Quantitative Modeling of Self-Assembly Growth of Luminescent

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C: Physical Processes in Nanomaterials and Nanostructures

Quantitative Modeling of Self-Assembly Growth of Luminescent Colloidal CHNHPbBr Nanocrystals 3

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Weizheng Wang, Yumeng Zhang, Wenhui Wu, Xiaoyu Liu, Xuanxuan Ma, Guixiang Qian, and Jiyang Fan J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b01339 • Publication Date (Web): 28 Apr 2019 Downloaded from http://pubs.acs.org on April 28, 2019

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The Journal of Physical Chemistry

Quantitative Modeling of Self-Assembly Growth of Luminescent Colloidal CH3NH3PbBr3 Nanocrystals Weizheng Wang,† Yumeng Zhang,† Wenhui Wu,† Xiaoyu Liu,† Xuanxuan Ma,† Guixiang Qian,‡ Jiyang Fan*,† †School

of Physics, Southeast University, Nanjing 211189, People’s Republic of

China ‡College

of Biological and Chemical Engineering, Anhui Polytechnic University,

Wuhu 241000, People’s Republic of China

ABSTRACT: The organic‒inorganic hybrid metal halide perovskites with different dimensions and diverse architectures are highly attractive materials for optoelectronic applications. However, people know little about the dynamics of their formation processes. Here, we study both experimentally and theoretically the self-assembly formation dynamics of the luminescent colloidal CH3NH3PbBr3 nanocrystals. We have observed their successive transformations from original spherical quantum dots to periodically stacked nanoplatelets when the primitive colloidal nanocrystals with high concentration were maintained in liquid for prolonged period of time. A theoretical dynamic collision model by taking into account the popular van der Waals force, the polarization force that is unique for the perovskites, and the electrostatic forces between particle surfaces in the presence of the surface ligands is proposed to explain the self-assembly process of the colloidal CH3NH3PbBr3 nanocrystals. The result reveals that the rather easy self-assembly of the organic‒inorganic hybrid perovskites with different morphologies in the absence of enough surface ligands is closely related to their intrinsic polarization force, whereas the presence of the surface 1

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ligands could hamper the self-assembly process. The proposed theoretical model is general and can be used to analyze the self-assembly dynamics of various types of colloidal nanostructures.

INTRODUCTION The colloidal lead halide perovskite nanocrystals (NCs) have attracted great attention owing to their unique optical versatility, high luminescence quantum yield, and facile synthesis.1–5 Spectroscopic studies and application explorations of the lead halide perovskite NCs strongly rely on high-quality NCs, and great efforts have been made to refine synthesis of such materials.6–10 In addition, controlling the sizes and shapes of the perovskite NCs is greatly desired in order to understand the fundamental size-dependent optical properties and optimize their device performances. The organic‒inorganic hybrid perovskites with cubic crystal structures can be described by the formula ABX3, where A is an organic cation (such as Cs+, formamidinium (FA+), and methylammonium (MA+)), B is a metal cation (such as Pb2+ and Sn2+), and X is a halogen ion (such as Cl–, Br–, and I–). Br– is chemically more stable than I– in the ambient conditions, and the colloidal CH3NH3PbBr3 (MAPbBr3) NCs can be readily synthesized. The organic‒inorganic halide perovskite NCs can be synthesized at room temperature because they have small formation energies.11–12 In addition, the cost of MAPbBr3 is lower than that of FAPbBr3 and CsPbBr3. Therefore, MAPbBr3 has been widely used in photovoltaic devices,13-16 photodetectors,17-19 and light-emitting devices.20-22 Recently, there has been great interest in studies of the colloidal 2


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perovskite nanoplatelets (NPLs), these quasi two-dimensional layered nanomaterials exhibit bright luminescence, tunable and narrow absorption and emission spectra, strongly confined exciton states, and facile colloidal synthesis.6,23–30 The perovskite NPLs are described by the formula L2An–1BnX3n+1, where L represents a long-chain cation (e.g., octylamine cation) which ensures structural stability of the NPLs.31 The subscript n represents the number of the interconnected metal halide octahedron layers. Many studies have pointed out that there exists self-organization process during crystal growth of the perovskite nanoplatelets in liquid medium.32–34 But little has been known about the dynamics of such self-assembly process. On the other hand, although the halide perovskite NCs with controllable dimensions and shapes can be synthesized through alteration of ligands and solvents,25, 35,36 however, the products are usually mixtures of various NCs and their sizes are on the micrometer scale, so their luminescence properties are very similar to that of bulk perovskites. There have been rare reports on short-wavelength (especially blue) luminescence of colloidal MAPbBr3 NCs with uniform sizes. Herein, we study the self-assembly dynamics of the different morphologies of MAPbBr3 NCs. The as-synthesized colloidal MAPbBr3 quantum dots (QDs) are transformed into nanowires (NWs) after long time of self-assembly in the liquid medium. Given more time, the NWs further evolve into layered perovskite NPLs and superlattices. In fact, other than the traditional ion-by-ion crystal growth mechanism as described in the standard textbook, in recent years, the researchers have found that there exists another very important crystal growth mechanism on the nanoscale: 3



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oriented attachment.37,38 It refers to self-assembly and fusion of a couple of smaller nanoparticles to produce a larger nanostructure. Taking into account this novel crystal growth mechanism, we establish a theoretical model and analyze the self-assembly dynamics of the MAPbBr3 NCs: from original QDs to the ultimate superstructures. The roles of the general van der Waals force and the unique spontaneous polarization force in the perovskites in driving their self-assembly processes are investigated. The repulsion force in the presence of the surface ligands is also taken into account. To the best of our knowledge, this is the first detailed theoretical study of the step-by-step self-assembly crystal growth of the MAPbBr3 superlattices.

