forming the air-toluene grid. The area of the represent this in the macroscopic model used for calculat- ðaÞ ’ θ portion of the NC kth facet in contact with air, i.e., Wk ,is ing E, we assigned a Young s contact angle given by obtained by summing the areas of all the triangles of the NC cos θ ¼ 0.30 on the NC f100g facets and by cos θ ¼ 0.64 surface grid belonging to the NC kth facet and above the on the NC f110g, f111g facets; see Sec. III. air-toluene grid (with the triangles in between the toluene- In this section, we present analogous results, but for a air interface split, and only the surface area of their portion PbSe NC fully covered by ligands; that is, we set cos θ ¼ above the air-toluene grid included). By repeating this 0.64 on all the NC facets. In Fig. 9, we show, for such a NC, procedure for different values of φ, ψ, zc, the energy the energy Eðφ; ψÞ [Eq. (B1)], minimized on zc (i.e., landscape Eðφ; ψ;zcÞ is obtained. allowing volume exchange between toluene and air when The computed minimum-E air-toluene interface shape computing the minimum-E shape of the toluene-air inter- forms an angle with the NC surface along the three-phase face). In contrast with the energy Eðφ; ψÞ of the NC with contact line that matches the input parameter θk (with k f100g facets ligands-free, which presented a deep well

021015-21 GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019)

FIG. 9. Interface-adsorption energy E [Eq. (B1)] of a PbSe NC fully covered by ligands, that is with Young’s contact angle θ given by cos θ ¼ 0.64 on all the NC facets, with respect to the NC orientation φ, ψ, see Fig. 2 (and minimized on the NC height zc), at the air- toluene interface plane z ¼ 0, computed using the numerical approach in Refs. [99–101]. The level E ¼ 0 corresponds to the NC desorbed from the interface and immersed in toluene. For the rotational symmetry of the NC shape in φ and ψ, E is shown only for φ ∈ ½0; π=2, ψ ∈ ½0; π=4, since outside these intervals is periodically repeated. The insets on the right-hand side show a 3D view, close to the NC, of the toluene-air interface minimum-E shape (blue grid, with the toluene below), computed by the numerical approach of Soligno et al. [99–101], for the various metastable-E orientations of the NC, that is, (a) φ ¼ 0.14π, ψ ¼ 0.25π, (b) φ ¼ 0.30π, ψ ¼ 0.25π, and (c) φ ¼ 0.40π, ψ ¼ 0.10π.

−20 (energy barriers ≃4 × 10 J) around the minimum-E position and orientation zc, φ, ψ of each NC is expected orientation (i.e., with a f111g facet parallel to z ¼ 0, see to be negligibly affected by multiparticle effects; see Fig. 2), the energy Eðφ; ψÞ of the NC fully covered by Ref. [99]. Hence, the minimum-E values of zc, φ, ψ of ligands is characterized by several minima with smaller each NC, computed for a single-adsorbed PbSe NC, are the energy barriers (≃2 × 10−20 J); see Fig. 9. This suggests same when many PbSe NCs are at the toluene-air interface. that, when fully covered by ligands, PbSe NCs at the However, when many NCs are adsorbed, if they induce toluene-air interface have essentially random orientations, capillary deformations in the equilibrium shape of the fluid- since the NC can easily jump from metastable to metastable fluid interface, the interface-adsorption energy E can vary, orientation by thermal motion. Note, however, that PbSe hence generate interactions, with respect to the reciprocal NCs fully covered by ligands are bonded to the air-toluene distances and reciprocal azimuthal orientations in the interface by an energy ∼10−19 J (see Fig. 9, where the interface plane between the adsorbed NCs. Indeed, the energy level E ¼ 0 corresponds to the NC desorbed from capillary deformations induced by the adsorbed NCs in the interface and immersed in toluene); hence, once the interface shape correspond to a larger surface area adsorbed at the interface, they are expected to remain at of the interface, i.e., the term S in Eqs. (10) and (B1). the interface, even if they can change orientation. For PbSe Therefore, adsorbed NCs inducing capillary deformations NCs with f100g facets free from ligands, the energy can arrange themselves to overlap capillary deformations bonding to the air-toluene interface is even stronger with the same sign (i.e., either rises or depressions in the −19 (≃2 × 10 J, see Fig. 2), so, in the experiments, when interface height profile), decreasing in this way S, and so E. ligands detach from the f100g facets of the PbSe NCs, the These interactions between adsorbed NCs are called NCs remain strongly bonded to the toluene-air interface. capillary interactions [102]. References [99,102] show that cubes and cantellated 3. Capillary interactions of PbSe NCs rhombicuboctahedrons can adsorb at a fluid-fluid interface We study in this section the capillary interactions with a f111g facet parallel to the interface plane, and in this between PbSe NCs adsorbed at a toluene-air interface, orientation they induce a hexapolar capillary deformation with the ultimate goal of showing that these interactions are in the interface height profile. From the capillary inter- negligible compared to the other forces involved in the NC actions induced by such capillary deformations, and taking self-assembly, for the experimental conditions of interest. into account also the entropy of the particles (using In Sec. III and Appendix B2, we studied the interface- approximated analytic expressions), the self-assembly into adsorption energy E [Eqs. (8)–(10), (B1)] for a single- periodic lattices with honeycomb, hexagonal, and square adsorbed PbSe NC at the toluene-air interface, with respect geometry is predicted. These lattices are different from the ’ to the NC position and orientation zc, φ, ψ [see Fig. 2(a)]at NCs superstructures observed experimentally and pre- the interface plane z ¼ 0. If many PbSe NCs are adsorbed dicted in this work with our coarse-grained MD model; at the interface, the energy E with respect to only the for example, the honeycomb lattice is not buckled, the

