#  Liquid phase fast electron tomography unravels the true 3D structure of
colloidal assemblies

Daniel Arenas Esteban  Da Wang  Guangdong Provincial Key Laboratory of Optical
Information Materials and Technology, Institute of Electronic Paper Displays,
South China Academy of Advanced Optoelectronics, South China Normal
University, Guangzhou 510006, China.  Ajinkya Kadu  Centrum Wiskunde &
Informatica (CWI), Amsterdam, The Netherlands.  Noa Olluyn  Ana Sánchez
Iglesias  Centro de Investigación Biomédica en Red, Bioingeniería,
Biomateriales y Nanomedicina, CIBER-BBN, Paseo de Miramon 182, 20009 Donostia-
San Sebastián, Spain.  Alejandro Gomez-Perez  Jesús Gonzalez Casablanca
Stavros Nicolopoulos  Luis M. Liz-Marzán  Centro de Investigación Biomédica en
Red, Bioingeniería, Biomateriales y Nanomedicina, CIBER-BBN, Paseo de Miramon
182, 20009 Donostia-San Sebastián, Spain.  Ikerbasque, Basque Foundation for
Science, 48013 Bilbao, Spain.  Cinbio, Universidade de Vigo, 36310 Vigo,
Spain.  Sara Bals

###### Abstract

Electron tomography has become a commonly used tool to investigate the three-
dimensional (3D) structure of nanomaterials, including colloidal nanoparticle
assemblies. However, electron microscopy is typically carried out under high
vacuum conditions. Therefore, pre-treatment sample preparation is needed for
assemblies obtained by (wet) colloid chemistry methods, including solvent
evaporation and deposition on a solid TEM support. As a result of this
procedure, changes are consistently imposed on the actual nanoparticle
organization. Therefore, we propose herein the application of electron
tomography of nanoparticle assemblies while in their original colloidal liquid
environment. To address the challenges related to electron tomography in
liquid, we devised a method that combines fast data acquisition in a
commercial liquid-TEM cell, with a dedicated alignment and reconstruction
workflow. We present the application of this method to two different systems,
which exemplify the difference between conventional and liquid tomography,
depending on the nature of the protecting ligands. 3D reconstructions of
assemblies comprising polystyrene-capped Au nanoparticles encapsulated in
polymeric shells revealed less compact and more distorted configurations for
experiments performed in a liquid medium compared to their dried counterparts.
On the other hand, quantitative analysis of the surface-to-surface distance of
self-assembled Au nanorods in water agrees with previously reported dimensions
of the ligand layers surrounding the nanorods, which are in much closer
contact when in similar but dried assemblies. This study, therefore,
emphasizes the importance of developing high-resolution characterization tools
that preserve the native environment of colloidal nanostructures.

## 1 Introduction

The most characteristic feature of nanomaterials is the stark dependence of
their properties on the size and shape of the nanostructured material.
However, manipulation of the properties of nanomaterials can also be achieved
by tuning interparticle distance and relative orientation[Guerrero-Martínez et
al., 2012]. In this context, a wide variety of techniques have been devised
toward obtaining nanostructured materials with well-defined dimensions and
interparticle arrangements. Although top-down methods, such as e-beam
lithography, can be used to design nanostructures with high precision, these
are typically limited to two dimensions (2D). On the other hand, bottom-up
strategies based on colloid science can be employed to obtain 2D or 3D
assemblies comprising combinations of (equal or dissimilar) nanosized
particles, with distinct properties determined by the size, shape, and
arrangement of the constituting elements[Glotzer and Solomon, 2007, Boles et
al., 2016]. The formation and behavior of such assemblies are governed by
interaction forces between nanoparticles (NPs), typically mediated by surface
charge, ligands, and the solvent. As such, colloids have long been used as
model systems for investigating fundamental phenomena in soft condensed
matter, such as nucleation and phase transitions. In the context of nanoscale
materials, assemblies additionally play a key role in shaping functional
(meta)materials at various scales.

Therefore, understanding the formation mechanisms and structure-determined
properties of colloidal assemblies requires quantitative 3D structural
characterization, including measurements of interparticle distances and
packing[Wang et al., 2021a, Wang et al., 2018, Wang et al., 2022]. Although
bulk scattering methods have often been employed with great success and high
precision, they can only provide average information over a huge number of
individual particles or clusters thereof. Accurate and comprehensive
information can be obtained by studying individual nanostructures in 3D, for
which electron tomography (ET) in scanning transmission electron microscopy
(STEM) has become an essential tool[Miao et al., 2016, Midgley and Dunin-
Borkowski, 2009, Altantzis et al., 2021, Bals et al., 2014]. Apart from
providing detailed reconstruction images (or movies), recent improvements have
been made to ET, both related to the acquisition and to the reconstruction
process, which, e.g., enable us to extract the positions of individual
nanoparticles, even in very large and/or dense nanoassemblies, where missing
wedge and streaking artifacts are likely to hinder relevant features[Zanaga et
al., 2016, Kavak et al., 2023].

However, all these investigations have been performed under conventional
conditions in a TEM, including ultra-high vacuum. Samples for ET are therefore
typically prepared by dropcasting the colloidal dispersion on a TEM grid. A
problem that has often been overlooked during this process is related to the
presence of soft materials within colloidal assemblies, such as ligands and
polymers[Lyu et al., 2023]. As a result, the drying process may result in
deformations of the assemblies, either by evaporation of the remaining solvent
or by contact with the support (grid), thereby altering their original 3D
configuration[Marchetti et al., 2023]. To mitigate this effect, cryogenic
electron tomography (cryo-TEM) and freeze-drying have been used, but the
experimental environment can still lead to subtle changes in the 3D
structures[Kumar et al., 2018, De Nijs et al., 2015, Zanaga et al., 2016, Wang
et al., 2021b]. It is therefore important to develop 3D characterization
approaches based on ET that enable the investigation of colloidal assemblies
in their natural environments, such as water or a different solvent.

Recent advancements in liquid cell electron microscopy have yielded new
insights into nanomaterial dynamics and structure in liquid environments[de
Jonge et al., 2019, Ross, 2015, De Yoreo and NAJM, 2016]. Initial attempts
utilized amorphous silicon nitride ($Si_{x}N_{y}$) microfluidic chambers as
liquid cells (LCs), but the holders based on such chambers often have a
limited tilt range (restricted to $\pm 30^{\circ}$). For ET, where a
sufficient angular sampling is desired, such a limited tilt range can cause
missing wedge artifacts, compromising reconstruction accuracy. Additionally,
the presence of the liquid layer and relatively thick $Si_{x}N_{y}$ windows
frequently reduces the signal-to-noise ratio (SNR) in the tilt-series
projection images, especially at higher angles where the total effective
thickness increases. To address these limitations, graphene liquid cells
(GLCs) have been employed to enhance SNR while allowing the study of
nanomaterial growth, self-assembly, and dynamics[Park et al., 2021]. Using the
3D SINGLE methodology, Park et al. demonstrated that the 3D structures of NPs
in GLC can be characterized by observing their translational and rotational
motions in a liquid environment[Park et al., 2015]. Using advanced
reconstruction algorithms, the structural disparities between individual NPs
could be discerned, even at the atomic scale[Kim et al., 2020]. However, the
spatial constraints of the GLC (commonly up to 100 nm) make the single-
particle method sub-optimal for 3D characterization of significantly larger
colloidal assemblies[Yang et al., 2019, Keskin et al., 2021].

We present herein an advanced Liquid-Phase (LP) fast electron tomography
workflow to characterize 3D structures of colloidal assemblies in their native
environments. This approach applies fast electron tomography[Albrecht and
Bals, 2020, Vanrompay et al., 2021, Koneti et al., 2019], a recently proposed
technique to significantly reduce the acquisition time for ET tilt series,
using a commercially available LC chip. To overcome experimental challenges
such as limited tilt range, image distortion, environmental background noise,
and potential intra-LC sample movement, advanced image processing techniques
and a dedicated reconstruction algorithm are proposed[Batenburg and Sijbers,
2011, Donoho, 2006], as well as the use of a novel prototype LC-chip
(Tomochip) that allows a higher tilt angle. To illustrate the importance of 3D
characterization by LP fast electron tomography, we investigated the structure
of colloidal clusters comprising hydrophobic Au NPs surrounded by a block-
copolymer shell that provides colloidal stability in water. Our analyses
reveal subtle structural differences when comparing colloidal clusters studied
in the liquid phase and in vacuum (i.e., dried state). As a further
demonstration of the importance of characterizing nanoassemblies in their
native environment, we studied bilayer assemblies of cetyltrimethylammonium
bromide (CTAB)-stabilized Au nanorods (NRs) in water, revealing the surface-
to-surface distance between Au NRs in water, in agreement with literature
values[Gomez-Grana et al., 2012]. These in situ 3D measurements are in
contrast to observations of NR assemblies characterized in a dried state and
for which significantly smaller distances were determined. Our results
therefore illustrate the importance of performing 3D characterization of NP
assemblies in a liquid environment. Based on such advancements, a more
comprehensive and accurate 3D analysis of colloidal assemblies in their native
conditions becomes possible.

## 2 Results

###  2.1 Challenges in 3D structural characterization of colloidal clusters by
electron tomography

As a model system for colloidal assemblies, we followed our previous
work[Sánchez-Iglesias et al., 2012], in which the self-assembly of polystyrene
(PS)-capped Au NPs was induced by adding water to a dispersion of the
(hydrophobic) Au-PS NPs in tetrahydrofuran (THF), and subsequently stabilizing
the obtained NP clusters by further addition of a polystyrene-b-polyacrylic
acid (PSS-PAA) block copolymer. Whereas the PS block interdigitates with the
PS ligands on Au NPs, the PAA block allows redispersion of the protected
(hydrophilic) assemblies in water. It should be noted that, aiming to enhance
the interdigitation of PS chains between the NPs inside the cluster, slight
heating was applied to help expel the remaining THF. Therefore, the NP
clusters redispersed in water are expected to be compact and allow minimum
internal dynamics. However, this hypothesis could not be tested by standard ET
in vacuum because sample preparation would lead to the complete evaporation of
any remaining solvent. This effect is likely to increase PS chain
interdigitation, thereby further reducing interparticle distance. As a result,
our reported tomography reconstructions typically showed a highly regular
organization of Au NPs, with interparticle distances regulated by the
dimension (molecular weight) of the PS ligands[Galván-Moya et al., 2014].