EXPERIMENTAL METHODS Materials. PbBr2 (Lead bromide, 99%), n-octylamine (≥ 99%), and oleic acid (≥ 90%) were purchased from Adamas-beta. Methylamine (CH3NH2, 33 wt% in water), hydrobromic acid (HBr, 48 wt% in water), octylamine (99%), toluene (99%), and N-dimethylformamide (99.5%) were purchased from Aladdin. Ethyl acetate was purchased from Macklin. Synthesis of MABr. MABr was synthesized via reaction of equivalent molar ratios of methylamine and hydrobromic acid.39 Firstly, 10 mL methylamine solution was cooled to 0 C, then 15 mL hydrobromic acid was added dropwise under vigorous stirring. The reaction was carried out in an ice bath at 0 C for 2 h until a clear solution formed. Then the solution experienced evaporation at 45 °C and under 0.1 MPa for 30 min in a rotary evaporator. The precipitate was washed with pure ethanol 4


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three times and recrystallized in diethyl ether twice, yielding white crystals. It was dried overnight in a vacuum oven before the usage. Synthesis and purification of MAPbBr3 QDs. The initial colloidal MAPbBr3 QDs were synthesized by using the modified ligand-assisted reprecipitation method.40 First, 12.5 mL toluene containing 1.875 mL oleic acid and 0.250 mL octylamine was prepared. 375 μL MABr (0.533 M) and 625 μL PbBr2 (0.4 M) in DMF were then introduced, resulting in formation of a light blue transparent colloidal solution. 5 mL resultant solution was stored in the dry environment waiting for the optical characterizations. The QD sample was washed with ethyl acetate to remove excessive long-chain surface ligands. Synthesis and purification of MAPbBr3 NWs. The MAPbBr3 QD solution was mixed with ethyl acetate at a volume ratio of 1:10 or 1:20 to induce the self-assembly process. There are two methods to synthesize the MAPbBr3 NWs. (i) After addition of ethyl acetate, the clear solution became cloudy immediately. Then the mixed solution was centrifuged at 10000 rpm for 10 min. After that, the precipitate was dried in vacuum at room temperature, and 20 mg resultant QD precipitate was redispersed in 5 mL toluene to form a high concentration (4.0 mg mL–1, obtained through measurement of the mass of the dry QD powder) solution. Then the solution was maintained still in vacuum and dark environment for 1 day. (ii) MAPbBr3 NWs can also be obtained through keeping the mixed solution containing the QDs and ethyl acetate (1:20) still for 2 h. To confirm that the above mentioned mass concentration of the MAPbBr3 QD 5



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solution (4.0 mg mL–1) is reasonable and that it does not contain the contribution of the residual ligands or impurities, we estimate the mass concentration of the MAPbBr3 QDs in toluene on the basis of the Beer–Lambert law. Firstly, we calculate the molar attenuation coefficient ε by measuring the UV–Vis absorption spectra of several successively diluted MAPbBr3 QD solutions (as reference samples). According to the Beer–Lambert law, 𝐴·ln10 = 𝜀𝑐𝐿, where A is the absorbance at the first exciton absorption peak (𝜆 = 448 nm), ε is the molar extinction coefficient in units of M ―1 cm ―1 (mol -1 L cm ―1), 𝑐 is the molar concentration of MAPbBr3 QDs in toluene (mol L–1), and 𝐿 = 1 cm is the optical path length in the QD solution (cross-section size of the cuvette). Figure S1 shows the plot of the measured 𝐴 versus 𝑐 as obtained from the UV–Vis absorption spectra of the dilute QD solutions. The slope of the linear fit of the curve gives 𝜀 = 1765 × ln10 M ―1 cm ―1. To make the result more accurate, we have prepared five similar MAPbBr3 QD solutions (4.0 mg mL–1) and obtained their absorbance values at 𝜆 = 448 nm from the UV–Vis absorption spectra (Figure S2). The average absorbance 𝐴 = 0.913 ± 0.002, so the molar concentration 𝑐 =

𝐴 ·ln10 𝜀𝐿

= (5.17 ± 0.01) × 10 -7 mol mL ―1. The number of

QDs in the sample equals the mole number multiplied by the Avogadro constant. The mass of a single MAPbBr3 QD 𝑚 = 𝜌𝑉, where 𝜌 is the density of bulk MAPbBr3 4

(4.92 g cm−3) , and the volume of a QD 𝑉 = 3π𝑅3, where the average radius of the QDs 𝑅 was obtained from the TEM observation. Then we derive that in 1 mL toluene the total mass of the MAPbBr3 QDs is 4.01 ± 0.01 mg, being in good agreement with the value obtained from direct measurement of the mass of the QD 6


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powder. This agreement indicates that the ethyl acetate behaves well in purifying the MAPbBr3 NCs and that the precipitate added to toluene was indeed composed of nearly pure MAPbBr3 QDs. Characterizations. Photoluminescence (PL) spectra were measured by using a Fluorolog3-TCSPC spectrophotometer (HORIBA JOBIN YVON) with a Xe lamp as the light source. Transmission electron microscopy (TEM) images and selected-area electron diffraction (SAED) patterns were obtained by using a Tecnai G2 T20 transmission electron microscope (FEI Company) operating at 200 kV. X-ray diffraction (XRD) patterns were measured by using a Smartlab (3) X-ray diffractometer. UV–Vis absorption spectra were recorded by using a HITACHI U-3900 UV–Vis spectrophotometer. RESULTS AND DISCUSSION Self-assembly of MAPbBr3 QDs into superlattices. Figure 1a shows the photographs of the MAPbBr3 QD solution under illumination of ambient light and 365-nm UV light, respectively. The QD solution is homogeneous and exhibits bright fluorescence under 365-nm UV irradiation. Figure 1b shows the TEM image of the synthesized nanoparticles. They are spherical, and the average size is 1.74 ± 0.34 nm (Figure S3a). The inset of Figure 1b shows the high-resolution TEM image of a typical nanoparticle, and it has good crystallinity. Figure 1c shows the XRD pattern of the QDs. The diffraction peaks roughly agree with the calculated diffraction peaks of the (001), (011), (002), (012), and (022) crystal planes of cubic MAPbBr3. The small difference between measured and calculated values may arise from deviation of the 7



crystal structure of the very small perovskite nanostructures from the standard bulk crystal structure.

(d) 1.0

(c) 1.0

1.2

0.5

15

20

25

30 35 2 (deg)

40

45

0.8

0.6

0.6

0.4

0.4

0.2

022 003

121

012

111

011

-0.5

002

0.0

0.8

50

0.0 405

Absorbance (a.u.)