021015-22 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019) particles in the square lattice are oriented with a f111g, rather than f100g, facet parallel to the interface plane, and in general the particles in these lattices are not attached by f100g facets. As a matter of fact, in Refs. [99,102] only capillary and hard interactions between adsorbed particles at fluid-fluid interfaces are considered, i.e., in the limit of no short-range forces. Short-range facet-specific attractions between NCs, however, are crucial for the self-assembly of PbSe NCs, as shown by the results with our coarse-grained FIG. 10. Contour plot (left) of the toluene-air interface minimum- E height profile h, as computed by the numerical method of Soligno MD model. Given, however, the fair similarity of the et al. [99–101], close to a PbSe rhombicuboctahedron NC (with honeycomb lattice predicted in Refs. [99,102] with the Young’s contact angle θ given by cos θ ¼ 0.64 on the NC f111g, PbSe NCs silicene-honeycomb superstructure experimen- f110g facets and cos θ ¼ 0.3 on the NC f100g facets) in its tally observed, and that the narrow range of parameters minimum-E orientation, i.e., with a f111g facet parallel to the predicted for obtaining the honeycomb lattice was com- interface plane (see Sec. III). In this orientation the NC induces a patible with the PbSe NCs experiments, in Refs. [99,102] hexapolar capillary deformation in the interface. On the right, a 3D it was speculated that NC-NC capillary interactions profile view is shown, with the blue grid being the toluene-air could play a major role in the self-assembly of NCs at fluid- interface (and the toluene below). fluid interfaces (although, as stated by Soligno et al., short-range forces needed to be included to verify this NC, i.e., with a f111g facet parallel to the interface plane. hypothesis). We show in this section that NC-NC capillary The center-of-mass height zc of each NC is aligned on the interactions are actually negligible in the NC self-assembly, same plane (parallel to the z ¼ 0 plane), and the minimum- – at least for the experiments considered here [73,85 88]. E shape of the toluene-air interface is computed allowing Note that, in Refs. [99,102], an estimated comparison volume exchange between air and toluene. The hexagonal- between the predicted NC-NC capillary interactions and lattice unit cell is a hexagon with each side having the same NC-NC van der Waals attractions was provided, showing interface height profile of its opposite side. A single NC is that, for typical NC sizes, van der Waals attractions are in the center of the hexagonal-lattice unit cell. The NC negligible compared to capillary interactions, except at azimuthal orientation in the lattice is shown in the contour almost-contact NC-NC distance where these two forces are plots of Fig. 11. The honeycomb-lattice unit cell is a of the same order. However, in the experiments considered rectangle with short sides having the same interface height here, the f100g facets of PbSe NCs are not covered by profile, and long sides having the same interface height ligands; hence, at short range, electrostatic-chemical inter- profiles but shifted of half-side. Each cell of the honeycomb actions between surface atoms of opposite f100g facets lattice has two NCs, positioned and azimuthally oriented in (∼10 times stronger than van der Waals forces) are also the lattice as shown in the contour plots of Fig. 11. An exact involved. Note also that in Refs. [99,102] it was assigned a definition of these two unit cells is reported in Ref. [99]. Young’s contact angle θ given by cos θ ¼ 0 on the whole In Figs. 11(a) and 11(b), we show the pair capillary- particle surface, while here we use more realistic surface interaction energy E for the honeycomb- and hexagonal- tensions for the experiments of interest. lattice unit cell, respectively, as computed with our We consider in this section PbSe NCs at a toluene-air numerical method and with respect to the center-of- interface, assigning cos θ ¼ 0.30 on their f100g facets and mass distance D between two closest-neighbor NCs in cos θ ¼ 0.64 on their f110g, f111g facets. As shown in the lattice, where E is the interface-adsorption energy E Sec, III, the minimum-E orientation of these NCs is with a [Eq. (B1)] divided by the number of capillary bonds f111g facet parallel to the interface plane. When adsorbed (between closest neighbors) in the lattice, and shifted to in this orientation, each NC induces in the toluene-air be zero for an infinite lattice spacing. That is, interface height profile a hexapolar capillary deformation; see Fig. 10. To estimate the capillary interactions induced E − NE1 E ≡ ; ðB2Þ by these hexapolar capillary deformations, we consider the n hexagonal and honeycomb NCs’ periodic lattices predicted in Refs. [99,102], i.e., which minimize E by capillary where n is the number of pairs between closest-neighbor interactions. To consider a periodic lattice of NCs at the NCs in the lattice, N is the number of NCs in the lattice, and interface, we apply our numerical method for computing E1 is the energy E [Eq. (B1)] of a single-adsorbed NC in the minimum-E interface shape (see Appendix B1)toa its minimum-E position and orientation at the interface. unit cell of the lattice, with periodic boundary conditions For the unit cells considered here, n ¼ 3, N ¼ 2 for the applied at the borders of the unit cell to the interface height honeycomb-lattice unit cell, and n ¼ 3, N ¼ 1 for the profile. Each NC in the lattices is oriented with the φ; ψ hexagonal-lattice unit cell. In Figs. 11(a) and 11(b), we also orientation of minimum E computed for a single-adsorbed show a fit of the obtained energy EðDÞ using