For high-angle annular dark-field (HAADF)-STEM tomography experiments, we
prepared colloidal clusters made of 12 nm Au NPs, with an overall average
cluster diameter of 80 nm (Fig. 1a; synthesis details are provided in the
Methods section). We first applied conventional ET in vacuum to Au@PS
clusters. For conventional ET, selected clusters comprising 4, 5, or 6 NPs,
and encapsulated within polymer shells, were thoroughly dried on a TEM grid
and imaged in vacuum. We noted that the colloidal clusters settled onto the
TEM grid upon drying, which resulted in a slightly deformed or flattened
structure, evident from 2D TEM projections at high tilt angles (Fig. 1b). The
2D projections from the tilt series suggest Au NP stacking into polyhedral
structures, e.g., tetrahedra for clusters with 4 NPs (Fig. 1b and
Supplementary Movie 1). Our analysis also showed a 2-3 nm reduction in the
overall size of the colloidal clusters post conventional ET experiments (Fig.
1c; Supplementary Fig. S1a-c), suggesting that the electron beam has a
significant impact on the structure, in turn posing additional challenges when
attempting to obtain accurate 3D reconstructions using conventional methods
(Supplementary Notes).

To mitigate electron beam damage during tilt series acquisition, we employed
the fast electron tomography method, maintaining a relatively low electron
dose per frame ($\sim 0.46~{}\text{e}^{-}/\text{\r{A}}^{2}$; see Methods
section for more details). In this approach, focusing and tracking are
executed concurrently while the sample is continuously tilted[Albrecht and
Bals, 2020, Koneti et al., 2019]. We devised advanced image processing and
alignment techniques, coupled with a reconstruction algorithm, to address
distortions from continuous tilting and the challenges of a low electron dose.
Further details are provided in the Methods section. The 3D reconstructions of
colloidal clusters comprising 4, 5, and 6 Au NPs distinctly showcased
tetrahedral, trigonal bipyramidal, and octahedral arrangements (Fig. 1d and
Supplementary Movie 2). Post fast electron tomography, although the overall
size of the colloidal clusters remained consistent, flattening of the
polymeric shell was observed (Fig. 1d and Supplementary Movie 2). It is
therefore likely that the capillary forces exerted during sample drying
affected the polymeric shell structures, again posing challenges for precise
3D reconstruction. Consequently, these findings underscore the need to
investigate the 3D arrangement of Au NPs in colloidal clusters in their native
environment.

Drawing inspiration from the 3D SINGLE methodology developed by Park et
al.[Park et al., 2015, Kim et al., 2020]. we aimed to achieve 3D
reconstruction by tracking the Brownian motion of NP clusters in a liquid
phase. Given the spatial constraints within the GLC, we housed the colloidal
cluster dispersion in the $Si_{x}N_{y}$ chamber of a commercial liquid TEM
holder (Supplementary Notes). After initiating flow within the commercial LC
TEM holder, we attempted to capture the dynamics of the colloidal clusters
(Supplementary Fig. S2a). Contrary to the expected translational motion due to
liquid flow, the colloidal clusters exhibited only minimal degree of rotation
within the LC chamber, irrespective of the flow rate. As a result, the
acquired angular samplings were inadequate for tomographic reconstruction.
Additionally, we noted partial aggregation, which may be attributed to
degradation of the protective polymer shell under electron beam exposure
(Supplementary Fig. S2b-d; Supplementary Movie 3). In summary, the reliability
of tomographic reconstructions for colloidal clusters in a liquid setting is
compromised by challenges related to limited angular projections and electron
beam-induced damage. We therefore developed an optimized workflow for the
acquisition, alignment, and reconstruction of accurate 3D representations of
colloidal NP clusters, which we present in the following sections.

Figure 1: Challenges in 3D characterization of colloidal clusters by electron
tomography. a) HAADF-STEM image showing an overview of Au@PS colloidal
clusters in vacuum, where the polymer shell can be observed as a grey shadow
around the bright NPs. b) Observation of the flattening effect of a colloidal
cluster at a high-tilting angle. c) 2D HAADF-STEM images of a colloidal
cluster before (top) and after (bottom) conventional electron tomography tilt
series acquisition, indicating volume change. See also Supplementary Movie 1.
d) 3D reconstructions of colloidal clusters containing 4, 5, and 6 Au NPs via
fast electron tomography in vacuum. The stacking of Au NPs within the
polymeric shells resembles a tetrahedron, a trigonal bipyramid, and an
octahedron, respectively. See also Supplementary Movie 2.

###  2.2 Fast tilt series acquisition for LP fast electron tomography

To increase the angular sampling during the collection of 2D projection images
and thereby minimize electron beam-induced structural damage in a liquid
environment, we performed fast electron tomography in a LC. We employed a
commercial monolithic LC (K-Kit from Bio MA-TEK) container with a window gap
of $0.5~{}\mu m$ (Fig. 2a). This device enables the structural
characterization of samples within a liquid environment and can be mounted
onto a standard single-tilt holder (Fig. 2b, c). In comparison to commercially
available LCs with a limited inclination angular range (usually not more than
$\pm 30\degree$), our setup reaches a slightly extended total tilt range of
approximately $\ 90\degree$ (i.e., $\pm 45\degree$ from the central axis). To
minimize beam damage during the tilt series acquisition, we adopted the fast
electron tomography acquisition methodology that we previously used in vacuum
conditions, as described above[Albrecht and Bals, 2020]. The electron doses
per frame were set as $0.46~{}\text{e}^{-}/\text{\r{A}}^{2}$ and
$2.31~{}\text{e}^{-}/\text{\r{A}}^{2}$ for liquid and vacuum conditions,
respectively (Fig. 2d). Due to the reduced beam current employed to preserve
the sample integrity and the limited tilting range in liquid, the total
electron dose of LP fast electron tomography was lower by one order of
magnitude, with values of $71~{}\text{e}^{-}/\text{\r{A}}^{2}$ and
$787~{}\text{e}^{-}/\text{\r{A}}^{2}$ for tilt series acquisition in liquid
and in vacuum, respectively (Fig. 2e). Importantly, no significant changes in
the interparticle distances between Au NPs within the colloidal clusters were
observed after fast tilt series acquisition (Supplementary Fig. S3a-f).

###  2.3 Optimization of tilt series denoising and alignment for LP fast
electron tomography

Mechanical movements of the goniometer during fast tilt series acquisition can
lead to scanning distortions like streaking artifacts in the final 3D
reconstruction, a phenomenon particularly evident in STEM mode[Vanrompay et
al., 2021]. Factors such as low-dose imaging, inherent distortions, solvent
presence, and the relatively thick $Si_{x}N_{y}$ window of the LC can
adversely affect the SNR of the raw tilt series (Fig. 2f and Supplementary
Movie 4).

In response to these challenges, we applied an advanced image processing and
alignment approach (Fig. 2f-i; see the Methods section for details). We began
with a self-supervised denoising technique utilizing convolutional
autoencoders (CAE)[Gondara, 2016] (Fig. 2g; Supplementary Fig. S4). This
technique exploits the inherent sequential patterns present in tilt series
images, which essentially are multiple representations of the object of
interest from different angles. By utilizing this redundancy, the method
effectively improves the SNR while retaining crucial structural details
(Supplementary Fig. S5). Following denoising, our iterative workflow consists
of three stages to refine the tilt series. In the first stage, robust
principal component analysis (RPCA)[Candès et al., 2011] (Fig. 2h) was applied
to detect and eliminate distortions from the tilt series. At its core, RPCA
decomposes the tilt series into two distinct matrices: a low-rank matrix and a
sparse matrix. The low-rank matrix encapsulates the dominant, consistent
features of the data, representing the underlying structure of the material.
In contrast, the sparse matrix pinpoints irregularities or distortions, often
arising from various sources during data acquisition. By isolating these
anomalies, RPCA enhances the fidelity of the tilt series, facilitating
improved registration, alignment, and 3D reconstruction. Next, we registered
the tilt series projections using the iterative closest point (ICP) method, as
depicted in Fig. 2h[Zhang et al., 2021]. The ICP method stands out from
conventional algorithms due to its iterative approach, to minimize the
difference between two clouds of points (computed from given tilt series and
their low-rank RPCA component), making it particularly adept at handling the
HAADF-STEM images. This iterative refinement ensures that even minor shifts or
rotations that occur during image acquisition are accounted for. The final
stage of each iteration focuses on aligning the tilt-axis for the tilt series.
Accurate tilt-axis alignment is crucial because it ensures that the 3D
reconstruction accurately represents the original structure without
introducing artifacts. Misalignment can lead to distortions in the
reconstructed volume, compromising the integrity of the analysis. This step
involves comparing the tilt series to forward projections from an initial 3D
reconstruction obtained using Filtered Back Projection (FBP), as shown in Fig.
2h[Houben et al., 2011]. The effectiveness of this step stems from its ability
to iteratively refine the alignment by leveraging the consistency in forward
projections, ensuring that each subsequent iteration brings the tilt series
closer to the true structural representation. This three-step procedure is
repeated until sufficient convergence is obtained. At each iteration, the
alignment of the tilt series images with their respective RPCA components is
progressively refined, ensuring peak registration and alignment by the fifth
cycle (Supplementary Fig. S6). The outcome is a finely aligned tilt series, as
illustrated in Fig. 2i and Supplementary Movie 5.

Figure 2: Liquid-phase fast electron tomography. a) Schematic illustration of
a K-Kit LC used for experimental investigations, highlighting the LC
dimensions. b-c) Optical micrographs of a K-Kit loaded on a single-tilt
tomography holder, with, b) $0\degree$ and c) $45\degree$ tilting view,
respectively. d) The LP fast electron tomography tilt series acquisition
method continuously tilts the sample while recording projection images of the
sample. e) Comparison of time and electron dose required for acquiring a
complete tilt series using fast electron tomography in liquid and vacuum and
conventional electron tomography in vacuum, highlighting the electron-beam
efficiency of the fast electron tomography on liquid. f-i) fast electron
tomography tilt series pre-processing workflow. f) Representation of the raw
tilt series stack with a sample image from the raw stack displayed. g)
Illustration of the self-supervised denoising using CAE. A sample image from
the denoised stack is displayed, demonstrating the effectiveness of the
autoencoder denoising compared to the original one. h) Schematic overview of
iterative process undertaken: refining the tilt series using RPCA, followed by
rigid registration using the ICP method, and then tilt-axis alignment via FBP.
i) The final processed stack, which is refined, aligned, and denoised, with a
representative image displayed for clarity.