PL intensity (a.u.)

1.0

001

Intensity (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.2

450

495 540 Wavelength (nm)

585

0.0

Figure 1. (a) Photos of MAPbBr3 QD solution under ambient light (left) and under 365-nm UV illumination (right). (b) TEM image of as-prepared spherical MAPbBr3 QDs. Inset shows the high-resolution TEM image of a typical nanoparticle. (c) XRD pattern of MAPbBr3 QDs. Vertical bars at bottom denote calculated X-ray diffraction peaks of cubic MAPbBr3. (d) UV–Vis absorption spectrum and PL spectrum (excitation: 360 nm) of MAPbBr3 QD solution.

8


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Figure 1d shows the UV‒Vis absorption spectrum of the colloidal MAPbBr3 QDs. It has an obvious exciton absorption peak at 446 nm (2.78 eV). In accordance, the PL spectrum consists of a single sharp band centered at around 460 nm (2.70 eV), and the full width at half maximum (FWHM) is 22 nm, being characteristic for cubic perovskite. The emission maximum shifts to blue by 83 nm compared to bulk MAPbBr3 (Figure S4). This is attributed to the strong quantum confinement effect, considering that the average size of the synthesized QDs is much smaller than the exciton Bohr diameter (4.0 nm) of MAPbBr3 crystal.41 The small Stokes shift (0.08 eV) suggests that the PL of the MAPbBr3 QDs originates from direct recombination of excitons or free charges in the shallow traps, not from defects or impurities acting as deep traps.1,42 The morphology evolution of the MAPbBr3 NCs in the solution with time was studied by using the TEM observation (Figure 2). The typical sample was obtained through aggregation of the original colloidal MAPbBr3 QDs, followed by dissolving in a nonpolar solvent (toluene or hexane) and subsequent storage in vacuum and dark environment. As shown in Figure 2a, a lot of needle-like nanowire arrays appeared after aging for 24 h. These NWs should be formed through aggregation and coalescence of the original MAPbBr3 QDs. In fact, there are still some residual QDs in the sample and their sizes are very close to the diameters of the NWs (1.82 ± 0.45 nm, Figure 2a and S3e). This phenomenon is similar to the previously observed spontaneous organization of the CdSe QDs into NWs.43 With longer storage time, these NWs exhibited trend of lateral oriented attachment 9



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and got close to each other, forming rectangle-area alignment. In addition, the original QDs gradually disappeared in the alignment areas. After aging for 96 h, most of the NW arrays had transformed into tiled NPLs (Figure 2c). The similar phenomenon had also been observed in the self-assembly process of the CsPbX3 nanorods.34 The SAED patterns (insets of Figure 2) indicate that the NWs and their aligned arrays are highly

Figure 2. TEM images of colloidal MAPbBr3 NCs (4.0 mg mL–1) after storage for (a) 24 h, (b) 72 h, and (c) 96 h at room temperature. Insets show corresponding SAED patterns. Schematic diagrams of oriented attachment self-assembly of (d) quantum dots, (e) nanowires, and (f) nanoplatelets.

crystalline. In the initial stage of the MAPbBr3 NPL formation (Figure 2a), the (001) and (002) diffraction rings are the most bright, suggesting that the alignment of the QDs prefers to take place along the orientation to form NWs. This may be 10


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attributed to the low surface energy of the (001) planes.44,45 Generally speaking, a single nanowire can only be formed through assembly of the nanoparticles along the same lattice orientation. Therefore, the needle-like nanowire could be formed through oriented attachment of the original MAPbBr3 QDs along the orientation. The arising of diffraction rings rather than diffraction spots in the SAED pattern is ascribed to the random distribution of orientations of the NWs. The schematic diagrams in Figure 2d–f describe the various self-assembly growth stages of the MAPbBr3 NCs. The QDs are assembled into NWs, then the NWs are bound together to form NPLs, and the NPLs further form arrays. There are some special structures in the resulting NPLs (Figure 3a), and their proportion increases with storage time. These structures are similar to the CsPbBr3 superlattices observed in previous studies. 46,47 These wire-like superlattices are formed by face-to-face stacking of the NPLs. The enlarged view of the stacked NPLs is shown in the inset of Figure 3a. The thicknesses of the NPLs are around 1.82  0.38 nm (Figure 3b), corresponding to three layers of Pb‒Br octahedra (a single metal halide octahedron layer is about 0.6 nm in thickness32,48). The fluorescence properties of these structures are mainly determined by the quantum confinement effect of the individual NPL components. Figure 3c shows the PL spectrum of these structures. It consists of a single emission band centered at 464 nm (2.67 eV), with a FWHM of 14 nm, being consistent with that of n = 3 layered two-dimensional perovskites. 49

11



(c)

(d)

1.6

1.0

1.2

0.6

0.8 0.4

0.4

0.2 0.0

Absorbance (a.u.)

0.8

400

Measured diffraction pattern (periodicity = 3.04) Calculated diffraction peaks with d = 2.7 nm

8.0k 4.0k 0.0

0.0 600

450 500 550 Wavelength (nm)

Intensity (a.u.)

12.0k PL intensity (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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5

10

15

20 25 2 (deg)

30

35

Figure 3. (a) TEM image of colloidal MAPbBr3 NCs (4.0 mg mL–1) after longtime storage. Inset shows magnified image. (b) TEM image showing face-to-face stacked nanoplatelets. (c) UV‒Vis absorption spectrum and PL spectrum and (d) XRD pattern of such nanostructures.

Next, we use a quantum well model to calculate the bandgap of the stacked MAPbBr3 nanoplatelet structure. According to the Kronig‒Penney effective-mass model, the separate dispersion relations for the electron and hole can be written as49

(

1

1

)

cos (𝑘𝐿QW)cosh (𝛼𝐿B) + 2 𝜑 ― 𝜑 sin (𝑘𝐿QW)sinh (𝛼𝐿B) = cos 𝛿.

