Kinetics of the self-assembly of nanocrystal Kinetics of ... · 1 Kinetics of the self-assembly of nanocrystal superlattices measured by real-time in situ X-ray scattering Mark C

1

Kinetics of the self-assembly of nanocrystal superlattices measured by real-

time in situ X-ray scattering

Mark C. Weidman1, Detlef-M. Smilgies2, William A. Tisdale1*

1Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA

02139, USA.

2Cornell High Energy Synchrotron Source (CHESS), Cornell University, Ithaca, NY 14850,

USA.

*e-mail: [email protected]

Contents: Supplementary Text 1 – 6 Supplementary Figures S1 – S14 Table S1 Supplementary Movies S1 – S3

Kinetics of the self-assembly of nanocrystalsuperlattices measured by real-time in situ

X-ray scattering

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NMAT4600

NATURE MATERIALS | www.nature.com/naturematerials 1

© 2016 Macmillan Publishers Limited. All rights reserved.

http://dx.doi.org/10.1038/nmat4600

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1. Experimental methods

Nanocrystal synthesis and purification

Monodisperse colloidal lead sulfide (PbS) nanocrystals were synthesized according to the

method detailed in our previous work22. Briefly, 12.5 g of lead chloride (PbCl2, Alfa Aesar,

99.999%) were combined with 75 mL of oleylamine (Sigma Aldrich, 98% primary amine) in a

three-neck flask. The lead solution was degassed on a Schlenk line under vacuum for 10 minutes

while stirring the solution. The three-neck flask was then pressurized with nitrogen and heated to

120°C. Meanwhile, 0.180 g of sulfur (Sigma Aldrich, ≥99.99%) was combined with 7.5 mL of

oleylamine and heated in an oil bath at 120°C for 20 minutes with nitrogen bubbling into the

solution. The sulfur solution was cooled to room temperature and the lead solution was allowed

to reach a steady value of 120°C. 5 mL of the sulfur solution were then swiftly injected into the

lead solution, which was being vigorously stirred. After ~1 minute of growth time, the reaction

was quenched by the rapid injection of 40 mL of cold hexanes and the addition of a water bath

around the three neck flask.

The reaction products were transferred to centrifuge tubes and the addition of methanol and

butanol caused the nanocrystals to precipitate from the suspension. The tubes were centrifuged

and the supernatant was discarded. The black precipitate (containing nanocrystals as well as

excess PbCl2 and sulfur) was redispersed in hexane. To the suspension, 200% by volume of oleic

acid (Sigma Aldrich, 90%) was added, which caused the nanocrystals to precipitate but not the

excess PbCl2 or sulfur. The tubes were centrifuged and the supernatant was discarded. Again, the

precipitate was redispersed in hexane and 200% by volume oleic acid was added. The tubes were

centrifuged and the supernatant was discarded. The remaining black precipitate is now mainly

nanocrystals, having removed most of the excess precursors and solvent from the reaction. The

precipitate was redispersed in hexane and centrifuged to remove any last PbCl2, which

precipitated as a white powder after centrifugation. The precipitated white powder PbCl2 was

discarded and the black supernatant was precipitated with methanol and butanol a final time to

remove any excess oleic acid. The purified PbS nanocrystal product was then brought into an air-

free, water-free glove box and dispersed in toluene at a concentration of 50 mg/mL.

Thin film preparation

Thin film samples, like that used for Fig. 2c, were prepared on 0.5”x0.5” glass slides. First the

glass slides were cleaned and treated overnight in a 0.02 M (3-mercaptopropyl)trimethoxysilane

solution in toluene to improve nanocrystal adhesion to the slide surface. ~40 µL of nanocrystal

suspension in toluene (at 50 mg/mL) were then pipetted onto the glass slide and spin coated in

the glove box for 30 seconds at 1500 rpm to create a uniform film with thickness ~100 nm.

Transmission electron microscopy (TEM)

Samples were prepared for TEM by dropping one drop of a nanocrystal suspension in toluene

onto a TEM grid with an amorphous carbon support layer. The lower magnification micrograph

in Fig. 1b was taken on a JEOL 2011 operating at 200 kV and the high resolution micrograph

showing the atomic planes of a single nanocrystal was taken on a JEOL 2010F FEG operating at

200 kV. See Fig. S1 for additional TEM images.