####  2.3.1 Advanced 3D reconstruction algorithm for LP fast electron
tomography

After pre-processing to mitigate distortions, misalignments, and noise, as
mentioned above, the challenge of the missing wedge due to a limited angular
range needs to be addressed. Therefore, we devised an advanced 3D
reconstruction algorithm. While conventional reconstruction algorithms, such
as the simultaneous iterative reconstruction technique (SIRT)[Gilbert, 1972],
maximum-likelihood expectation-maximization (ML-EM)[Moon, 1996], and total-
variation minimization (TVM)[Goris et al., 2012], fall short in addressing
these challenges, the discrete algebraic reconstruction technique
(DART)[Batenburg and Sijbers, 2011] has been widely adopted in electron
tomography. DART capitalizes on the premise that materials are distinct and
maintain constant intensity. Building on this foundation, we introduced an
enhanced algorithm: Compressed-Sensing DART (CS-DART). CS-DART, an evolution
of the standard DART, introduces additional refinements by integrating
compressed sensing principles, thereby enhancing its ability to reconstruct
images with greater accuracy, especially in scenarios with limited data, such
as the missing wedge challenge in LP fast electron tomography. This method
incorporates a shape smoothness prior, as detailed in the Methods section.
Colloidal clusters, made up of Au NPs and polymeric shells, exhibit two
distinct grey values in line with the DART reconstruction criteria. We
hypothesize that these components exhibit smooth geometries, which play a
crucial role in reducing the remaining noise and alignment errors during
reconstruction (Supplementary Fig. S7). Once the 3D reconstruction with CS-
DART is complete, we quantify the structural characteristics of the NP-formed
polyhedra using quantitative descriptors such as interparticle distance,
surface area, volume, and regularity index (see Supplementary Notes for
details).

###  2.4 Quantitative 3D analysis for colloidal clusters in liquid phase

Our study primarily focused on the 3D arrangement of Au NPs within polymeric
shells. Specifically, we observed that clusters containing 4, 5, and 6 Au NPs
predominantly formed tetrahedral, trigonal bipyramidal, and octahedral
structures, respectively (Fig. 3a; Supplementary Movies 6-8). However,
reconstructing the polymeric shells proved challenging due to the low
Z-contrast arising from the water layer and the $Si_{x}N_{y}$ membrane of the
LC chip.

To delve deeper into the structural nuances influenced by the environment, we
compared the polyhedral structures obtained from fast electron tomography
reconstructions in both vacuum and liquid settings. The radar charts in Fig.
3b-d visually contrast four quantitative descriptors (mean interparticle
distance, volume, surface area, and regularity index) for the polyhedra formed
in vacuum versus liquid across three assemblies (N = 4, 5, 6) (Table 1;
Supplementary Movies 6-8). Our quantitative analysis revealed that in a liquid
environment, the average interparticle distance for clusters containing 4, 5,
and 6 Au NPs was expanded by $13\%$, $10\%$, and $15\%$, respectively,
compared to their vacuum counterparts. Similarly, the surface area and volume
of these polyhedra in liquid were larger by varying percentages, indicating a
more spacious arrangement of the Au NPs within the polymeric shells in liquid.

To further understand the structural regularity, we introduce a regularity
index, quantifying the similarity between an experimentally assembled 3D
cluster formed by the Au NPs in different environments and an idealized,
regular polyhedron (Table 1). In a vacuum condition, the arrangement of Au NPs
more closely resembles regular and compact polyhedra than in a liquid
environment (Fig. 3). This suggests that capillary forces acting on the
polymeric shells during drying compress the Au NPs into regular
configurations. We observed a more pronounced 3D structural disparity between
clusters in vacuum and liquid, with increasing numbers of Au NPs. This may be
attributed to the tetrahedron (N = 4) having the highest packing fraction and
thus, the least free volume among the studied polyhedra. As the number of Au
NPs increases, they likely have increased mobility within the polymeric shell
when dispersed in water. This disparity in structures between the liquid phase
and vacuum is likely attributed to the presence of remaining solvent (THF)
within the clusters when still in the liquid phase, which is removed during
drying in vacuum, additionally leading to deformation of the polymeric shell
by capillary forces during the same drying process prior to ET.

In conclusion, our findings emphasize the significant influence of the
experimental environment on the structural characterization of colloidal
clusters. We advocate for electron tomography in a liquid environment as it
avoids capillary forces, offering a more authentic and representative 3D
structural characterization.

Figure 3: Quantitative structural comparison between 3D reconstructions of colloidal clusters with different numbers of particles, implemented in liquid and vacuum conditions. a) Polyhedra computed from the centroid positions of the Au NPs obtained through CS-DART reconstructions of three clusters containing N = 4, 5, 6 Au NPs in a liquid environment. b-d) Quantitative normalized structural comparison between polyhedra formed by the stacking of Au NP obtained from fast electron tomography in vacuum (depicted in orange) and liquid (depicted in blue), including mean interparticle distance (Mean ID), surface area, volume, and regularity index for b) N = 4, c) N = 5, and d) N = 6, respectively. Importantly, the Au NPs demonstrated a notable tendency to adopt regular but more condensed configurations when observed in a vacuum environment. N | mean ID (nm) | SA ($10^{3}$ nm2) | Volume ($10^{3}$ nm3) | RI
---|---|---|---|---
| Vacuum | Liquid | Vacuum | Liquid | Vacuum | Liquid | Vacuum | Liquid
4 | 26.51 |  29.96 ($+13\%$) | 1.31 |  1.54 ($+15\%$) | 2.42 |  3.07 ($+21\%$) | 7.27 |  6.56 ($-10\%$)
5 | 29.21 |  32.16 ($+10\%$) | 2.17 |  2.41 ($+10\%$) | 6.23 |  6.79 ($+8\%$) | 5.56 |  11.46 ($+52\%$)
6 | 28.88 |  33.33 ($+15\%$) | 2.66 |  3.18 ($+16\%$) | 9.96 |  12.29 ($+19\%$) | 3.44 |  16.63 ($+79\%$)
Table 1: Characteristics of Au NP-formed polyhedra in vacuum versus liquid
conditions. The table presents metrics, such as the mean interparticle
distance (Mean ID), surface area (SA), volume, and regularity index (RI) of
the polyhedrons derived from CS-DART reconstructions, highlighting the
influence of imaging conditions on the reconstructed structures. The RI
indicates the degree of regularity in the arrangement of the Au NPs, with a
smaller value suggesting the arrangement is closer to a regular polyhedron.

###  2.5 Characterizing bilayer assemblies of Au NRs in liquid phase

Apart from colloidal assemblies, NPs are often organized on solid substrates.
A well-known example is the organization of Au NRs for exploiting their unique
(and anisotropic) localized surface plasmon resonance (LSPR) properties, e.g.,
in sensing based on surface-enhanced Raman scattering (SERS)[Langer et al.,
2019]. It has been shown that the formation of ordered Au NR multilayers can
lead to highly efficient SERS substrates[Solís et al., 2017, Hamon et al.,
2015]. However, the excitation of individual vs. collective LSPR modes in Au
NR assemblies not only depends on their degree of organization but also (and
very strongly) on the interparticle distances within the assembly. Again,
usual practice involves TEM or SEM (rarely ET) on dry samples in vacuum, which
is likely to affect both the structure and the interparticle distance. In this
context, we decided to investigate bilayers of self-assembled CTAB-coated
single crystalline Au NRs. For measurements in liquid, we used a Tomochip,
i.e., a modified LC chip based on the monolithic K-Kit LC design[Das et al.,
2022, Gonzalez Casablanca et al., 2020]. Although this LC has a smaller window
gap of 100 nm, it allows achieving a significantly larger tilt range (up to
$\pm 70\degree$; Supplementary Fig. S8a-c). This configuration not only
ensures a thinner liquid layer but also improves the SNR and angular coverage
for tilt series in liquid conditions, minimizing the increase of the effective
thickness at higher angles. We additionally dropcasted a dispersion of the
same Au NRs on a TEM grid and dried it for structural analysis in vacuum.

We conducted fast electron tomography experiments in both vacuum and liquid
environments, examining self-assembled bilayers of Au NRs, either dried on a
TEM grid or encapsulated within the Tomochip (Fig. 4a,b; Supplementary Movie
9). The improved angular coverage and SNR facilitated the use of the
conventional ML-EM algorithm[Moon, 1996] to achieve precise 3D reconstructions
in both settings. These reconstructions clearly allowed us to observe well-
defined rod shapes for all Au NRs in either environment, organized in an AB-
stacking pattern (Fig. 4c-f; Supplementary Movie 10). To further understand
the environmental influence on Au NR stacking, we focused on the central
region of the assemblies and measured the diameter of the Au NRs from 2D
projections of the tilt series in vacuum and liquid, separately, ensuring our
analysis was not affected by edge distortions (Supplementary Fig. S9a-b). We
then calculated the surface-to-surface distance by subtracting the radii of
the Au NRs from the distance between their centers of mass (Supplementary Fig.
S9c-d, Supplementary Table S1). This mode of examination highlighted clear
differences in surface-to-surface distances of both settings: 2-4 nm in vacuum
vs. 6-8 nm in liquid (Fig. 4e-f). The length of a fully stretched CTAB
molecule is around 2.2 nm[Weidemaier et al., 1997]. The surface-to-surface
distance in liquid environment corresponds to almost four times the length of
a CTAB molecule, where the two adjacent rods are expected to share a layer of
counterions[Sau and Murphy, 2005]. We remark that, although the expected CTAB
layer thickness on Au NRs has been characterized as approximately 3.2
nm[Gomez-Grana et al., 2012, Meena and Sulpizi, 2013], its precise structure —
whether interdigitated bilayer or isolated micelles — is still under debate,
in particular considering the influence of the experimental
environment[Mosquera et al., 2023]. In contrast to the observation in liquid
environment, the shorter surface-to-surface distance between NRs measured in
vacuum indicates compression of the CTAB ligands. This is likely due to
plastic deformation from capillary forces during sample drying, high vacuum
conditions, or a combination of both. Our results highlight how LP fast
electron tomography preserves the 3D structures of self-assembled Au NRs,
avoiding distortions from capillary forces, in contrast with previous reports.

Figure 4: Quantitative structural comparison of Au NR bilayer assemblies in
vacuum and liquid-phase. 2D HAADF-STEM images of self-assembled Au NR
superlattices measured in a) vacuum and b) liquid. c,d) 3D reconstructions and
e,f) orthogonal views of superlattices assembled from two layers of Au NRs in
c,e) vacuum and d,f) liquid. Insets of panels e) and f): zoomed-in views of
two adjacent Au NRs, showing the disparity between surface-to-surface
distances in vacuum and in liquid environment. Note that the transparency of
the 3D renderings was increased for visual clarity.