(1)

The widths of barrier (𝐿B) and quantum well (𝐿QW) can be obtained from the TEM 12


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observation (Figure 3b), which is 1.11  0.42 nm and 1.82  0.38 nm, respectively. The effective masses (𝑚Be(h) and 𝑚QW e(h)) and the confinement potential 𝑉CB(VB) are taken from the literature.50 The modified energy gap is given by the following formula 𝐸g = 𝐸3D g + 𝐸e + 𝐸h.

(2)

The single-particle (electron or hole) ground-state energy can be calculated from eqn (1), Ee = 0.15 eV, Eh = 0.27 eV. 𝐸3D g = 2.28 eV. So 𝐸g= 2.70 eV. This value is close to the bandgap of the face-to-face stacked MAPbBr3 NPLs (Figure 3c). The XRD peaks of the stacked MAPbBr3 NPLs show an approximate periodicity of 3.04  0.06° (Figure 3d). This XRD pattern is characteristic of the nanostructures of face-to-face stacked NPLs.51,52 According to Bragg's Law, 2𝑑sin𝜃 = 𝑗𝜆, where 𝜃 is the diffraction angle, 𝜆 = 1.5406 Å is the wavelength of Cu K, and 𝑗 is a positive integer. To match the XRD pattern (see Figure 3d for comparison of measured and calculated diffraction peaks based on Bragg’s law), we derive the spacing between the NPLs, 𝑑 = 2.66 ± 0.06 nm, being roughly consistent with the observed periodicity of the face-to-face stacked MAPbBr3 NPLs (~2.93 nm, Figure 3b).

Theoretical Simulation of Self-Organization of MAPbBr3 NCs. To understand the process of self-organization of MAPbBr3 QDs into superstructures, we perform theoretical simulations. According to the classical DLVO theory, the van der Waals force and electrostatic interaction between particles could dominate the self-assembly process of nanocrystals.53 The van der Waals force has been regarded as the most 13



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general interaction on the nanoscale. The self-assembly process can be controlled by using stabilizing surface ligands or appropriate solvents, which assist in formation of 2D and 3D superstructures such as those composed of NPLs54 and nanorods.55,56 In our simulation, the van der Waals force between two nearest neighbor nanoparticles is calculated based on the Hamaker integration approximation. In the case of the MAPbBr3 NCs, the dipole-dipole interaction may play important roles in all stages of crystal growth. Most inorganic perovskites have spontaneous electric polarization, arising from breaking of the crystal centro-symmetry, as a result of the B cation moving away from the center of the BX6 octahedron.57 This phenomenon is particularly pronounced in the organic‒inorganic hybrid halide perovskites, where the asymmetry of the organic cations leads to absence of an inversion center in the structure. Some studies indicated that the electric polarization may arise from rotation of the dipolar MA+.58–60 The recent density-functional theory calculations demonstrated that the orientation order of the methylammonium cations influences the magnitude of the MAPbX3 perovskite polarization.61,62 A recent simulation study of polarization in CH3NH3PbI3 adopted the latter polarization mechanism because the dipole rotation is easier to handle than the cooperative ionic displacement.63 In our theoretical calculation, each MAPbBr3 QD is treated as a single dipole. For the initial stage of self-assembly involving neighboring MAPbBr3 QDs, the dipole‒dipole interaction has to be taken into account because the dipolar force has an interaction distance even longer than that of the van der Waals force.64 The dipole vectors will eventually align in the same direction as a result of the dipole‒dipole 14


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interaction. This type of interaction leads to self-organization of the CdTe, CdSe, PbSe, PbS, Fe3O4, and Au NCs.65–68 Being driven by this force, two nearest QDs will get closer and closer. In this growth stage, the Ostwald ripening helps the QDs to fuse together, forming surface-smooth NWs (Figure 2a). For simplicity of the calculation, the NWs can be supposed to be nanosized cylinders. In addition, the dipole‒dipole interaction between two approaching NWs is attractive only when their polarization vectors are nearly anti-parallel (see Figure S5b), because in this case the separated charges in the corresponding locations of two adjacent NWs are opposite. However, the dipole vectors cannot be ensured to align in this way without external electric field. If the dipole vectors are parallel, as shown in Figure S5a, there will be repulsive force between two close NWs. This will significantly reduce the probability of polarization attraction between two NWs. So for the calculation of the self-assembly of the MAPbBr3 NWs, only the van der Waals force was taken into account. Further ripening effect will make these NW arrays grow into NPLs with smooth surfaces (Figure 2c). In the final stage, the NPLs will be stacked up together to form superstructures.

Because

there

exists

strong

spontaneous

polarization

in

MAPbBr3,58,59,63,69 charge separation in it will be enahnced.29 In addition, research has shown that the separated charges are more likely to be distributed on the surface of the perovskite crystals.70 Therefore, being driven by the polarization interaction, two neighboring NPLs attract and rotate until they face each other with two oppositely charged faces, then they move towards each other faster and faster owing to the strong attraction force between them. Owing to the presence of the organic ligands on the 15



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surfaces, the NPLs cannot fuse into a 3D single crystal, and instead, they form the analogues of Ruddlesden–Popper perovskite phases (Figure 3a and b). We now study theoretically the role of the van der Waals force and the polarization force in driving self-assembly of the MAPbBr3 NCs. The theoretical model is based on the classical Newton equation applied to the case of the liquid medium.71 Consider first the role of the polarization force in self-assembly of two MAPbBr3 QDs. In this case, each MAPbBr3 QD undergoes a total force 𝐹total = 𝐹polarization ― 𝑓drag. The first term refers to the dipole‒dipole force between two neighboring QDs, and 𝑓drag is the drag force of the fluid. The polarization force can be derived from the interaction energy of two dipoles,72 and it is expressed as 𝜇1𝜇2𝐶12