3

Absorption and photoluminescence

Absorption measurements were taken using a Cary 5000 UV-vis-NIR spectrophotometer.

Suspensions of nanocrystals were prepared in tetrachloroethylene for these measurements. The

photoluminescence was recorded using a Bayspec NIR spectrometer for samples also in

tetrachloroethylene.

X-ray diffraction (XRD)

XRD was performed on a Rigaku Smartlab with Cu Kα source operating at 45 kV and 200 mA.

The sample was rotated during the measurement and data was collected from 20° to 80°.

Inductively coupled plasma optical emission spectroscopy (ICP-OES)

ICP-OES was performed by Evans Analytical Group to determine the relative ratio of lead to

sulfur atoms in the nanocrystals studied. 10 mg of dried nanocrystals were digested in nitric acid

for analysis. The atomic ratio of Pb:S was determined to be 1.35:1.


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2. Nanocrystal size and shape

To determine the average nanoparticle diameter, we fit the azimuthally integrated GISAXS

pattern from measurements made while the nanoparticles were still in colloidal form. The data fit

well to the form factor of spherical particles having an average diameter of 5.6 nm. This is in

good agreement with the sizing curve for PbS nanocrystals published in our previous work (as

determined using both TEM and solution SAXS measurements). According to that correlation,

our studied nanocrystals, which absorb with a peak at 1275 nm, will have a diameter of 5.5 nm22.

A 5.6 nm spherical PbS nanocrystal has ~3430 atoms (based on a rock salt lattice constant of

5.936 Å), we use a polyhedron with 3040 atoms as it most closely matches the number of atoms

while still having all Pb-terminated {111}NC faces (The presence of these Pb-terminated {111}NC

faces is what leads to the experimentally observed 1.35:1 Pb:S ratio of our nanocrystals). Of

these 3040 atoms, 1655 are Pb and 1385 are S, giving a Pb:S ratio of ~1.2:1. This is in good

agreement with the results of Choi et al. for a 5.6 nm PbS nanocrystal and our ICP-OES data23.

We note that in the work of Choi et al. they also found that ICP-OES measurements produced

slightly higher Pb:S atomic ratios than their models and X-ray photoelectron spectroscopy

measurements23. An image of our atomic representation is presented in Fig. 1c.


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3. GISAXS/GIWAXS data collection

Grazing-incidence small-angle X-ray scattering (GISAXS) and grazing-incidence wide-angle X-

ray scattering (GIWAXS) measurements were performed at the D1 beamline of the Cornell High

Energy Synchrotron Source (CHESS). The X-ray beam was produced by a hardbent dipole

magnet and a Mo:B4C multilayer double-bounce monochromator with the radiation having

wavelength of 1.157 Å at a bandwidth of 1.5%. The GISAXS patterns were collected on a

DECTRIS Pilatus3 200K detector and the GIWAXS patterns were collected on a DECTRIS

Pilatus 100K detector. The sample-to-detector distances were calibrated using a silver behenate

standard and a cerium oxide (CeO2) standard for the GISAXS and GIWAXS detectors,

respectively. GISAXS and GIWAXS data were both collected for 1 second exposure time.

Because the GISAXS detector has a ‘blind spot’ of ~10 pixels which appears as a horizontal

stripe in the collected data, the patterns presented in the text are actually two GISAXS patterns

which have been combined together after we moved the detector upwards so as to change the

position of the blind spot. In this way we can image the entire scattering pattern. We find the

slight delay caused by changing the detector position and recording another exposure does not

affect the data or the data interpretation.