## 3 Discussions

In this work, we have developed a methodology to characterize colloidal
assemblies in 3D, in their native liquid environment. We addressed technical
challenges, predominantly associated with commercial LCs, such as self-
rotation and restricted tilt range. This was achieved through the use of
advanced image processing algorithms, along with the development of the
dedicated CS-DART 3D reconstruction algorithm. Moreover, we demonstrated that
designing specialized LC devices specifically for ET can effectively mitigate
the constraints of a limited tilt range. This approach has the potential to
reduce the computational load required for precise data analysis and
characterization. We successfully applied our approach to explore the 3D
structural characterization of Au@PS colloidal clusters in their native liquid
environment, contrasting these findings with structures observed in vacuum.
Notably, we identified that capillary forces during the drying phase in vacuum
conditions induced structural compression, leading to a more compact
arrangement of Au NPs. Conversely, in liquid conditions, the NPs adopted a
more expansive polyhedral shape. These structural variations can significantly
influence the assemblies’ optoelectronic properties. Furthermore, we adapted
our technique to study CTAB-stabilized Au NRs in bilayer self-assembly using a
specialized Tomochip for 3D liquid characterization. In vacuum, the surface-
to-surface distances of these bilayers are shorter, indicative of capillary
effects. Yet, in a liquid setting, these distances align with both bulk
measurements and theoretical predictions, emphasizing the importance of the
experimental environment in electron microscopy studies.

Our study underscores the importance of analyzing colloidal assemblies in
conditions that closely resemble their natural or application-specific
environments. Analyzing them in non-native settings can introduce structural
changes, potentially affecting our understanding of the assemblies’ true
nature and properties. This information is crucial when adjusting these
assemblies, especially in areas where specific changes, such as plasmonic
effects, are desired. Moreover, we expect that our methodology can be applied
to study the structure of a broad range of colloidal assemblies, giving
insight into the structure formation mechanisms and structure-property
relations of nanomaterials. While we have achieved notable progress, refining
the structural analysis of colloidal assemblies remains an ongoing effort.
Upcoming advancements, such as improved Tomochips and low-dose TEM
techniques[Lazić et al., 2016, Yu et al., 2022], aim to boost both precision
and range in characterizations, proving essential for studying diverse
assemblies, from varied sizes to those with low atomic numbers.

## 4 Methods

###  4.1 Chemicals

Gold (III) chloride hydrate (HAuCl4, $\geq 99.9\%$),
hexadecyltrimethylammonium bromide (CTAB, $\geq 99\%$), cetyltrimethylammonium
chloride solution (CTAC, 25 wt. % in H2O), sodium borohydride (NaBH4, $\geq
96\%$), L-ascorbic acid (AA, $\geq 99\%$), silver nitrate (AgNO3, $\geq
99\%$), sodium oleate (NaOL, $\geq 99\%$), hydrochloric acid (HCl, $37\%$) and
tetrahydrofuran (THF, $\geq 99\%$) were purchased from Sigma-Aldrich-Merck.
Thiol-terminated polystyrene (PS509-SH, MW: 53K) and poly (styrene-b-acrylic
acid) (PS403-b-PAA62) were purchased from Polymer Source. All chemicals were
used without further purification. Milli-Q water (resistivity
$18.2\,\text{M}\Omega\cdot\text{cm}$ at $25^{\circ}\text{C}$) was used in all
experiments. All glassware was washed with aqua regia, rinsed with Milli-Q
water, and dried before use.

###  4.2 Preparation of Au@PS colloidal clusters

Gold seeds ($\sim 1.5~{}nm$) were prepared by fast reduction of HAuCl4
($5~{}mL$, $0.25~{}mM$) with freshly prepared NaBH4 ($0.3~{}mL$, $10~{}mM$) in
aqueous CTAB solution ($100~{}mM$), under vigorous stirring[Zheng et al.,
2014]. The solution color changed from yellow to brownish yellow and the seed
solution was aged at $27^{\circ}\text{C}$ for $30~{}min$ before use, to
promote the decomposition of sodium borohydride. The seed was used for the
growth of gold nanospheres and nanorods. An aliquot of gold seed solution
($0.6~{}mL$) was added under vigorous stirring to a growth solution containing
CTAC ($100~{}mL$, $100~{}mM$), HAuCl4 ($0.36~{}mL$, $50~{}mM$), and ascorbic
acid ($0.36~{}mL$, $100~{}mM$). The mixture was left undisturbed for 12 h at
$25^{\circ}\text{C}$. Upon synthesis, the solution containing $12~{}nm$ Au NPs
was centrifuged ($9000~{}rpm$, 2 h) to remove excess of CTAC and ascorbic
acid, and finally redispersed in water to a final gold concentration of
$5~{}mM$. The average diameter determined from TEM images was $12\pm 1~{}nm$.
To replace the cationic surfactant with a hydrophobic polymer, thiolated
polystyrene (PS-SH) was used. The colloidal dispersion containing $12~{}nm$ Au
NPs ($2~{}mL$, $5~{}mM$) was added dropwise under sonication to a dispersion
of PS-SH (1 molecule of PS-SH per $nm^{2}$ of Au surface) in THF ($20~{}mL$).
The solution was left for 15 min in an ultrasonic bath. To ensure ligand
exchange, the resulting mixture was left undisturbed for 12 h, and then
centrifuged twice ($8000~{}rpm$, 30 min). The particles were finally dispersed
in THF to a final gold concentration of $2.5~{}mM$. The clustering of PS-
functionalized Au NPs was carried out according to our previously
reported[Sánchez-Iglesias et al., 2012]. An aliquot of water ($0.8~{}mL$) was
added dropwise to the PS-functionalized Au NPs in THF ($3.2~{}mL$) under
magnetic stirring. The final concentration of metallic gold in the mixture was
$0.25~{}mM$. The solution was left undisturbed for 5 min, and then a solution
of PS403-b-PAA62 in THF ($0.4~{}mL$, $6~{}mg/mL$) was added dropwise under
magnetic stirring. Subsequently, the water content was increased up to
$35~{}wt\%$, followed by increasing the temperature up to
$70^{\circ}\text{C}$, which was maintained for 30 min. The clusters dispersion
was centrifuged twice ($8500~{}rpm$, 30 min) and redispersed in water.

###  4.3 Synthesis of Au NRs

Au NRs were prepared through the seeded growth method, based on the reduction
of HAuCl4 on CTAB-stabilized gold seeds in the presence of silver
ions[Xingchen et al., 2013]. To prepare the growth solution, $1.8~{}g$ of CTAB
and $0.25~{}g$ of NaOL were dissolved in $100~{}mL$ of warm Milli-Q water
($50^{\circ}\text{C}$). Once sodium oleate was completely dissolved, the
mixture was cooled down to $30^{\circ}\text{C}$ and AgNO3 ($4.8~{}mL$,
$4~{}mM$) was added under stirring. The mixture was kept at
$30^{\circ}\text{C}$ for 15 min after which HAuCl4 was added ($0.5~{}mL$,
$100~{}mM$) under vigorous stirring. The mixture became colourless after 20
min at $30^{\circ}\text{C}$ and then HCl ($0.42~{}mL$, $37\%$) was introduced.
After 15 min of stirring, AA ($0.25~{}mL$, $64~{}mM$) was added, and the
solution was vigorously stirring for 30 seconds. Finally, a certain volume of
seed solution ($0.16~{}mL$, $0.25~{}mM$) was injected into the growth solution
under vigorous stirring for 5 minutes, and then the solution was left
undisturbed al $30^{\circ}\text{C}$ for 12 h. An aliquot of the Au NR
dispersion ($2~{}mL$) was centrifuged twice ($8000~{}rpm$, 30 min) to remove
excess reactants, and then redispersed in a dilute CTAB aqueous solution
($0.25~{}mL$, $0.2~{}mM$). The length and width of the obtained NRs were
$67\pm 2~{}nm$ and $19\pm 1~{}nm$, respectively.

###  4.4 Electron microscopy sample preparation and measurements

For a typical sample preparation for conventional and fast electron tomography
measurement in vacuum, $2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{L}$ of
colloidal dispersion was dropcast on a Quantifoil (2/2, 200 mesh) copper grid
and was dried under an ambient environment. For a fast liquid electron
tomography experiment on the Au@PS colloidal asseamblies,
$2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{L}$ of the colloidal dispersion
was loaded into commercial monolithic LC (K-Kit from Bio MA-TEK) with a window
gap of $0.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ under capillary
forces, followed by being sealed with water-resistant glue. For a typical
experiment on the Au NRs bilayer assemblies, the dispersion was loaded in a
similar manner but using a Tomochip 0.1 µm window gap.

###  4.5 Fast electron tomography in vacuum and liquid

All tilt series were obtained from a “cubed” aberration-corrected Thermo
Fisher Titan microscope microscope at room temperature with an acceleration
voltage of 200 $kV$. A Fischione model 2020 single tilt holder was used for
the fast acquisition of the tilt series both in vacuum and liquid. For fast
electron tomography tilt series acquisitions in vacuum, a tilt range of $\pm
72^{\circ}$ and $\pm 74^{\circ}$ was applied for colloidal clusters ($N=4$,
$N=5$ and $N=6$) and self-assembled Au NRs, respectively. For fast electron
tomography tilt series acquisitions in liquid, a tilt range from $-48^{\circ}$
to $46^{\circ}$, $-48^{\circ}$ to $44^{\circ}$, $-46^{\circ}$ to $46^{\circ}$,
and $-56^{\circ}$ to $58^{\circ}$ was applied for colloidal clusters of $N=4$,
$N=5$, $N=6$, and self-assembled Au NRs, respectively. The total dose employed
for each case was calculated by multiplying the dose per frame by the number
of frames. Table 2 summarizes the experimental conditions from each of the
tilt series acquired in this study.

| Sample |  Min. angle |  Max. angle |  Total tilt- series frames |  Current (pA) |  dwell time ($\mu$s) |  pixel size (pm) |  dose per frame ($\text{e}^{-}/\text{\r{A}}^{2}$) |  Total dose ($\text{e}^{-}/\text{\r{A}}^{2}$)
---|---|---|---|---|---|---|---|---|---
Vacuum | N = 4 | -72 | 72 | 341 | 40 | 1 | 1040 | 2.31 | 787
N = 5 | -72 | 72 | 341 | 40 | 1 | 1040 | 2.31 | 787
N = 6 | -72 | 72 | 341 | 40 | 1 | 1040 | 2.31 | 787
AuNRs | -74 | 74 | 242 | 50 | 1 | 736 | 5.76 | 1394
Liquid | N = 4 | -46 | 46 | 154 | 2 | 0.5 | 367 | 0.46 | 71
N = 5 | -48 | 44 | 156 | 2 | 0.5 | 367 | 0.46 | 72
N = 6 | -46 | 46 | 154 | 2 | 0.5 | 367 | 0.46 | 71
AuNRs | -56 | 58 | 163 | 5 | 0.5 | 367 | 1.16 | 189

Table 2: Summary of vacuum and liquid fast electron tomography experimental
conditions.