6

6𝜅

3𝜅2

𝜅3

𝐹polarization = 4𝜋𝜀0𝜀eff 𝑒 ―𝜅𝑌(𝑡)[𝑌(𝑡)4 + 𝑌(𝑡)3 + Y(𝑡)2 + 𝑌(𝑡)], 3𝑒𝜅𝑅

𝐶1 = 2 + 2𝜅𝑅 + (𝜅𝑅)2 + 𝜀 (1 + 𝜅𝑅)/𝜀 , 0

(3) (4)

eff

where 𝜇1 = 𝜇2 = 𝑃 ∙ 𝑉 = 1.18 × 10 ―26 C m are the dipole moment of the QD.63 𝜀0 is the vacuum dielectric constant (8.854 × 10 ―12 F m ―1), 𝜀eff is the relative permittivity of the surrounding medium, and 1/𝜅 is the Debye screening length (calculation details of 𝜀eff and 𝜅 are in Supporting Information). 𝑅 = 0.87 ± 0.17 nm is the average radius of the MAPbBr3 QDs. 𝑌(𝑡) = 𝑥(𝑡) +2𝑅 is the distance 1

between the centers of two approaching QDs. The drag force 𝑓drag = 2𝐶𝜌f𝐴p𝑣2,73 where 𝑣 is the velocity of the QD, 𝐶 is the drag coefficient, 𝐶 ≈ 1 for spheres and plates, and 𝜌f is the mass density of the fluid, 𝜌f_toluene = 0.87 × 103 kg m ―3. 𝐴p is the shadow area of the QD on the screen normal to its velocity. The MAPbBr3 QD can be treated as a sphere, so 𝐴p equals the largest cross-section area of the sphere 16


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𝑆. Substitution of the expressions of forces into the equation of Newton's second law of motion leads to 𝜇1𝜇2𝐶12

6

6𝑘

3k2

𝑘3

1

𝑑𝑣(𝑡)

𝑒 ―𝑘𝑌(𝑡)[𝑌(𝑡)4 + 𝑌(𝑡)3 + 𝑌(𝑡)2 + 𝑌(𝑡)] ― 2𝐶𝜌f𝑆[𝑣(𝑡)]2 = 𝜌𝑉 𝑑𝑡 , 4𝜋𝜀0𝜀eff

(5)

4

where 𝑆 = 𝜋𝑅2, 𝑉 = 3𝜋𝑅3, and the mass density (MAPbBr3) 𝜌 = 4.92103 kg m ―3.74

Figure 4. Schematic diagram depicts the integration method used to calculate the polarization force between two nanoplatelets.

The polarization force between two neighboring NPLs can be calculated by using the integration method, as depicted in Figure 4. We divide each NPL into 𝑁1 × 𝑁2 parts, and assume the area of each part ∆𝑆 = 𝛿𝑎𝛿𝑏 = 0.087 ± 0.020 nm2. The coordinates of an area element on the left NPL surface are (𝑎, 𝑏, 𝑐) and those on the right NPL surface are (𝑎′,𝑏′,𝑐′). According to the formula of the electrostatic force, we obtain the polarization force between two area elements 1 (𝜎 ∙ ∆𝑆)2 𝐹 = 4𝜋𝜀eff𝜀0 [(𝑎 ― 𝑎′)2 + (𝑏 ― 𝑏′)2 + (𝑐 ― 𝑐′)2],

(6)

where 𝜎 is the polarization charge density on the surface of the NPL and 𝜎 = 𝑃 = 17



Page 18 of 42

0.38 C m ―2.63 Then we obtain the force component parallel to the velocity of the NPL (𝜎 ∙ ∆𝑆)2

𝑥(𝑡)

Fcomponent force = 4𝜋𝜀eff𝜀0 [[(𝑎 ― 𝑎 )2 + (𝑏 ― 𝑏 )2 + (𝑐 ― 𝑐 )2]3/2]. ′

′

′

(7)

Similarly, we can calculate the force component between the area elements (𝑎,𝑏,𝑐) and (𝑎′ + 𝛿𝑎′ , 𝑏′ + 𝛿𝑏′), and so on, until all the force components between the area element (𝑎,𝑏,c) and all the parts of another NPL surface are calculated. Then these force components are summed to get the overall polarization force component exerted on the area element (𝑎,𝑏,c) by a whole surface of another NPL. Then we integrate over all the area elements (𝑎,𝑏,c) to derive the total polarization force between two approaching charged surfaces. This force is a function of the distance between the nearest surfaces of two NPLs 𝑥(𝑡), as shown in Figure S6. By using such an integration method and considering that there are two oppositely charged surfaces for each NPL, we could obtain the total polarization force between two approaching NPLs at any separation distance 𝑥(𝑡). We fit the polarization force versus separation distance curve and obtain the approximate analytical expression Ftotal = 𝐶𝑂𝐸𝐹 × {𝑥(𝑡)𝑝 + [𝑥(𝑡) + 2𝐿]𝑝 ―2[𝑥(𝑡) + 𝐿]𝑝},

(8)

where 𝐶𝑂𝐸𝐹 and 𝑝 are two fitting constants, 𝐶𝑂𝐸𝐹 = 9.15 × 10 ―24, 𝑝 = ―1.957, and 𝑥(𝑡) and 𝐹total are in units of m and N, respectively. The first two terms represent the attraction force between two nearest oppositely charged surfaces and that between two farthest oppositely charged surfaces of two interacting NPLs, whereas the third term represents the repulsive force between two positively charged surfaces plus that between two negatively charged surfaces of the two NPLs. According to the equation of the Newton's second law of motion, we have 18


Page 19 of 42

1

𝐶𝑂𝐸𝐹 × {𝑥(𝑡)𝑝 + [𝑥(𝑡) + 2𝐿]𝑝 ― 2[𝑥(𝑡) + 𝐿]𝑝} ― 2𝐶𝜌f𝑆𝑣2 = 𝜌𝐿𝑆

𝑑𝑣(𝑡) 𝑑𝑡 ,

(9)

where 𝑆 is the area of the charged surfaces of the NPLs, and 𝐿 is the average thickness of the NPLs. Figure S3b, c shows the length and thickness distribution of the NPLs. Assume the initial velocities of the two approaching NPLs are zero, then we derive the movement duration time and instantaneous velocity of the two attracting MAPbBr3 QDs or NPLs as functions of the interparticle distance 𝑥(𝑡) during the process of their approaching in the liquid medium by using eqn (5) and (9) (Figure 5a and c). The calculated result indicates that the dipole–dipole force seems to

(a)

QDs NPLs

(b)

6 5 4 3 2 1 0

Time (ms)

Time (ns)

x0.01

x10

0 5 10 15 20 25 30 35 40 45 50

0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

(c)

NWs NPLs QDs

x0.01

0 5 10 15 20 25 30 35 40 45 50

Interparticle distance (nm)


QDs NPLs

(d)

QDs NWs

NPLs 4

3 2

1 0

0

5 10 15 20 25 30 35 40 45 50


Velocity (m s-1)

Velocity (103m s-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

x0.1 0

5 10 15 20 25 30 35 40 45 50


Figure 5. Calculated movement duration time and velocity of two attracting MAPbBr3 nanocrystals (QDs, NWs, and NPLs) driven by the polarization force (a, c) and by the van der Waals force (b, d).