021015-23 GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019)

FIG. 11. (a),(b) Pair capillary interaction energy E [Eq. (B2)], with respect to the center-of-mass distance D between two closest- neighbor NCs in a honeycomb- (a) and hexagonal-lattice (b) unit cell (see text), as computed through the numerical approach illustrated in Appendix B, for PbSe NCs of size 6 nm at a toluene-air interface, with Young’s contact angle θ given by cos θ ¼ 0.30 on the NC f100g facets, cos θ ¼ 0.64 on the f110g, f111g facets. Each NC is oriented at the interface with a f111g facet parallel to the interface plane, hence generating a hexapolar capillary deformation; see Fig. 10. The black curves show a fit using fðDÞ [Eq. (B3)], which gives uc ¼ −16 6 −16 6 4.5 × 10 Jnm for the honeycomb lattice and uc ¼ 5.7 × 10 Jnm for the hexagonal lattice. The vertical dotted lines indicate the NC-NC contact distance. (c) Analogous calculation to (b), i.e., for a honeycomb-lattice unit cell, but for a fixed lattice spacing (D ¼ 7.8 nm) and rotating the azimuthal angle α in the lattice of one of the two NCs. The black curve shows a fit using gðαÞ [Eq. (B4)], −16 6 which gives uc ¼ 4.6 × 10 Jnm . The insets show contour plots of the equilibrium interface height profile hðx; yÞ, as computed by the numerical method of Soligno et al. [99–101], for some of the unit cells considered.