###  4.6 Tilt-series processing

####  4.6.1 Denoising tilt-series images with convolutional autoencoders

Imaging nanoparticles in liquid environments, especially under low-dose
conditions with water surrounding the particles, often introduces significant
noise. This noise can degrade the quality and accuracy of tilt series images.
Recognizing that these images represent the same object from different angles,
we can exploit inherent patterns for denoising. To enhance the SNR of these
images, we employed a self-supervised denoising mechanism using CAE[Gondara,
2016]. CAEs, a specialized architecture in unsupervised machine learning, are
designed to reconstruct their input data and consist of an encoder, which
compresses the input into a latent representation, and a decoder, which
reconstructs the input from this latent space. For our application, the CAE is
specifically tailored to serve as a denoising tool, trained to generate noise-
free images from their noisy counterparts. The mathematical representation of
the denoising autoencoder is:

$\displaystyle\text{Encoder: }\qquad\boldsymbol{z}$
$\displaystyle=\mathcal{E}_{\boldsymbol{\lambda}}(\boldsymbol{y}),$ (1)
$\displaystyle\text{Decoder: }\qquad\boldsymbol{y}^{\prime}$
$\displaystyle=\mathcal{D}_{\boldsymbol{\gamma}}(\boldsymbol{z}),$ (2)
$\displaystyle\text{Loss Function:
}\qquad\mathcal{L}(\boldsymbol{\lambda},\boldsymbol{\gamma})$
$\displaystyle=-\sum_{i}y_{i}\log(y^{\prime}_{i})+(1-y_{i})\log(1-y^{\prime}_{i}),$
(3)

where $\boldsymbol{y}\in\mathbb{R}^{n}$ denotes the noisy input image,
$\mathcal{E}_{\boldsymbol{\lambda}}$ represents the encoder function
parameterized by $\boldsymbol{\lambda}$, $\mathcal{D}_{\boldsymbol{\gamma}}$
is the decoder function parameterized by $\boldsymbol{\gamma}$, and
$\mathcal{L}$ is the negative log likelihood, tailored to be robust against
Poisson noise. Our CAE architecture was implemented using the PyTorch library
as shown in Supplementary Fig. S4. Training was conducted over 50 epochs with
a batch size of 32, using the Adam optimizer with a learning rate of
$10^{-4}$. By harnessing the patterns present in sequential tilt series
images, the CAE effectively reconstructs noise-free versions. This approach
not only elevates the SNR but also preserves the crucial structural details
inherent to the images. In a comparative analysis against traditional
denoising techniques, such as Gaussian smoothing, our methodology showcased a
superiority in both noise attenuation and preservation of structural
intricacies (see Supplementary Fig. S5).

####  4.6.2 Refining tilt series using robust principal component analysis

High-quality tilt series images are crucial for accurate 3D reconstructions.
However, the intricacies of fast image acquisition can introduce distortions
and anomalies that compromise subsequent analyses. While denoising techniques,
such as the CAE we employed, are effective in enhancing the SNR, they
primarily target random noise. Systematic distortions, outliers, or structured
anomalies, which can arise due to various factors in the imaging process, may
still persist post-denoising. These structured anomalies can have a pronounced
impact on the accuracy of 3D reconstructions. To specifically address and
rectify these structured distortions, we further refined our tilt series using
RPCA[Candès et al., 2011]. RPCA, an advanced extension of classical Principal
Component Analysis (PCA), offers a robust approach to data decomposition.
While PCA assumes the observed data is a mix of low-rank components and
Gaussian noise, RPCA decomposes a data matrix
$\boldsymbol{Y}\in\mathbb{R}^{n\times a}$ (with $a$ being number of images)
into a low-rank matrix $\boldsymbol{L}\in\mathbb{R}^{n\times a}$ and a sparse
matrix $\boldsymbol{S}\in\mathbb{R}^{n\times a}$:

$\underset{\boldsymbol{L},\boldsymbol{S}}{\mbox{minimize}}\quad\|\boldsymbol{L}\|_{*}+\lambda\|\boldsymbol{S}\|_{1}\qquad\text{subject
to}\qquad\boldsymbol{L}+\boldsymbol{S}=\boldsymbol{Y}.$ (4)

Here, $\|\cdot\|_{*}$ denotes the nuclear norm, approximating the rank of
$\boldsymbol{L}$. $\|\cdot\|_{1}$ is the $\ell_{1}$ norm, a convex
approximation for the count of non-zero elements in $\boldsymbol{S}$. For our
implementation, we utilized the augmented Lagrange multiplier method for
optimization, a popular approach for RPCA. The algorithm was run for a maximum
of $100$ iterations with a convergence criterion set at $10^{-4}$. The
parameter $\lambda$ was empirically set at $1/\sqrt{\max(a,n)}$, where $a$ and
$n$ are the dimensions of $\boldsymbol{Y}$, ensuring a balance between the
low-rank and sparse components. Applying RPCA to the tilt series allows us to
discern the primary patterns within the series, highlighting important
structures and relationships between individual images. We then establish a
threshold based on a specific percentile of the distribution of projection
scores from RPCA. Images deviating significantly from this threshold are
considered outliers with potential distortions. These outliers are excluded
from the tilt series, ensuring a refined dataset ready for further processing.

####  4.6.3 Image alignment using iterative closest point method

Accurate 3D reconstructions from tilt series critically depend on the precise
alignment of the tilt series images. Given that particles can rotate in the
liquid during acquisition, it’s imperative to account for both translational
and rotational misalignments. Merely Only using cross-correlation to register
shifts is insufficient, as it predominantly addresses translational
misalignments. Therefore, a more comprehensive approach, like rigid
registration, becomes indispensable to ensure that each image in the series is
aligned to a reference (in this case, we use low-rank component
$\boldsymbol{L}$ obtained from RPCA). The ICP method offers a robust solution
for this challenge[Zhang et al., 2021]. Originally designed for 3D point cloud
alignments, ICP’s iterative approach is well-suited for tilt series image
registration. At each iteration, the algorithm identifies pairs of closest
points between two images. These points are derived from prominent features
within the images, extracted using Speeded-Up Robust Features (SURF). The
optimal transformation (rotation and translation) is then calculated to
minimize the distance between these point pairs. This transformation
progressively aligns the images. The iterations continue until the algorithm
converges to a minimal distance between corresponding points in the datasets
or until a predefined number of iterations (typically set to 100) is reached.
The mathematical objective of ICP is:

$\underset{\boldsymbol{R},\boldsymbol{t}}{\mbox{minimize}}\quad\sum_{i}\|\boldsymbol{p}_{i}-(\boldsymbol{R}\boldsymbol{q}_{i}+\boldsymbol{t})\|^{2},$
(5)

where $\boldsymbol{R}\in\mathbb{R}^{2\times 2}$ is the rotation matrix,
$\boldsymbol{t}\in\mathbb{R}^{2}$ is the translation vector, and
$\boldsymbol{p}_{i}\in\mathbb{R}^{2}$ and
$\boldsymbol{q}_{i}\in\mathbb{R}^{2}$ are the corresponding points from the
two images being aligned. For the optimization, we employed the proximal
gradient algorithm, which is adept at handling non-linear least squares
problems. We introduced constraints on the shifts and rotations to ensure
physically meaningful alignments. These constraints were set based on prior
knowledge of the maximum possible misalignments during the imaging process.
Rigid registration, as facilitated by ICP, is essential because it compensates
for any minor shifts or rotations that might occur during image acquisition.
This ensures that the images are consistently oriented and overlaid with
precision, which is fundamental for generating coherent and accurate 3D
reconstructions.

####  4.6.4 Alignment of tilt-axis using filtered back projection

Tilt-axis alignment is a critical step for accurate three-dimensional (3D)
reconstruction from electron tomography tilt series. Even minor misalignments
can lead to artifacts, diminishing the fidelity of the reconstructed 3D
structures. Therefore, achieving precise tilt-axis alignment is imperative for
obtaining reliable 3D reconstruction. To facilitate this, we employ the
Filtered Back Projection (FBP) method. FBP effectively reconstructs an object
from its projections by utilizing the Radon transform and its inverse. This
process generates a rapid 3D reference model. For the tilt-axis alignment, our
approach begins with creating forward projections from the 3D model obtained
via FBP. These projections are then quantitatively compared with the original
tilt series images. The comparison employs a similarity metric, specifically
the Structured Similarity Index Measure (SSIM), to identify any misalignments.
Identifying discrepancies between the FBP-generated projections and the
original tilt series is crucial. These discrepancies indicate the extent and
nature of misalignments. To rectify these misalignments, we use an algorithmic
refinement process[Houben et al., 2011]. In each iteration of this process,
the tilt-axis orientation is adjusted based on the observed discrepancies from
the preceding iteration. This iterative refinement employs a gradient descent
optimization strategy, systematically reducing the discrepancy metric to reach
an optimal alignment. The number of iterations for convergence is typically
around 10, although this can vary depending on the initial degree of
misalignment and the quality of the tilt images. Upon convergence, the tilt-
axis is accurately aligned.

####  4.6.5 Iterative workflow

In the pre-processing of electron tomography tilt series, adopting an
iterative workflow is essential due to the complex nature of image acquisition
and the possibility of sample movements during the process. Our methodology
encompasses three fundamental steps: distortion correction using RPCA, image
alignment through ICP algorithm, and precise tilt-axis alignment. The accuracy
of each step is critical because it significantly affects the subsequent
stages. Initially, the RPCA method identifies and corrects distortions,
producing a more accurate tilt series for the next phase. Subsequently, the
ICP algorithm aligns the images, and this alignment is further refined through
tilt-axis adjustment. This sequence of steps is repeated across five
iterations for optimal results. The decision to limit the process to five
iterations stems from empirical observations. Specifically, we monitor a
similarity metric that compares the tilt series with the low-rank component
$\boldsymbol{L}$ (referenced in Supplementary Figure S6). As the iterations
proceed, this metric generally shows improvement, signaling better alignment
and registration quality. However, after the fifth iteration, we observe a
plateau in this metric, implying that additional iterations would yield
minimal further improvements. This plateau indicates the point where the
balance between computational effort and alignment quality is optimized,
thereby justifying the choice of five iterations in our process.