19



Page 20 of 42

be much stronger for the QDs than for the NPLs. This is partially because that the drag force plays a more important role in the case of motion of the NPLs owing to their much larger surface areas. It should be noted that to make the simulation more accurate, we have calculated the electrostatic force between two charged surfaces of the two approaching MAPbBr3 NPLs by using the integration method. For simplicity of calculation, one can assume that this force approximately equals the force between two total point charges separately distributed on two surfaces. Figure S7 shows comparison between the accurately and approximately calculated results. It can be seen that the approximate method highly overestimates the movement duration time before collision of the two NPLs, that is, it underestimates the role of the polarization force in driving the collision. This is understandable because the approximate method takes into account only the electrostatic force between each group of two nearest unit areas of charged surfaces of the two NPLs and hence the total force is significantly underestimated. The current accurate integration method developed for calculation of the polarization force between two charged surfaces can be generalized to study the interaction dynamics of various types of charged NPLs. Next, we consider the role of the van der Waals force in driving self-organization of the MAPbBr3 NCs. We first consider the case of two spherical MAPbBr3 QDs with equal radius 𝑅. When the distance between them 𝑥(𝑡) is larger than 0.1(𝑅 + ℎ0), where ℎ0 is the equilibrium length of the capping ligand, the van der Waals force is 𝐹QDs1 =

2𝐴131 3

(

𝑅2[𝑥(𝑡) + 2𝑅]

1

2

𝑥(𝑡)

1

2

)

― 𝑥(𝑡)2 + 4𝑅𝑥(𝑡) + 4𝑅2 ， + 4𝑅𝑥(𝑡)

(10)

In the case of 𝑥(𝑡) < 0.1(𝑅 + ℎ0), the van der Waals force is 20


Page 21 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2𝐴123

𝐴232

𝑅

𝐴131

𝐹QDs2 = 12[ ― 𝑥(𝑡)2 + [𝑥(𝑡) + ℎ ]2 ― [𝑥(𝑡) + 2ℎ ]2]. 0

(11)

0

In the above two equations, 𝐴131, 𝐴232, and 𝐴123 are the Hamaker constants, and the subscript 1, 2, and 3 represent MAPbBr3, n-octylamine, and toluene, respectively. The calculation detail of the effective Hamaker constant is given in the Supporting Information, and we have 𝐴131 = 4.69 × 10 ―20 J, 𝐴232 = 4.29 × 10 ―22 J, 𝐴123 = 1.05 × 10 ―21 J. According to the Newton’s second law of motion, we have 2𝐴131 3

(

𝑅2[𝑥(𝑡) + 2𝑅]

1

2

)

1

― 𝑥(𝑡)2 + 4𝑅𝑥(𝑡) + 4𝑅2 𝑥(𝑡)2 + 4𝑅𝑥(𝑡)

1

― 2𝐶𝜌f𝑆𝑣2t = 𝜌 ∙

4𝜋𝑅3𝑑𝑣(𝑡) 3 𝑑𝑡 ,

(12)

[𝑥(𝑡) > 0.1(𝑅 + ℎ0)] 𝑅 12[

𝐴232

2𝐴123

𝐴131

1

― 𝑥(𝑡)2 + [𝑥(𝑡) + ℎ ]2 ― [𝑥(𝑡) + 2ℎ ]2] ― 2𝐶𝜌𝑓𝑆𝑣2𝑡 = 𝜌 ∙ 0

0

4𝜋𝑅3𝑑𝑣(𝑡) 3 𝑑𝑡 ,

(13) [𝑥(𝑡) < 0.1(𝑅 + ℎ0)]

where R = 0.87  0.17 nm is the average radius of the MAPbBr3 QDs, and 𝑆 is their largest cross-section area. For the case of two attracting NWs, we use the van der Waals force of two parallel rods with radius 𝑅 (0.91 ± 0.22 nm) to mimic the self-assembly growth from the NWs to the NPL: 𝐹NWs = 8

𝐴eff𝐿 2[𝑥(𝑡)]

𝑅 1/2

(2)

5/2

,

(14)

where 𝑥(𝑡) is the separation of the nearest surfaces of the two NWs, 𝐴eff is the effective Hamaker constant, and in the absence of the surface ligands, we have 𝐴eff = 𝐴131. In the presence of the surface ligands, we have 𝐹NWs = 8

𝐿

1/2

𝑅 2( 2 )

𝐴232

2𝐴123

𝐴131

[ ― [𝑥(𝑡)]5/2 + [𝑥(𝑡) + ℎ ]5/2 ― [𝑥(𝑡) + 2ℎ ]5/2], 0

0

(15)

where 𝐿 = 25.59 ± 6.47 nm is the average length of the NWs (Figure S3f), ℎ0 is the equilibrium length of the capping ligand. The van der Waals force driving self-assembly of the two NPLs can be calculated by using the formula for two plates 21



Page 22 of 42

with thickness 𝐿: 𝐹NPLs =

𝐴eff𝑆

[

1

6 𝑥(𝑡)3

1

+ [𝑥(𝑡) + 2L]3 ―

2

].