≡ − uc to play a role in the self-assembly of PbSe NCs at the fðDÞ 6 ; ðB3Þ D toluene-air interface. Instead, as shown by our coarse- grained MD model, the NC self-assembly is regulated by which is the expected potential Uc between two interacting hexapolar capillary deformations, see Eq. (A41) in the interplay between interface-adsorption forces binding 3 α − α α 1 and orienting the NCs at the interface and the attractive Appendix A4, for cos½ ð 1 2 þ φ1;φ2 Þ ¼ . Finally, electrostatic-chemical forces between specific facets of NCs in Fig. 11(c), we show the energy E [Eq. (B2)]ofa U honeycomb-lattice unit cell, for a fixed lattice spacing at close distance. The pair capillary potential c is imple- (D ¼ 7.8 nm) and with respect to the azimuthal orientation mented in our coarse-grained MD model, see Appendix A4, α in the lattice unit cell of one of the two NCs (keeping and used in all the simulations presented in this work setting 4 10−16 6 U fixed the azimuthal orientation in the lattice of the other uc ¼ × Jnm . However, c does not play a role in ’ U NC). In Fig. 11(c), we also show a fit of the obtained energy the NCs dynamics. Indeed, the total potential c, i.e., EðαÞ using summed over all NC-NC pairs, in all the simulations is essentially always constant with respect to the potentials Uz, u α ≡ − c 3 α π 2 Uφ, Uψ (summed over all NCs) and U (summed over all gð Þ 6 cos½ ð þ = Þ; ðB4Þ LJ ð7.8 nmÞ bead-bead pairs). Even in the simulations where no NC attachment by 100 facets is achieved, see results in U f g which is the potential c [Eq. (A41)] at the NC-NC Appendix C, the NCs’ dynamics are essentially unaffected distance 7.8 nm. by Uc. The parameter uc sets the strength of the forces due to Uc. The estimated values of uc in the three cases shown in −16 6 APPENDIX C: ADDITIONAL SIMULATION Figs. 11(c)—that is, respectively, uc ¼5.7×10 Jnm , −16 6 −16 6 DETAILS uc ¼ 4.5 × 10 Jnm , and uc ¼ 4.6 × 10 Jnm — are, as expected, similar. Note that, in this approach to Here, we report additional simulation details for the estimate the pair capillary interactions between PbSe NCs at results shown in Figs. 3–5 of Sec. IV. the toluene-air interface, we are neglecting multiparticle For the interface-adsorption potential Uz [Eq. (3)], we set effects. That is, the estimated pair capillary interaction −19 −18 z0 ¼ −1 nm, zd ¼ 4 nm, E1 ¼−2 ×10 J, Ea ¼ 10 J, potential U is actually stronger than in the case of an c Es ¼ 0 (the latest three parameters do not matter for the isolated pair of NCs. Despite this overestimation, the simulation, they just set the offsets of Uz for the energy plots). maximum bonding energy between two capillary-interact- For the NC-NC pair capillary potential Uc introduced in ing NCs, that is, fðDÞ [Eq. (B3)] in the hexagonal lattice at Appendix A4, see Eq. (A41),wesetu ¼ 4 × 10−16 Jnm6, −20 c the contact distance D ≃ 6 nm, is ≃1.3 × 10 J, i.e., which is close to the value we estimated for PbSe NCs negligible compared to the total bonding energy E ∼ at a toluene-air interface; see Appendix B3. As extensively −19 10 J expected from the electrostatic-chemical atomic discussed in Appendix B3, for such a value of uc, the NC-NC interactions between NCs attached by opposite f100g pair potential Uc does not play a role in the NC self-assembly, facets. Hence, capillary interactions are not expected since the capillary interactions between NCs oriented with

021015-24 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019)

FIG. 12. For the assembled-phase diagram of Fig. 5, in the case of total bonding energy between two NCs attached by f100g facets given by E ¼ −0.7 × 10−19 J, we show here a top view of the interface at the end of each simulation (t ¼ 2 μs) for the various values considered of up, uz (i.e., the parameters setting the strength of the interface-adsorption forces orienting and keeping the NCs at the interface plane, respectively). The black line borders indicate the periodic boundary conditions. The graphs above each snapshot show, μ U 10−17 for each simulation and with respect to t= s, the Lennard-Jones potential (light blue line) LJ= J summed over all the bead-bead −17 −17 pairs, the interface-adsorption potentials (green line) Uz=10 J and (fuchsia line) ðUφ þ Uψ Þ=10 J, summed over all NCs, and the −17 total potential internal energy (black line) Upot=10 J; see Eq. (7).

021015-25 GIUSEPPE SOLIGNO and DANIEL VANMAEKELBERGH PHYS. REV. X 9, 021015 (2019)

FIG. 13. For the assembled-phase diagram of Fig. 5, in the case of total bonding energy between two NCs attached by f100g facets given by E ¼ −1.0 × 10−19 J, we show here a top view of the interface at the end of each simulation (t ¼ 2 μs) for the various values considered of up, uz (i.e., the parameters setting the strength of the interface-adsorption forces orienting and keeping the NCs at the interface plane, respectively). The black line borders indicate the periodic boundary conditions. The graphs above each snapshot show, μ U 10−17 for each simulation and with respect to t= s, the Lennard-Jones potential (light blue line) LJ= J summed over all the bead-bead −17 −17 pairs, the interface-adsorption potentials (green line) Uz=10 J and (fuchsia line) ðUφ þ Uψ Þ=10 J, summed over all NCs, and the U 10−17 total internal potential energy (black line) pot= J; see Eq. (7).

021015-26 UNDERSTANDING THE FORMATION OF PbSe … Phys. Rev. X 9, 021015 (2019)

FIG. 14. For the assembled-phase diagram of Fig. 5, in the case of total bonding energy between two NCs attached by f100g facets given by E ¼ −1.3 × 10−19 J, we show here a top view of the interface at the end of each simulation (t ¼ 2 μs) for the various values considered of up, uz (i.e., the parameters setting the strength of the interface-adsorption forces orienting and keeping the NCs at the interface plane, respectively). The black line borders indicate the periodic boundary conditions. The graphs above each snapshot show, μ U 10−17 for each simulation and with respect to t= s, the Lennard-Jones potential (light blue line) LJ= J summed over all the bead-bead −17 −17 pairs, the interface-adsorption potentials (green line) Uz=10 J and (fuchsia line) ðUφ þ Uψ Þ=10 J, summed over all NCs, and the U 10−17 total internal potential energy (black line) pot= J; see Eq. (7).

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