###  4.7 Advanced reconstruction method from tilt series

To ensure consistency and avoid potential bias, the tilt series in vacuum and
liquid was processed using the methodology outlined above. This involved
denoising, alignment, and distortion removal. Notably, given the broad tilting
angular range of $140\degree$ achieved in vacuum, the ML-EM algorithm
implemented using the ASTRA Toolbox[Van Aarle et al., 2015] was employed for
reconstruction. Subsequent 3D visualizations were rendered using Amira 5.4.0.

In the context of electron tomography, particularly for samples imaged in
liquid, challenges arise due to the restricted tilting range, often limited to
about $90\degree$. This limitation poses significant challenges even for
advanced reconstruction algorithms like DART[Batenburg and Sijbers, 2011].
DART can struggle with incomplete datasets, as a limited tilt range often
fails to provide comprehensive angular coverage, leading to ambiguities and
potential inaccuracies in the reconstructed 3D volume. To address these
challenges, we have developed the Compressed Sensing Discrete Algebraic
Reconstruction Technique (CS-DART). This advanced algorithm enhances the
conventional DART by incorporating a shape smoothness prior. The key
innovation in CS-DART lies in its use of level-set methods for material
discretization, representing material intensities through a combination of
level-set functions. By integrating smoothness into the material surfaces and
leveraging discrete cosine transform (DCT) basis for regularizing the
reconstruction problem, CS-DART effectively overcomes the limitations posed by
incomplete angular coverage. The mathematical formulation of CS-DART is as
follows:

$\begin{split}\underset{\boldsymbol{\alpha}}{\mbox{minimize}}\quad\|\boldsymbol{W}\boldsymbol{x}(\phi(\boldsymbol{\alpha}))-\boldsymbol{y}\|_{2}^{2},\qquad\mbox{subject
to}\quad\|\boldsymbol{\alpha}\|_{1}\leq\tau,\end{split}$ (6)

where $\boldsymbol{W}$ is the tomographic operator that discretizes the Radon
transform, $\boldsymbol{\alpha}$ are the DCT coefficients of the object under
reconstruction, $\boldsymbol{y}$ is the acquired, processed tilt series, $n$
are the number of voxels, and $\tau$ is the regularization parameter. Here, we
have modeled the intensity as

$\boldsymbol{x}(\boldsymbol{\phi})=c_{\text{soft}}H(\boldsymbol{\phi}-c_{\text{soft}})+c_{\text{hard}}H(\boldsymbol{\phi}-c_{\text{hard}}),\qquad\boldsymbol{\phi}=\boldsymbol{\Psi}\boldsymbol{\alpha},$
(7)

where $c_{\text{soft}}$ and $c_{\text{hard}}$ are the intensities of soft
(polymeric) and hard (Au NPs) material, $H$ is the Heaviside function ensuring
the imposition of discreteness, $\boldsymbol{\phi}$ is the discretized level-
set function, and $\boldsymbol{\Psi}$ is a DCT basis. This condensed approach
forms the basis of our method, which we have termed the Compressed Sensing
Discrete Algebraic Reconstruction Technique (CS-DART). CS-DART utilizes the
DCT basis to efficiently compress shape information in level-set functions,
allowing for effective handling of large datasets by focusing on key shape
features. This approach not only enhances reconstruction accuracy but also
significantly reduces computational complexity, as the number of functions in
the basis is much lower than the voxel count. By capturing the essential
features of the sample and minimizing redundancy, CS-DART streamlines the
reconstruction of complex structures from incomplete datasets. The
optimization problem is solved iteratively via the well-known Fast Iterative
Shrinkage-Thresholding Algorithm[Beck and Teboulle, 2009]. This method
efficiently accommodates the non-differentiability introduced by the
$\ell_{1}$-norm in our objective function. Additionally, we use an
approximation of the Heaviside function to compute the gradient of the loss
function[Kadu et al., 2016]. In each iteration, the DCT coefficients,
$\boldsymbol{\alpha}$, are updated based on the gradient of the data fidelity
term[Kadu et al., 2017]. Moreover, the Radon transform was implemented using
ASTRA Toolbox[Van Aarle et al., 2015]. CS-DART algorithm accurately
reconstructs heterogeneous structures from incomplete datasets, making it
particularly beneficial for analyzing these colloidal assemblies under limited
tilt ranges.

We remark that our advanced reconstruction algorithm recreates the morphology
of the assembly while overcoming the challenges posed by missing wedge
artifacts and noise. A comparison of the 3D reconstructions obtained using our
proposed method and conventional reconstruction methods is presented in
Supplementary Fig. S7. Our results underline the enhanced performance and
accuracy of our proposed CS-DART method, especially in preserving the fine
structural details of the colloidal assemblies.

Due to enhanced angular sampling and improved SNR realized by Tomochip
($130\degree$ angular range), bilayer assemblies of Au NRs were reconstructed
using ML-EM algorithm implemented in ASTRA Toolbox[Van Aarle et al., 2015].

## Acknowledgement

S.B., D.A.E., D.W., N.O., and A.K. acknowledge financial support from ERC
Consolidator Grant Number 815128 REALNANO and Horizon Europe MSCA-SE no.
101131111 – DELIGHT. D.W. acknowledges an Individual Fellowship funded by the
Marie-Sklodowska-Curie Actions (MSCA) in Horizon 2020 program (grant 894254
SuprAtom). L.M.L.M. acknowledges financial support from Project
PID2020-117779RB-I00, State Research Agency of Spain, Ministry of Science and
Innovation.