[𝑥(𝑡) + 𝐿]3

(16)

Substituting eqn (14), (15), and (16) as well as the expression for the drag force into the equation of the Newton's second law of motion, we derive the movement duration time and velocity of the two attracting MAPbBr3 NWs or NPLs as functions of the separation between them (Figure 5b and d). The result shows that the spontaneous polarization force is much stronger than the van der Waals force in driving self-assembly of the MAPbBr3 NCs. In the case of the van der Waals force, the movement duration time before the two MAPbBr3 QDs reach a small enough distance of 0.2 nm is 0.124 ms (Figure 5b); in contrast, the movement during time in the case of the polarization force decreases to as small as 0.163 ns (Figure 5a). The movement velocity at 0.2 nm is 4.15 × 103 m s–1 (Figure 5c) in the case of polarization force, whereas the velocity is only 11.97 m s–1 in the case of van der Waals force (Figure 5d). This difference becomes more obvious in the case of approaching of two MAPbBr3 NWs or NPLs. The role of the drag force exerted on the QDs is less important than that exerted on the NWs and NPLs, and this partially explains why the velocity of the QDs right before their collision is much larger than that of the NWs or NPLs in the case of van der Waals force. Considering the effect of the surface ligand groups, ultimately, the NPLs tend to be self-assembled into periodic structures with fixed spacing. The above calculations focus on the roles of the dominant attracting forces in driving self-assembly of the MAPbBr3 NCs. Next we consider the role of the 22


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repulsion force generated mainly by the surface ligands on the basis of the DLVO theory.75–77 The total energy of the colloidal MAPbBr3 QD system comprises contributions from both attraction forces and repulsion forces when there are enough surface ligands on the QD surface. The attractive energy arises from the van der Waals force and the polarization force. In the case of 𝑥(𝑡) < 0.1(𝑅 + ℎ0), we have 𝜇2

𝑊A = Wp ― p + WvdW = ― 4𝜋𝜀 𝜀

𝑅 𝐴232

2𝐴123

[2 + 2𝜅𝑌(𝑡) + (𝜅𝑌(𝑡))2]𝑒 ―𝜅𝑌(𝑡)𝐶21 ― 12(𝑥(𝑡) ― 𝑥(𝑡) + ℎ0 + 𝑌(𝑡) 3

0 eff

𝐴131

𝑥(𝑡) + 2ℎ0),

(17) In the case of 𝑥(𝑡) > 0.1(𝑅 + ℎ0), we have 𝑊A = ― 4𝜋𝜀 𝜀

𝜇2

[2 + 2𝜅𝑌(𝑡) + (𝜅𝑌(𝑡))2]𝑒 ―𝜅𝑌(𝑡)C21 𝑌(𝑡)

0 eff

𝑥(𝑡)2 + 4𝑅𝑥(𝑡)

𝐴131

3

3

𝑅2

𝑅2

1

[𝑥(𝑡)2 + 4𝑅𝑥(𝑡) + 𝑥(𝑡)2 + 4𝑅𝑥(𝑡) + 4𝑅2 + 2ln (

(18)

)]，

𝑥(𝑡)2 + 4𝑅𝑥(𝑡) + 4𝑅2

C1 =

3e𝜅𝑅

.

(19)

2 + 2𝜅𝑅 + (𝜅𝑅)2 + 𝜀0(1 + 𝜅𝑅)/𝜀eff

The pair repulsion energy arising from the net charges and dipoles as well as the steric repulsion of organic ligands can be calculated by using the following expressions64,77 𝑄2

𝑄𝜇

WR = 𝑊qq + 𝑊pq + 𝑊RS = 4𝜋𝜀0𝜀eff𝑒 ―𝜅 𝑌(𝑡)𝐶20 + 2𝜋𝜀0𝜀eff𝑌(𝑡)2𝑒 ―𝜅𝑌(𝑡)[1 + 𝜅𝑌(𝑡)]C0𝐶1 + 9

1

1

[ ― ln(𝑢) ― 5(1 ― 𝑢) + 3(1 ― 𝑢3) ― 30(1 ― 𝑢6)],

where 𝑢 =

𝑌(𝑡) ― 2𝑅 2ℎ0 ,

𝜋3𝑅𝑘𝑏𝑇Γℎ30 6𝑁𝐵2

(20)

𝑒𝜅𝑅

C0 = 1 + 𝑘𝑅, 𝑄 is the charge of a MAPbBr3 QD, Γ is the grafting

density of the organic ligands, 𝑁 is the number of the Kuhn monomers, and 𝐵 is the characteristic length of the ligand chain. In addition, it is necessary to take into account the steric energy of the surface ligands only when 𝑥(𝑡) ≤ 2ℎ0 in that the steric repulsion is a short-range force. The detail concerning calculation based on the DLVO theory is given in the Supporting Information. Figure 6 shows the calculated 23



Page 24 of 42

interaction energy as a function of interparticle distance, where WA, W𝑅, and W𝑇 represent the attraction energy (from van der Waal and polarization forces), repulsion energy (from double-layer repulsion force), and total energy, respectively. For the two

Figure 6. Calculated interaction energy (in units of 𝑘B𝑇 with 𝑇 = 300 K) as function of interparticle distance for two MAPbBr3 QDs (a, b), two NWs (c), and two NPLs (d). 𝑊A, 𝑊R, 𝑊T denote the attraction (van der Waals plus polarization forces), repulsion (surface charges), and total interaction energies, respectively.

interacting MAPbBr3 QDs, in the presence of a large number of surface ligands 24


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(OA+), the repulsion force dominates at large separation (Figure 6a), whereas the attraction force dominates at short distance, and there is an energy barrier which prohibits the approaching of two MAPbBr3 QDs when they are close enough. Therefore, the oriented attachment of two MAPbBr3 QDs can occur only when enough energy is achieved such that the system can cross the energy barrier. The height of the energy barrier is about 0.078 eV ( ≈ 3.0 𝑘B𝑇) and the energy barrier maximum occurs at a separation of around 4 nm. It is not easy to acquire such a thermal energy for the two-QD system at room temperature. This explains why the self-assembly process of the MAPbBr3 QDs is rarely observed in experiment in the presence of many surface ligands. Next we consider the interaction of two MAPbBr3 QDs with negligible number of surface ligands (corresponding to the case in our experiment). The grafting density Γ, Kuhn monomer number 𝑁, charge 𝑄, and other associated parameters in the calculation should be adjusted to minimize the effect of surface ligands. Figure 6b displays the calculated result. It can be seen that the attraction energy dominates now and it continuously drives self-assembly of the MAPbBr3 QDs until they collide and bind. This prediction agrees with our experiment. Because most surface ligands have been got rid of in our experiment, to simulate the interaction of the MAPbBr3 NWs, we assume a smaller ligand concentration (with Γ = 0.1 nm ―2, 𝑁 = 1). Figure 6c shows the calculated result. It can be seen that the repulsion force dominates at large separation of the two NWs, whereas the attraction force becomes dominate when the separation is smaller than 30 nm. In addition, the 25