The experimental apparatus was a custom build chamber designed for controlled evaporation of

a liquid sample during in situ measurements. The chamber has two Kapton windows which allow

for the incident X-ray beam to impinge on the sample and for the scattering to pass through the

windows and onto the detectors. For the measurements, a blank glass slide was placed in the

evaporation chamber and aligned to have an angle of 0.25° with respect to the X-ray beam. We

then placed 20 µL of pure toluene in the bottom of the chamber and sealed the chamber, leaving

the vapor pressure to equilibrate for several minutes. This allowed the toluene vapor to saturate

the chamber such that when we added a drop of our nanocrystal suspension, evaporation only

began when we started flowing dry helium gas. To add the nanocrystal sample, we removed the

chamber lid and added 20µL of a 10 mg/mL suspension of nanocrystals in toluene onto the glass

slide, being careful not to move the glass slide. The chamber lid was quickly replaced and the

collection of GISAXS and GIWAXS data began. To start the evaporation process, we flowed

dry, inert helium gas into the cell at a flow rate of ~25 SCCM. At this minimal flow rate the

evaporation rate is slowed sufficiently such that we captured all of the kinetics. For the

measurements we collected the GISAXS and GIWAXS patterns every 18 seconds, or 0.3

minutes. Overall measurement time was 30.6 minutes for a total of 308 X-ray beam exposures.

Nevertheless, we observe no beam damage caused by the X-ray beam, as determined by

comparison to thin film samples on their first X-ray exposure.


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4. GISAXS data treatment and analysis

GISAXS patterns were indexed using software provided by Detlef Smilgies34,35. The software

allowed for selection of the superlattice type (FCC, BCC, etc.), the values for the a, b, and c axes

lengths of the superlattice, and the plane of the superlattice which is parallel to the plane of the

substrate (see Fig. S3-4). The software then overlaid the expected scattering pattern onto the

experimental pattern for comparison. All patterns were indexed by eye to ensure that both the

high intensity nearest-neighbor peaks and the lower intensity, higher-order reflections were fit

well. The locations of the indices were then exported and used for Matlab processing of the

experimental images. See Fig. S3 for an example of a pattern with and without the indexed fit.

The experimental scattering patterns were formatted and calibrated in Matlab. All GISAXS

patterns are presented on a base-10 logarithmic scale (typically from 1.35 to 4.5). The integration

plots presented in the insets of Fig. 2 and Fig. S5 were performed in Matlab by azimuthally

integrating around the main beam location from 30° to 50°. The azimuthally integrated GISAXS

pattern for all time steps are presented in Fig. S5.

Superlattice characterization

The data can be indexed to a superlattice type starting at 12.0 minutes, when we first see the

beginning of crystallization. The indexing parameters at each time point are provided in Table

S1. The exponential decay line shown in Fig. 4a is fit from 13.5 minutes to 30.6 minutes (last

time point) to the equation:

𝑐 (𝑛𝑚) = 3.75𝑛𝑚 ∙ e−0.6068(t−13.5) + 9.05 𝑛𝑚

giving the decay a time constant of 1/0.6068 = 1.65 minutes.

Surface-to-surface distance

To convert from c axis length to the surface-to-surface distance data presented in Fig. 4c we use

the equation:

𝑖𝑛𝑡𝑒𝑟𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑠𝑝𝑎𝑐𝑖𝑛𝑔 (𝑛𝑚) = √(12.8 𝑛𝑚)2 + 𝑐2

2− 4.5 𝑛𝑚

The numerator gives the face diagonal length of the FCC unit cell or, equivalently, the body

diagonal length of the BCC unit cell, as the two cells are interlaced (see Fig. S6). The

denominator divides that length by two as it is the length of two nanocrystal cores plus their

ligand lengths, making the number the length of one nanocrystal core plus the ligands on each

side of the core. We then subtract 4.5 nm, which is twice the distance from the nanocrystal center

to the (111)NC face, to just leave us with the interparticle spacing (surface to surface separation of

neighboring nanocrystals). We use the more accurate value of 4.5 nm instead of the average

‘diameter’ of 5.6 nm for these calculations. The interparticle spacing data can be fit to an

exponential decay with equation:

𝑖𝑛𝑡𝑒𝑟𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑠𝑝𝑎𝑐𝑖𝑛𝑔 (𝑛𝑚) = 1.2𝑛𝑚 ∙ 𝑒−0.6384(𝑡−13.5) + 3.4 𝑛𝑚

giving the decay a time constant of 1/0.6384 = 1.57 minutes.