## References

  * [Albrecht and Bals, 2020] Albrecht, W. and Bals, S. (2020).  Fast electron tomography for nanomaterials.  The Journal of Physical Chemistry C, 124(50):27276–27286.
  * [Altantzis et al., 2021] Altantzis, T., Wang, D., Kadu, A., Van Blaaderen, A., and Bals, S. (2021).  Optimized 3d reconstruction of large, compact assemblies of metallic nanoparticles.  The Journal of Physical Chemistry C, 125(47):26240–26246.
  * [Bals et al., 2014] Bals, S., Goris, B., Liz-Marzán, L. M., and Van Tendeloo, G. (2014).  Three-dimensional characterization of noble-metal nanoparticles and their assemblies by electron tomography.  Angewandte Chemie International Edition, 53(40):10600–10610.
  * [Batenburg and Sijbers, 2011] Batenburg, K. J. and Sijbers, J. (2011).  Dart: a practical reconstruction algorithm for discrete tomography.  IEEE Transactions on Image Processing, 20(9):2542–2553.
  * [Beck and Teboulle, 2009] Beck, A. and Teboulle, M. (2009).  A fast iterative shrinkage-thresholding algorithm for linear inverse problems.  SIAM journal on imaging sciences, 2(1):183–202.
  * [Boles et al., 2016] Boles, M. A., Engel, M., and Talapin, D. V. (2016).  Self-assembly of colloidal nanocrystals: From intricate structures to functional materials.  Chemical reviews, 116(18):11220–11289.
  * [Candès et al., 2011] Candès, E. J., Li, X., Ma, Y., and Wright, J. (2011).  Robust principal component analysis?  Journal of the ACM (JACM), 58(3):1–37.
  * [Das et al., 2022] Das, Partha Pratim and Cookman, Jennifer and Pérez, Alejandro Gómez and Plana-Ruiz, Sergi and López-Haro, Miguel and Calvino, José J and Casablanca, Jesús Gonzalez and Grivas, Evangelos and Antipas, Georgios SE and Ntallis, Nikolaos A and Nicolopoulos Stavros (2022).  High Angle Liquid Cell TEM Tomography for In Situ Observation and 3D Reconstruction in Liquid  Microscopy and Microanalysis, 28(S):854–856.
  * [de Jonge et al., 2019] de Jonge, N., Houben, L., Dunin-Borkowski, R. E., and Ross, F. M. (2019).  Resolution and aberration correction in liquid cell transmission electron microscopy.  Nature Reviews Materials, 4(1):61–78.
  * [De Nijs et al., 2015] De Nijs, B., Dussi, S., Smallenburg, F., Meeldijk, J. D., Groenendijk, D. J., Filion, L., Imhof, A., Van Blaaderen, A., and Dijkstra, M. (2015).  Entropy-driven formation of large icosahedral colloidal clusters by spherical confinement.  Nature materials, 14(1):56–60.
  * [De Yoreo and NAJM, 2016] De Yoreo, J. J. and NAJM, S. (2016).  Investigating materials formation with liquid-phase and cryogenic tem.  Nature Reviews Materials, 1(8):1–18.
  * [Donoho, 2006] Donoho, D. L. (2006).  Compressed sensing.  IEEE Transactions on information theory, 52(4):1289–1306.
  * [Galván-Moya et al., 2014] Galván-Moya, J. E., Altantzis, T., Nelissen, K., Peeters, F. M., Grzelczak, M., Liz-Marzán, L. M., Bals, S., and Van Tendeloo, G. (2014).  Self-organization of highly symmetric nanoassemblies: a matter of competition.  ACS nano, 8(4):3869–3875.
  * [Gilbert, 1972] Gilbert, P. (1972).  Iterative methods for the three-dimensional reconstruction of an object from projections.  Journal of theoretical biology, 36(1):105–117.
  * [Glotzer and Solomon, 2007] Glotzer, S. C. and Solomon, M. J. (2007).  Anisotropy of building blocks and their assembly into complex structures.  Nature materials, 6(8):557–562.
  * [Gomez-Grana et al., 2012] Gomez-Grana, S., Hubert, F., Testard, F., Guerrero-Martínez, A., Grillo, I., Liz-Marzán, L. M., and Spalla, O. (2012).  Surfactant (bi) layers on gold nanorods.  Langmuir, 28(2):1453–1459.
  * [Gondara, 2016] Gondara, L. (2016).  Medical image denoising using convolutional denoising autoencoders.  In 2016 IEEE 16th international conference on data mining workshops (ICDMW), pages 241–246. IEEE.
  * [Gonzalez Casablanca et al., 2020] Gonzalez Casablanca, Jesús and Nicolopoulos Stavros (2020).  Devices and methods for high angle liquid electron tomograpy  WO2021123305A1.
  * [Goris et al., 2012] Goris, B., Van den Broek, W., Batenburg, K. J., Mezerji, H. H., and Bals, S. (2012).  Electron tomography based on a total variation minimization reconstruction technique.  Ultramicroscopy, 113:120–130.
  * [Guerrero-Martínez et al., 2012] Guerrero-Martínez, A., Grzelczak, M., and Liz-Marzán, L. M. (2012).  Molecular thinking for nanoplasmonic design.  ACS nano, 6(5):3655–3662.
  * [Hamon et al., 2015] Hamon, C., Novikov, S. M., Scarabelli, L., Solís, D. M., Altantzis, T., Bals, S., Taboada, J. M., Obelleiro, F., and Liz-Marzán, L. M. (2015).  Collective plasmonic properties in few-layer gold nanorod supercrystals.  ACS photonics, 2(10):1482–1488.
  * [Houben et al., 2011] Houben, Lothar and Sadan, M Bar (2011).  Refinement procedure for the image alignment in high-resolution electron tomography  Ultramicroscopy, 111(9-10):1512–1520.
  * [Kadu et al., 2017] Kadu, A., van Leeuwen, T., and Batenburg, K. J. (2017).  A parametric level-set method for partially discrete tomography.  In Discrete Geometry for Computer Imagery: 20th IAPR International Conference, DGCI 2017, Vienna, Austria, September 19–21, 2017, Proceedings 20, pages 122–134. Springer.
  * [Kadu et al., 2016] Kadu, A., van Leeuwen, T., and Mulder, W. A. (2016).  Salt reconstruction in full-waveform inversion with a parametric level-set method.  IEEE Transactions on Computational Imaging, 3(2):305–315.
  * [Kavak et al., 2023] Kavak, S., Kadu, A. A., Claes, N., Sánchez-Iglesias, A., Liz-Marzán, L. M., Batenburg, K. J., and Bals, S. (2023).  Quantitative 3d investigation of nanoparticle assemblies by volumetric segmentation of electron tomography data sets.  The Journal of Physical Chemistry C.
  * [Keskin et al., 2021] Keskin, S., Pawell, C., and de Jonge, N. (2021).  Verification of water presence in graphene liquid cells.  Micron, 149:103109.
  * [Kim et al., 2020] Kim, B. H., Heo, J., Kim, S., Reboul, C. F., Chun, H., Kang, D., Bae, H., Hyun, H., Lim, J., Lee, H., et al. (2020).  Critical differences in 3d atomic structure of individual ligand-protected nanocrystals in solution.  Science, 368(6486):60–67.
  * [Koneti et al., 2019] Koneti, S., Roiban, L., Dalmas, F., Langlois, C., Gay, A.-S., Cabiac, A., Grenier, T., Banjak, H., Maxim, V., and Epicier, T. (2019).  Fast electron tomography: Applications to beam sensitive samples and in situ tem or operando environmental tem studies.  Materials Characterization, 151:480–495.
  * [Kumar et al., 2018] Kumar, J., Eraña, H., López-Martínez, E., Claes, N., Martín, V. F., Solís, D. M., Bals, S., Cortajarena, A. L., Castilla, J., and Liz-Marzán, L. M. (2018).  Detection of amyloid fibrils in parkinson’s disease using plasmonic chirality.  Proceedings of the National Academy of Sciences, 115(13):3225–3230.
  * [Langer et al., 2019] Langer, J., Jimenez de Aberasturi, D., Aizpurua, J., Alvarez-Puebla, R. A., Auguié, B., Baumberg, J. J., Bazan, G. C., Bell, S. E., Boisen, A., Brolo, A. G., et al. (2019).  Present and future of surface-enhanced raman scattering.  ACS nano, 14(1):28–117.
  * [Lyu et al., 2023] Lyu, Z., Yao, L., Chen, W., Kalutantirige, F. C., and Chen, Q. (2023).  Electron microscopy studies of soft nanomaterials.  Chemical Reviews, 123(7):4051–4145.
  * [Marchetti et al., 2023] Marchetti, A., Gori, A., Ferretti, A. M., Esteban, D. A., Bals, S., Pigliacelli, C., and Metrangolo, P. (2023).  Templated out-of-equilibrium self-assembly of branched au nanoshells.  Small, 19(12):2206712.
  * [Meena and Sulpizi, 2013] Meena, S. K. and Sulpizi, M. (2013).  Understanding the microscopic origin of gold nanoparticle anisotropic growth from molecular dynamics simulations.  Langmuir, 29(48):14954–14961.
  * [Miao et al., 2016] Miao, J., Ercius, P., and Billinge, S. J. (2016).  Atomic electron tomography: 3d structures without crystals.  Science, 353(6306):aaf2157.
  * [Midgley and Dunin-Borkowski, 2009] Midgley, P. A. and Dunin-Borkowski, R. E. (2009).  Electron tomography and holography in materials science.  Nature materials, 8(4):271–280.
  * [Moon, 1996] Moon, T. K. (1996).  The expectation-maximization algorithm.  IEEE Signal processing magazine, 13(6):47–60.
  * [Mosquera et al., 2023] Mosquera, J., Wang, D., Bals, S., and Liz-Marzán, L. M. (2023).  Surfactant layers on gold nanorods.  Accounts of Chemical Research, pages 1472–1477.
  * [Park et al., 2015] Park, J., Elmlund, H., Ercius, P., Yuk, J. M., Limmer, D. T., Chen, Q., Kim, K., Han, S. H., Weitz, D. A., Zettl, A., et al. (2015).  3d structure of individual nanocrystals in solution by electron microscopy.  Science, 349(6245):290–295.
  * [Park et al., 2021] Park, J., Koo, K., Noh, N., Chang, J. H., Cheong, J. Y., Dae, K. S., Park, J. S., Ji, S., Kim, I.-D., and Yuk, J. M. (2021).  Graphene liquid cell electron microscopy: Progress, applications, and perspectives.  ACS nano, 15(1):288–308.
  * [Ross, 2015] Ross, F. M. (2015).  Opportunities and challenges in liquid cell electron microscopy.  Science, 350(6267):aaa9886.
  * [Sánchez-Iglesias et al., 2012] Sánchez-Iglesias, A., Grzelczak, M., Altantzis, T., Goris, B., Perez-Juste, J., Bals, S., Van Tendeloo, G., Donaldson Jr, S. H., Chmelka, B. F., Israelachvili, J. N., et al. (2012).  Hydrophobic interactions modulate self-assembly of nanoparticles.  ACS nano, 6(12):11059–11065.
  * [Sau and Murphy, 2005] Sau, T. K. and Murphy, C. J. (2005).  Self-assembly patterns formed upon solvent evaporation of aqueous cetyltrimethylammonium bromide-coated gold nanoparticles of various shapes.  Langmuir, 21(7):2923–2929.
  * [Solís et al., 2017] Solís, D. M., Taboada, J. M., Obelleiro, F., Liz-Marzán, L. M., and García de Abajo, F. J. (2017).  Optimization of nanoparticle-based sers substrates through large-scale realistic simulations.  ACS photonics, 4(2):329–337.
  * [Sorzano et al., 2004] Sorzano, C., Marabini, R., Velázquez-Muriel, J., Bilbao-Castro, J. R., Scheres, S. H., Carazo, J. M., and Pascual-Montano, A. (2004).  Xmipp: a new generation of an open-source image processing package for electron microscopy.  Journal of structural biology, 148(2):194–204.
  * [Van Aarle et al., 2015] Van Aarle, W., Palenstijn, W. J., De Beenhouwer, J., Altantzis, T., Bals, S., Batenburg, K. J., and Sijbers, J. (2015).  The astra toolbox: A platform for advanced algorithm development in electron tomography.  Ultramicroscopy, 157:35–47.
  * [Vanrompay et al., 2021] Vanrompay, H., Skorikov, A., Bladt, E., Béché, A., Freitag, B., Verbeeck, J., and Bals, S. (2021).  Fast versus conventional haadf-stem tomography of nanoparticles: advantages and challenges.  Ultramicroscopy, 221:113191.
  * [Wang et al., 2021a] Wang, D., Dasgupta, T., van der Wee, E. B., Zanaga, D., Altantzis, T., Wu, Y., Coli, G. M., Murray, C. B., Bals, S., Dijkstra, M., and van Blaaderen, A. (2021a).  Binary icosahedral clusters of hard spheres in spherical confinement.  Nature Physics, 17(1):128–134.
  * [Wang et al., 2018] Wang, D., Hermes, M., Kotni, R., Wu, Y., Tasios, N., Liu, Y., De Nijs, B., Van Der Wee, E. B., Murray, C. B., Dijkstra, M., et al. (2018).  Interplay between spherical confinement and particle shape on the self-assembly of rounded cubes.  Nature communications, 9(1):2228.
  * [Wang et al., 2022] Wang, D., Hermes, M., Najmr, S., Tasios, N., Grau-Carbonell, A., Liu, Y., Bals, S., Dijkstra, M., Murray, C. B., and van Blaaderen, A. (2022).  Structural diversity in three-dimensional self-assembly of nanoplatelets by spherical confinement.  Nature communications, 13(1):6001.
  * [Wang et al., 2021b] Wang, D., van der Wee, E. B., Zanaga, D., Altantzis, T., Wu, Y., Dasgupta, T., Dijkstra, M., Murray, C. B., Bals, S., and van Blaaderen, A. (2021b).  Quantitative 3d real-space analysis of laves phase supraparticles.  Nature communications, 12(1):3980.
  * [Lazić et al., 2016] Lazić, I., Bosch, E, GT and Lazar, S. (2016)  Phase contrast STEM for thin samples: Integrated differential phase contrast  Ultramicroscopy, 160:265–280.
  * [Yu et al., 2022] Yu, C.P., Friedrich, T., Jannis, D., Van Aert, S. and Verbeeck, J. (2022).  Real-time integration center of mass (riCOM) reconstruction for 4D STEM  Microscopy and Microanalysis, 28(5):1526–1537.
  * [Weidemaier et al., 1997] Weidemaier, K., Tavernier, H., and Fayer, M. (1997).  Photoinduced electron transfer on the surfaces of micelles.  The Journal of Physical Chemistry B, 101(45):9352–9361.
  * [Yang et al., 2019] Yang, J., Alam, S. B., Yu, L., Chan, E., and Zheng, H. (2019).  Dynamic behavior of nanoscale liquids in graphene liquid cells revealed by in situ transmission electron microscopy.  Micron, 116:22–29.
  * [Zanaga et al., 2016] Zanaga, D., Bleichrodt, F., Altantzis, T., Winckelmans, N., Palenstijn, W. J., Sijbers, J., de Nijs, B., van Huis, M. A., Sánchez-Iglesias, A., Liz-Marzán, L. M., et al. (2016).  Quantitative 3d analysis of huge nanoparticle assemblies.  Nanoscale, 8(1):292–299.
  * [Zhang et al., 2021] Zhang, J., Yao, Y., and Deng, B. (2021).  Fast and robust iterative closest point.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(7):3450–3466.
  * [Zheng et al., 2014] Zheng, Yiqun and Zhong, Xiaolan and Li, Zhiyuan and Xia, Younan (2022).  Successive, Seed-Mediated Growth for the Synthesis of Single-Crystal Gold Nanospheres with Uniform Diameters Controlled in the Range of 5–150 nm  Particle & Particle Systems Characterization, 31(2):266–273.
  * [Xingchen et al., 2013] Ye, Xingchen and Zheng, Chen and Chen, Jun and Gao, Yuzhi and Murray, Christopher B (2013).  Using binary surfactant mixtures to simultaneously improve the dimensional tunability and monodispersity in the seeded growth of gold nanorods  Particle & Particle Systems Characterization, 13(2):765–771.