Page 26 of 42

energy barrier is smaller than 0.25 𝑘B𝑇. Therefore, the system of two interacting MAPbBr3 NWs can easily cross the energy barrier and thus trigger the self-assembly process. The inset of Figure 6c shows the energy versus distance curve for two MAPbBr3 NWs in the absence of surface ligands. In this case, the energy barrier disappears and the attraction force drives more easily the self-assembly process. Some previous studies have indicated that the periodic spacing of MAPbBr3 NPLs might be attributed to the surface ligands used in experiment.33,78,79 So we have considered the role of surface ligands in self-organization of the MAPbBr3 NPLs. Considering the size of the octylamine ions and the stoichiometric ratio in this model, we can obtain the values of grafting density Γ, Kuhn monomer number 𝑁, and other related parameters. Figure 6d displays the calculated energy as function of interparticle distance. The result shows that for the NPLs, at large distances, the attraction interaction that consists of the polarization force and van der Waals force is much stronger than the repulsion force arising from the surface ligands, as a result, the two NPLs approach each other. The total energy reaches a minimum at 𝑥(𝑡) = 1.01 nm, and then the interaction becomes repulsive; the equilibrium thickness of the ligand layer between two MAPbBr3 NPLs is 2ℎ0 = 1.1 nm, and the small difference between above two values suggests that the ligand layer has some elasticity. The strong repulsion force at 𝑥(𝑡) < 1.01 nm prohibits further approaching of the two NPLs. This agrees with the average NPL spacing of ~1.1 nm as revealed by the TEM observation. The two-NPL system is the most stable at this distance. The calculated result shown in Figure 5 indicates that it takes short time for 26


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self-assembly of the MAPbBr3 QDs into NWs if there are no surface ligands. In our experiment, the MAPbBr3 NWs were observed after 24 h (starting from the colloidal MAPbBr3 QDs after removing most surface ligands). In practice, there have been rare reports on transition from MAPbBr3 QDs to NWs. On the one hand, this is because that this process occurs very fast, and reaction could be completed in a few hundred seconds at high reaction temperature.80 On the other hand, in most experiments, the many surface ligands generated during synthesis of the MAPbBr3 QDs greatly hamper their self-organization growth. The above calculation reveals it is difficult for the MAPbBr3 QDs to overcome the repulsion force caused by surface ligands at room temperature. So in experiments, it is necessary to elevate the reaction temperature80–82 or reduce the surface ligand ratio83 in order to overcome the ligand repulsion interaction in self-assembly processes. The calculation (Figure 5) also indicates that with consideration of the van der Waals force, the second stage of self-assembly (from MAPbBr3 NWs to NPLs) needs much longer time (4.45 ms) than the first stage (from MAPbBr3 QDs to NWs). In experiment, the complete conversion of MAPbBr3 NWs into NPLs was observed after 72 h of storage of the pristine colloidal MAPbBr3 QDs, much longer than the first stage. We can derive the kinetic energy of the MAPbBr3 NCs by using their masses and calculated velocities (Figure 5). The kinetic energies are 9.710–22 J, 2.410–22 J, and 1.310–20 J for two MAPbBr3 QDs, NWs, and NPLs at 𝑥(𝑡) = 0.2 nm, respectively. The MAPbBr3 nanowire system has the lowest kinetic energy; therefore, it is more stable compared with the MAPbBr3 QD and NPL systems. This explains 27



Page 28 of 42

why it is rare to observe self-assembly of the MAPbBr3 NWs in experiments. In contrast, the MAPbBr3 NPLs have often been observed to form superstructures made of stacked NPLs32,45,79,84 in that there is strong polarization attraction force between the NPLs. The calculation indicates that the self-assembly of the NPLs is much faster than the self-assembly of the NWs, and this agrees with our observation that there have been some titled NPLs at the end of the second stage of self-assembly (Figure 3a). It also agrees with the observation that after the second stage of self-assembly, it takes only 4 more hours for complete transformation of the tiled MAPbBr3 NPLs into superstructures through self-assembly.

CONCLUSION In summary, we have studied both experimentally and theoretically the self-assembly dynamics of the fluorescent MAPbBr3 QDs, NWs, and NPLs by taking into account the general van der Waals force, the special polarization force, and the probable double-layer (surface ligand) force. The result reveals that the intrinsic polarization force between these organic‒inorganic hybrid perovskite NCs is more powerful in driving their self-assembly than the usually considered van der Waals force. In contrast, the repulsion force generated by the surface ligands can strongly hamper the self-assembly process of the MAPbBr3 NCs, and it also accounts for arising of the stable and periodically spaced NPL superstructures. The calculation explains the experimental observation of different times taken for self-assembly of the MAPbBr3 QDs, NWs, and NPLs. To calculate the polarization force between the 28


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surface-charged nanoplatelets, we have developed the integration method which is much more accurate than the simplified point-charge model. These studies improve our understanding of the self-organization mechanisms of the perovskite nanocrystals. The theoretical model developed here can be generalized to study the self-assembly dynamics of various types of colloidal nanocrystals.

ASSOCIATED CONTENT Supporting information. UV‒Vis absorption and PL spectra of MAPbBr3 bulk crystals, calculated and fitted polarization force versus particle distance curves, and size distribution of MAPbBr3 NCs.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] (J.F.).

ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China Nos. 11874106 and 11574047. It was also supported by the Natural Science Foundation of Anhui Province No. 1508085QB40.

REFERENCES (1) Schmidt, L. C.; Pertegas, A.; Gonzalez-Carrero, S.; Malinkiewicz, O.; Agouram, 29



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Quantitative Modeling of Self-Assembly Growth of Luminescent

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