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Superlattice tilt relative to substrate Our indexation of the superlattice structures shows that the (111)SL plane of the FCC cell, which

is the same as the (110)SL plane of the BCC cell, is parallel to the substrate plane during the

superlattice changes. As shown in Fig. 5, this results in a rotation of the superlattice relative to

the substrate as it contracts. We characterize the angle of the superlattice relative to the substrate

plane using the equation:

𝑠𝑢𝑝𝑒𝑟𝑙𝑎𝑡𝑡𝑖𝑐𝑒 𝑎𝑛𝑔𝑙𝑒 (°) = tan−1 (𝑐 𝑎𝑥𝑖𝑠 𝑙𝑒𝑛𝑔𝑡ℎ

9.05 𝑛𝑚)

where c ranges from 12.8 nm initially (54.7°) to 9.05 nm in the final state (45°). We use the c

axis values with time given in Table S1 to plot the superlattice angle as shown in Fig. 4d.


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5. GIWAXS data treatment and analysis

GIWAXS patterns were indexed using software provided by Detlef Smilgies34,35. For lead sulfide

we use an FCC unit cell with lattice constant of 5.936 Å and the atomic plane which is parallel to

the substrate being used to determine the nanocrystal orientation. The locations of the indices

were then exported and used for Matlab processing of the experimental images. The

experimental scattering patterns were formatted and calibrated in Matlab. All GIWAXS patterns

are presented on a linear scale with the limits chosen to best emphasize the pattern (typically

from 400 to 4000). We integrate the patterns azimuthally around the main beam and find the

results match well with X-ray diffraction measurements on the same nanocrystals as well as the

expected reference peak locations for lead sulfide (see Fig. S7).

Peak alignment calculation

GIWAXS patterns were integrated along the scattering paths where q=1.75 ± 0.05 Å-1 and

q=2.10 ± 0.05 Å-1 (for the 111NC and 200NC Bragg reflections, respectively) as a function of

angle, χ, relative to the main beam location. For this work we define 0° as the angle closest to the

x-axis, with angle increasing counterclockwise towards the y-axis (see Fig. S8a). We integrate

over angles of 0° to 70° as that is the detection range given by the position of the GIWAXS

detector (see Fig. S8a,c). We determine the time dynamics of the nanocrystal atomic alignment

by looking for peaks in these angle resolved integrations. In the colloidal state, where

nanocrystals are randomly oriented with respect to one another, we found that scattering counts

are higher at larger angles (Fig. S8a,b). We do not believe this is a result of nanocrystal

orientation but rather an artifact of the experimental geometry – that is, we see more scattering

out of the plane of the sample than in the plane of the sample. As a result, we find there is a

linear increase in intensity as a function of angle even in the colloidal state. To extract the peaks

we fit both a line through 25-40° for the 111NC reflection and through 15-30° for the 200NC

reflection. We then take the counts above this linear slope over the angle of 45-65° for the 111NC

reflection and 30-70° for the 200NC reflection. See Fig. S8b and d for the data fitting procedure.

The angle-integrated peaks at each time step are shown in Fig. S9. By fitting the counts to a

linear baseline, we are able to account for the artifact of higher counts at larger angles.

We fit the peak counts as a function of time to a logistic function (an “S” shape sigmoidal curve)

with the form (see Fig. S8e):

𝑝𝑒𝑎𝑘 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 (𝑐𝑜𝑢𝑛𝑡𝑠) =𝐿

1 + 𝑒−𝑘(𝑡−𝑡0)

where L is the maximum peak counts, k is the steepness of the curve, and t0 is the time at which

the curve reaches 50% of its maximum value.

The {111}NC plane alignment can be normalized (to span from 0 to 100%) with the following:

{111}𝑁𝐶 𝑝𝑙𝑎𝑛𝑒 𝑎𝑙𝑖𝑔𝑛𝑚𝑒𝑛𝑡 (%) =100

1 + 𝑒−4.43(𝑡−13.7)

from which we can say the {111}NC have reached 95% of their alignment by 14.4 minutes.