## Supplementary Notes

## S1 Electron beam damage comparison on colloidal assemblies: fast vs.
conventional acquisition methodologies

To better understand the impact of electron beam exposure on colloidal
assemblies, we compared different acquisition methodologies, specifically fast
vs. conventional methods. This comparison is crucial, as prolonged exposure to
the electron beam could potentially alter the structure of the studied
samples. Presented in Supplementary Fig. S1, HAADF-STEM images of polymeric
assemblies with different structures ($N=4,5,6$), before and after both
acquisition approaches, are presented. It is interesting to note the
difference in the size of the assemblies post-acquisition. Whereas the fast
acquisition method seems to preserve the original size with minimal
alterations, the conventional methodology, on the other hand, demonstrates
evident shrinkage, by 2 - 3 nm on average. These findings confirm the
importance of carefully selecting acquisition methods, depending on the
sensitivity of the samples under investigation[Vanrompay et al., 2021].

## S2 Dynamic behavior of colloidal assemblies under flow in a liquid phase
TEM holder

To gain a deeper understanding of colloidal assembly dynamics, especially
under flow conditions, we employed a liquid phase TEM holder to capture the
real-time behavior of these assemblies. Supplementary Fig. S2a shows the flow-
induced liquid phase setup (Stream from DENSsolutions) utilized to observe the
colloidal clusters. As observed in the subsequent HAADF-STEM image series
(Supplementary Fig. S2b-d), distinct colloidal clusters demonstrated
significant translational and rotational motion over time. Examples of the
dynamic behavior are indicated by the dashed yellow circle. However, a
critical observation was the occurrence of electron beam damage after 33
seconds of exposure. This phenomenon resulted in unintended aggregation of the
colloidal clusters. A more detailed visual representation is presented in
Supplementary Movie 3.

## S3 Quantitative Indicators

Regarding 3D tomographic imaging of colloidal assemblies of Au nanoparticles
(NPs) within a polymeric shell, surface area, volume, and regularity index
serve as vital metrics for characterizing assembly structures. By analyzing
the polyhedron formed by the assembled Au NPs inside the polymeric shell,
these metrics are calculated.

Surface area measures the combined area of the polyhedron’s faces, providing
insight into the extent of surface interactions between NPs and the polymeric
shell, which can influence the stability of the colloidal assembly. To
calculate surface area, the areas of all the polyhedron’s faces are summed up:

$S=\sum A_{i},$ (1)

where $A_{i}$ is the area of the $i^{\text{th}}$ face of the polyhedron.

The volume quantifies the space enclosed by the polyhedron formed by the Au
NPs. This metric offers information about the density and packing of the NPs
within the polymeric shell, which can affect the assembly’s mechanical,
optical, and electronic properties. We calculate the volume ($V$) by dividing
the polyhedron into smaller parts, such as tetrahedrons or cubes, and summing
their volumes:

$V=\sum V_{i},$ (2)

where $V_{i}$ is the volume of the $i^{\text{th}}$ tetrahedron or cube.

The regularity index measures the polyhedron’s regularity. This metric
provides information about the degree of order and symmetry in the assembly,
which can influence physical and chemical properties of the colloidal
assembly. We calculate the regularity index ($R$) by comparing the polyhedron
to a regular polyhedron with the same number of faces and vertices and
measuring the deviation from the regular shape in terms of the angles and
lengths of the edges and faces:

$R=\frac{{\sum_{i}((\theta_{i}-\theta_{r})^{2}+(l_{i}-l_{r})^{2})/n}}{{\sum_{i}(d_{i}^{2}/n)}},$
(3)

where $\theta_{i}$ and $\theta_{r}$ are the angles of the $i^{\text{th}}$ and
regular faces of the polyhedron, $l_{i}$ and $l_{r}$ are the lengths of the
$i^{\text{th}}$ and regular edges of the polyhedron, $d_{i}$ is the distance
of the $i^{\text{th}}$ vertex from the center of the polyhedron, and $n$ is
the number of faces of the polyhedron. For reference, the ideal or ‘regular’
polyhedron considered has identical face and vertex counts as the Au NP-formed
polyhedron. The closer the value of $R$ is to zero, the more regular or
symmetric the polyhedron. A lower RI indicates a structure closely resembling
an ideal polyhedron, suggesting high order and symmetry. Conversely, a higher
RI points to irregularity.

## S4 Determining Au NRs assembly surface-to-surface distances

To understand the assembly patterns of Au NRs, we examined the surface-to-
surface distances under both dry and liquid conditions. As evident from the
HAADF-STEM projection images (Supplementary Fig. S9), distinct assembly
configurations were observed for both vacuum and liquid environments. These
images not only allowed us to visualize the assembly patterns but also
facilitated accurate identification and measurement of individual rod
diameters. To obtain accurate measurements, we took an orthogonal projection
from the 3D reconstruction that was directly facing the rods. Using MATLAB’s
‘imfindcircles’ function, we pinpointed the center and diameter of each rod.
With this data, we calculated the surface-to-surface distance between rods by
measuring the direct distance between their centers and adjusting for their
size. The surface-to-surface distance between the different rods (R, G, B) was
tabulated and compared under dry and liquid conditions, as detailed in
Supplementary TableS1. The table showcases variations in the distances between
the rods under the two conditions, providing essential insights into the role
of environment on the self-assembly behavior of NPs.

## References

  * [Vanrompay et al., 2021] Vanrompay, H., Skorikov, A., Bladt, E., Béché, A., Freitag, B., Verbeeck, J., and Bals, S. (2021).  Fast versus conventional haadf-stem tomography of nanoparticles: advantages and challenges.  Ultramicroscopy, 221:113191.

## Supplementary Figures

Figure S1: Comparison of electron beam damage on colloidal clusters. HAADF-
STEM images acquired before, after fast acquisition, and after conventional
tomography tilt series acquisition, for colloidal assemblies with different
numbers of Au NPs, a) $N=4$, b) $N=5$ and c) $N=6$. It can be observed that
fast acquisition shows a minimal size change, but conventional acquisition
shows a shrinkage of 2 - 3 nm.

Figure S2: Challenges in 3D characterization of colloidal clusters by tracking
translational and rotational motions. Series of HAADF-STEM images of clusters
obtained in a commercial Si3N4 LC chamber under flowing at different time
lapses: a) 0 seconds, b) 15 seconds, c) 33 seconds. Some colloidal clusters
were observed to translate and rotate freely, as denoted by the dashed yellow
circle. However, electron-beam damage is evident, even after only 33 seconds
of exposure time, leading to aggregation of the colloidal clusters. Note that
the presence of a liquid phase and thick Si3N4 LC windows resulted in the
polymeric shells being invisible. The dynamic process can be better
appreciated in Supplementary Movie 3.

Figure S3: Comparison of the structure, before and after fast tilt series
acquisition in liquid. HAADF-STEM images of different assemblies investigated
in liquid environment, acquired before (a, b and c) and after (d, e and f) the
tilt series acquisition for N = 4, 5 and 6 respectively. No significant
structure difference is observed after tilt series acquisition.

Figure S4: Schematic representation of a Convolutional Autoencoders (CAE)
architecture. The CAE consists of an encoder and a decoder. The encoder
progressively reduces the spatial dimensions of the input image through
convolutional and pooling layers, capturing its salient features in a
compressed latent space. The decoder then reconstructs the original image from
this compressed representation by using deconvolutional layers and upsampling.
The input to the CAE is a grayscale image of size $1024\times 1024$, and the
output is a reconstructed (denoised) image of the same size.

Figure S5: A side-by-side comparison of two image denoising techniques. The
leftmost column presents the original images, the center column displays the
outcomes of Gaussian denoising, and the rightmost column reveals the results
using CAE denoising. Each row represents a different colloidal assembly, all
imaged at a $0^{\circ}$ tilt-angle. The quality of denoising is quantified
using the Naturalness Image Quality Evaluator (NIQE) score, displayed at the
bottom right of each image. A lower NIQE score indicates superior image
clarity and denoising efficacy.

Figure S6: Structure similarity index measure for the iterative workflow.
Progression of the similarity metric (SSIM) across five iterations, comparing
the refined tilt series to its low-rank component, $\boldsymbol{L}$. A higher
SSIM value indicates a closer match between the refined series and its low-
rank counterpart, suggesting improved quality.

Figure S7: Comparison of 3D orthoslices from ML-EM and CS-DART reconstructions
of colloidal assemblies. The top row showcases ML-EM reconstructions, whereas
the bottom row presents those from CS-DART. From left to right, the panels
correspond to colloidal assemblies containing 4, 5, and 6 particles,
respectively.

Figure S8: Available tilt range of Tomo-chip. Optical micrographs of a Tomo-
chip loaded on a single-tilt tomography holder, with a) $0\degree$, b)
$45\degree$, and c) $70\degree$ tilting view, respectively.

Figure S9: Determining the diameters of Au NRs in vacuum and liquid
environments. Projection images captured from the tilt series of the
assemblies under investigation in a) vacuum and b) liquid environment,
respectively. The individual NRs selected for interparticle measurements were
identified within each image, and their diameters were measured independently.
This procedure ensured precise interparticle distance measurements.
Orthoslices from the reconstructed volume of the assemblies under c) vacuum
and d) liquid environment, where the 3 NRs selected for interparticle
measurements are highlighted in RGB code, as well as the distances between the
center of mass of each NR.

| Dry | Liquid
---|---|---
| R | G | B | R | G | B
R | 0 | 4.4 | 2.2 | 0 | 7.9 | 8.2
G | 4.4 | 0 | 6.6 | 7.9 | 0 | 5.9
B | 2.2 | 6.6 | 0 | 8.2 | 5.9 | 0
Table S1: Surface-to-surface distance between Au NRs (R, G, B) under dry and
liquid conditions. All distances are in nm. For coloring of rods (i.e., Red
(R), Green(G), Blue(B)), please refer to Supplementary Fig. S9.