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The {100}NC plane alignment can be normalized (to span from 0 to 100%) with the following:

{100}𝑁𝐶 𝑝𝑙𝑎𝑛𝑒 𝑎𝑙𝑖𝑔𝑛𝑚𝑒𝑛𝑡 (%) =100

1 + 𝑒−5.91(𝑡−14.0)

from which we can say that the {100}NC have reached 95% of their alignment by 14.5 minutes.

The {100}NC planes reach 50% alignment about 0.3 minutes more slowly than the {111}NC

planes, but they have a steeper slope so that both the {111}NC and {100}NC reach full alignment

at nearly the same time. We believe that the slight discrepancy between their alignment rates is

not significant to our physical interpretation.

Peak width (nanocrystal tilt) calculation

The same analysis described above used for the peak alignment calculation was also used to

calculate the peak width and the nanocrystal tilt (Fig. S8a-d). The only difference was that,

instead of taking the total peak counts as a function of angle, we fit the peak to a Gaussian

function and looked at how the standard deviation of the Gaussian peak changed with time (Fig.

S8g). We found that while the 111NC peak intensity increases with time, its peak width is nearly

constant. In contrast, the 200NC peak width decreases significantly over time and with nearly the

same exponential decay time constant we found for the decay rate of the c axis. See Fig. S10 for

a comparison of how the two scattering peaks change as the superlattice contracts.

We fit the 200NC peak standard deviation values to the exponential decay equation:

200𝑁𝐶 𝑝𝑒𝑎𝑘 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (°) = 6.5° ∙ 𝑒−0.6553(𝑡−13.5) + 5.1°

giving a decay rate of 1/.6553 = 1.53 minutes, which closely follows the c axis decay rate of 1.65

minutes. Looking at the 200NC peak width during the self-assembly, we see that the 200NC peak

at earliest times is actually a double peak. As discussed in the main text, this pattern results when

the nanocrystals exhibit a uniform tilt. We characterize this tilt by indexing the nanocrystals such

according to some low symmetry plane such as (750)NC as opposed to their final state where the

(110)NC plane of the nanocrystals are aligned with the (110)SL plane of the BCC superlattice,

corresponding to no overall tilt. For example, a (750)NC plane alignment corresponds to a tilt of

9.5° because the (750)NC plane is 9.5° off from the (110)NC plane. See Fig. S11 for examples of

characterizing the nanocrystal tilt using various low symmetry planes. Note that with this

nanocrystal tilt, the predicted 111NC peak location is unaffected and does not show splitting using

our indexing software. As expected, the experimental 111NC scattering peak shows little changes

once it emerges.

As a verification of our interpretation, we can look at a different batch of similarly sized PbS

nanocrystals (absorption peak = 1350 nm, diameter = 5.8 nm) which we found to assemble in an

FCC superlattice with similar GIWAXS patterns to the ones we observed at early times of

evaporation (see Fig. S12). Because this was a spin coated film which assembled in this manner,

we observed stronger scattering, particularly in the GIWAXS pattern and therefore can more

easily index the pattern. We find that the FCC superlattice is fit well to an FCC pattern with a = b

= c = 12.8 nm while the GIWAXS pattern clearly shows two peaks in the 200NC scattering which

are fit well to a nanocrystal tilt of ~9.7°.


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We quantify the nanocrystal tilt from the 200NC peak width values by setting the maximum peak

standard deviation (11°) to represent a value of 9.7° nanocrystal tilt and the minimum peak

standard deviation (5°) to a value of 0° nanocrystal tilt. This mapping is shown in Fig. S8g-h. We

find that the change in nanocrystal tilt, determined by GIWAXS, follows closely with the rate of

the superlattice tilt, determined by GISAXS, leading to the conclusion that the nanocrystals and

superlattice maintain their relative orientation as the superlattice contracts.

In summary, we find that the total GIWAXS scattering counts from the 111NC and 200NC

reflections reach their maximum early on in the self-assembly process, indicating that the

{111}NC and {100}NC nanocrystal facets are aligned throughout the majority of the self-assembly

process as other changes to the superlattice are still happening on slower timescales. We find that

the width of both the 111NC and 200NC GIWAXS scattering peaks change in a way which is

commensurate with a uniform nanocrystal tilt. Specifically, the 200NC peak starts as two closely

spaced peaks which eventually combine into one while the 111NC peak does not show any

changes in width.


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6. Effect of mass transport limitations on the overall kinetics of self-assembly

We have observed an exponential time dependence for the major structural changes to the

superlattice, which we hypothesize to be related to the total concentration of the solvent (toluene)

within the matrix of nanocrystals. Our experimental measurements showed that the superlattice

has a single, well-defined structure at each time point during the structural rearrangement. Due to

the grazing-incident angle of the X-ray beam, the scattering pattern is collected from an area of

the film approximately 0.5mm x 23mm, and therefore the observed superlattice changes are

occurring at the same rate throughout the film. We believe this is an indication that the toluene

concentration in the nanocrystal matrix is spatially homogeneous at any given time within the

film. Because the nanocrystal film is thin (𝐿 ≤ 500 nm), the toluene diffusion resistance in the

nanocrystal matrix is small compared to the toluene transfer resistance across the vapor phase

boundary layer above the nanocrystal surface. This relationship is quantified through the Biot

number,

𝐵𝑖𝑚 =ℎ𝑚𝐿

𝐷𝑡:𝑛𝑐,

where hm is the mass transfer coefficient in the gas phase, L is the nanocrystal film thickness, and

Dt:nc is the diffusivity of toluene in the nanocrystal matrix. While we do not know the diffusivity

in the nanocrystal matrix, it is likely comparable to the diffusivity of toluene in semi-crystalline

polyethylene, 𝐷 ≈ 10−10 m2s−1 (Lützow et al. Polymer 40, 2797-2803 (1999)). Using an

estimated free convection mass transfer coefficient of ℎ𝑚 ≈ 10−4 m/s (Incropera, Fundamentals

of Heat and Mass Transfer, 7th Edition), the corresponding Biot number is 𝐵𝑖𝑚 ≤ 0.1. For Bim <

1, external mass transfer is limiting and the concentration of toluene in the nanocrystal matrix is

spatially homogeneous and decreases exponentially with time (Incropera, Fundamentals of Heat

and Mass Transfer, 7th Edition),

𝐶𝑡𝑜𝑙𝑢𝑒𝑛𝑒(𝑡) = 𝐶𝑖 ∙ exp(−𝐵𝑖𝑚 ∙ 𝐹𝑜𝑚)

𝐵𝑖𝑚 =ℎ𝑚𝐿

𝐷𝑡:𝑛𝑐= 𝐵𝑖𝑜𝑡 𝑛𝑢𝑚𝑏𝑒𝑟

𝐹𝑜𝑚 =𝐷𝑡:𝑛𝑐𝑡

𝐿2= 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝑛𝑢𝑚𝑏𝑒𝑟

where Ci is the initial concentration of toluene in the nanocrystal matrix and L is the thickness of

the nanocrystal film. This behavior is illustrated schematically in the figures below.


12


13

Supplementary Figures

Figure S1 | Electron microscopy characterization of nanocrystals. a, Transmission electron

micrograph of a monolayer of the PbS nanocrystals used in this study. b, High resolution

micrographs of single nanocrystals imaged looking at either the (111)NC face (top row) or

(100)NC face (bottom row).


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Figure S2 | Size and size dispersity analyses of nanocrystals. a, GISAXS pattern from a

colloidal suspension of the nanocrystals in toluene and the integrated pattern, which fits with a

spherical form factor of diameter 5.6 nm ± 4.4%. b, TEM image size analysis, performed using

ImageJ, yielding an average nanocrystal diameter of 5.1 nm ± 4.9%


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Figure S3 | GISAXS pattern indexing. An example of a GISAXS pattern (left) and a GISAXS

pattern which has been indexed (right), showing the excellent agreement between the

experimental peaks and the indices for a BCC superlattice with a = b = c = 9.05 nm and (110)SL

plane parallel to the substrate. This is the final time point measured during the in situ experiment.


16

Figure S4 | Indexing orientation relative to substrate. a, GISAXS patterns for the final BCC

superlattice overlaid with the expected scattering peaks (white dots) for BCC superlattices with

different planes parallel to the substrate plane. The (110)SL parallel case matches the

experimental scattering peaks. b, GIWAXS patterns for the final orientationally-aligned BCC

superlattice overlaid with the expected scattering peaks (white dots) for an atomic PbS crystal

with different planes parallel to the substrate plane. The (110)PbS parallel case matches the

experimental scattering peaks.


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Figure S5 | Temporal evolution of integrated GISAXS patterns. Azimuthally integrated

GISAXS data spanning the entire measurement time, showing the distinct changes from a

colloidal suspension (early times) to a well-ordered superlattice (late times).


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Figure S6 | Visualizing the FCC to BCC transition with unit cells. Simplified model showing

the transition between a, an FCC superlattice and b, a BCC superlattice. Two FCC unit cells are

shown in blue while the BCC unit cell is shown in red.


19

Figure S7 | Azimuthal integration of GIWAXS patterns. The GIWAXS pattern in a was

integrated over the area shown in b to give the integrated intensity as a function of scattering

vector q, shown in c. In d, we provide the XRD pattern for the same nanocrystals. The gray

vertical lines are the reference locations for bulk PbS.


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Figure S8 | Angle-dependent integration of GIWAXS patterns and data treatment to

extract meaningful parameters. a, Example of the angle integration along the scattering

locations for the 111NC and 200NC Bragg reflections in a colloidal suspension. b, Integrated

intensity as a function of angle, where the raw data is shown in the solid color, the linear fit is

shown in the dotted color, and the counts above this linear fit are shown in gray. For a colloidal

suspension, there is no preferential orientation of the atomic planes so the gray lines are near

zero (c and d show the same information but for the final BCC superlattice of the nanoparticles).

e, Peak counts based on the total counts in gray shown in b and c. g, Peak standard deviation

based on the width of the gray traces in b and c as functions of time for the 111NC and 200NC

peaks. f, h, Conversion of the parameters in e and g to more meaningful properties such as plane

alignment and nanocrystal tilt.


21

Figure S9 | Temporal evolution of angle integrated GIWAXS patterns. Angle integrated

GIWAXS patterns with time along the 111NC scattering direction (top) and along the 200NC

scattering direction (bottom). The data show that the 111NC peak standard deviation is constant

while the 200NC peak starts off as a double peak with large standard deviation and gradually

shifts to its final peak width.


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Figure S10 | Peak width comparison of angle integrated GIWAXS patterns. Angle

integrated GIWAXS patterns scaled to have a value of 1 at the final peak location (55° for the

111NC and 46° for the 200NC) and offset for clarity. The data from 13.5 to 20.1 minutes are

presented. The black vertical lines are guides for the eye.


23

Figure S11 | Characterizing nanocrystal tilt relative to the substrate by using low symmetry

atomic planes. The tilt is calculated based on the angle between the low symmetry plane and the

nanocrystal (110)NC plane.


24

Figure S12 | X-ray scattering patterns from a crystallographically aligned FCC

superlattice. GISAXS and GIWAXS patterns for a different batch of PbS nanocrystals than

those used in the main text (peak absorption = 1350 nm, diameter = 5.8 nm), which exhibit an

FCC superlattice with (111)SL parallel to the substrate and a GIWAXS pattern which indexes to

the (750)NC plane parallel to the substrate, or a ~9.5° nanocrystal tilt. The integrated GIWAXS

pattern clearly shows the double-peak nature of the 200NC peak. For reference, in gray we show

the 200NC integration for nanocrystals with a single 200NC peak which results when the (110)NC

planes are parallel to the substrate.


25

Figure S13 | Renderings of initial and final superlattice states. Real-space rendering of

nanocrystals for the a, initial FCC superlattice arrangement and b, final BCC superlattice

arrangement. The side views illustrate that having the (111)SL plane in a be parallel to the

substrate requires the nanocrystals to tilt by 9.7° as compared with the final structure in b.


26

Figure S14 | Renderings of final BCC superlattice state with unit cell axes. Real-space

rendering of two layers of the BCC nanocrystal superlattice as they sit on the glass substrate with

the (110)SL and (110)NC planes parallel to the substrate plane. a, Perspective view. b, Top view. c,

Side view.


27

Table S1. FCC superlattice unit cell parameters used for indexing the GISAXS patterns in time.

*note: s-factor is a parameter that accounts for film shrinkage relative to the surface normal of the substrate, which is typically

seen in spin coated and drop cast films. An s-factor of 0.94 represents a 6% shrinkage in the of the nanocrystal film normal to the

substrate plane.

time (minutes) a (nm) b (nm) c (nm) s-factor

12.6 12.8 12.8 12.80 0.97

12.9 12.8 12.8 12.80 0.97

13.2 12.8 12.8 12.80 0.97

13.5 12.8 12.8 12.80 0.97

13.8 12.8 12.8 12.10 0.97

14.1 12.8 12.8 11.50 0.97

14.4 12.8 12.8 11.10 0.97

14.7 12.8 12.8 10.85 0.97

15.0 12.8 12.8 10.55 0.97

15.3 12.8 12.8 10.30 0.97

15.6 12.8 12.8 10.15 0.97

15.9 12.8 12.8 9.95 0.97

16.2 12.8 12.8 9.80 0.97

16.5 12.8 12.8 9.70 0.97

16.8 12.8 12.8 9.60 0.97

17.1 12.8 12.8 9.50 0.97

17.4 12.8 12.8 9.45 0.97

17.7 12.8 12.8 9.40 0.97

18.0 12.8 12.8 9.35 0.97

18.3 12.8 12.8 9.30 0.97

18.6 12.8 12.8 9.25 0.97

18.9 12.8 12.8 9.18 0.97

19.2 12.8 12.8 9.18 0.97

19.5 12.8 12.8 9.15 0.97

19.8 12.8 12.8 9.15 0.97

20.1 12.8 12.8 9.10 0.97

20.4 12.8 12.8 9.10 0.97

20.7 12.8 12.8 9.05 0.97

21.0 12.8 12.8 9.05 0.97

21.3 12.8 12.8 9.05 0.96

21.6 12.8 12.8 9.05 0.96

21.9 12.8 12.8 9.05 0.96

22.2 12.8 12.8 9.05 0.96

22.5 12.8 12.8 9.05 0.96

22.8 12.8 12.8 9.05 0.96

23.1 12.8 12.8 9.05 0.96

23.4 12.8 12.8 9.05 0.96

23.7 12.8 12.8 9.05 0.96

24.0 12.8 12.8 9.05 0.96

24.3 12.8 12.8 9.05 0.95

24.6 12.8 12.8 9.05 0.95

24.9 12.8 12.8 9.05 0.95

25.2 12.8 12.8 9.05 0.95

25.5 12.8 12.8 9.05 0.95

25.8 12.8 12.8 9.05 0.95

26.1 12.8 12.8 9.05 0.95

26.4 12.8 12.8 9.05 0.95

26.7 12.8 12.8 9.05 0.95

27.0 12.8 12.8 9.05 0.95

27.3 12.8 12.8 9.05 0.94

27.6 12.8 12.8 9.05 0.94

27.9 12.8 12.8 9.05 0.94

28.2 12.8 12.8 9.05 0.94

28.5 12.8 12.8 9.05 0.94

28.8 12.8 12.8 9.05 0.94

29.1 12.8 12.8 9.05 0.94

29.4 12.8 12.8 9.05 0.94

29.7 12.8 12.8 9.05 0.94

30.0 12.8 12.8 9.05 0.94

30.3 12.8 12.8 9.05 0.94

30.6 12.8 12.8 9.05 0.94


Documents

Kinetics of the self-assembly of nanocrystal Kinetics of ... · 1 Kinetics of the self-assembly of nanocrystal superlattices measured by real-time in situ X-ray scattering Mark C