11institutetext: Faculty of Technology, Policy and Management,
Delft University of Technology, The Netherlands
11email<EMAIL_ADDRESS>

# Using Natural Language Processing and Networks to Automate Structured
Literature Reviews: An Application to Farmers Climate Change Adaptation

Sofia Gil-Clavel 11 0000-0003-4707-849X
Tatiana Filatova 11 0000-0002-3546-6930

###### Abstract

The fast-growing number of research articles makes it problematic for scholars
to keep track of the new findings related to their areas of expertise.
Furthermore, linking knowledge across disciplines in rapidly developing fields
becomes challenging for complex topics like climate change that demand
interdisciplinary solutions. At the same time, the rise of Black Box types of
text summarization makes it difficult to understand how text relationships are
built, let alone relate to existing theories conceptualizing cause-effect
relationships and permitting hypothesizing. This work aims to sensibly use
Natural Language Processing by extracting variables relations and synthesizing
their findings using networks while relating to key concepts dominant in
relevant disciplines. As an example, we apply our methodology to the analysis
of farmers’ adaptation to climate change. For this, we perform a Natural
Language Processing analysis of publications returned by Scopus in August
2022. Results show that the use of Natural Language Processing together with
networks in a descriptive manner offers a fast and interpretable way to
synthesize literature review findings as long as researchers back up results
with theory.

###### Keywords:

Natural Language Processing Networks Literature Review Interpretative

##  1 Introduction

The fast expansion of research articles hinders researchers from keeping track
of the emerging findings in their areas of expertise [13]. While systematic
literature reviews are labor-intensive, the median time for them to go
outdated is 5.5 years [22]. Given the current pace of publishing, it is
increasingly challenging to cover the escalating pool of articles and to
monitor the latest developments in a particular field. Automated content
analysis [18] and extractive text summarization systems [5] have been argued
as key tools to support literature synthesis and to keep track of research
trends.

Automated content analysis for literature reviews commonly aims to map text
into topics using the articles’ abstracts. [18] show how automated content
analysis can synthesize big volumes of literature’s abstracts from ecology and
evolutionary biology. In the Climate Change domain, [7] use machine learning
and experts evaluations to map key messages relevant to climate change
mitigation using around 100,000 articles’ abstracts related to demand-side
mitigation. Regarding climate change adaptation, [14] use the abstracts and
metadata from around 7000 articles to map the evolution of climate change
adaptation science across time.

Text summarization systems are normally used for more in-depth literature
reviews that use the full articles’ text. In this context, a summarization
system takes a predefined set of documents and generates a summary, which can
be extractive (the system extracts and ranks the sentences from the original
text) or abstractive (the system extracts knowledge from the original text and
reconstructs them into a new piece) [26]. In this sense, [26] offers a
systematic review of the different methodologies for Electronic Health Records
and biomedical literature summarizations. Based on their systematic review,
83% of the articles analyzed used at least the full text as input. As summary
output, 81% of the articles adopted an extractive approach, while 17% of the
articles used an abstractive approach – the remaining 2% compared both
outputs.

However, automated content analysis and extractive text summarization systems
stem from machine learning and deep learning methodologies, commonly referred
to as Black Box. This is because they make it difficult to theorize based on
their results and to understand how text relationships are built [4, 20]. As
they tend to rank and re-organize the text, which hinders interpretation. On
the one hand, this hinders the understanding of the role of the extracted
piece in the sentence [26]. On the other hand, they might contain redundant
information, which may result in a deficiency of information [26].
Furthermore, these methodologies can also suffer from opacity [6], namely
opacity in terms of what text features the algorithm relied on to perform
classifications. This leads to calls to stop using Black Box methodologies in
favor of interpretable algorithms [20] grounded in theory and that use
tailored databases, as opposed to trained over all the data found in the web
[4].

In this work, we argue that using Natural Language Processing (NLP)
descriptively (as opposed to predictively) offers interpretable methodologies
to perform literature reviews when the results are sensibly visualized using
networks. On the one hand, NLP is a range of computational techniques whose
main goal is to provide machines with the capability to understand natural
language by understanding its semantics. This means that “the machine
understand[s] not just the statistical properties but also the meaning and
context of, say, a certain word” [11, p. 2]. On the other hand, in its
simplest form, a network is a collection of points (nodes) joined together in
pairs by lines (edges). In research, many systems can be represented as
networks, for example, the internet, social interactions, and collections of
computers. Network analysis can be done through their different structures,
properties, and metrics that can help uncover system patterns [16].

The joint use of NLP and networks to summarize text has been done in the past.
In their pioneering work [22] use NLP and networks to generate visual
overviews of unstructured text from novels and blogs to investigate different
research questions, such as “Who was involved in the debate about competing
scientific ideas?” In more recent work, [23] use NLP and networks to provide
insights on policy and scientific discourse around Sustainable Development
Goals. In their network, they use NLP to identify topics in the text and then
build the network using nodes to represent the topics and edge weights to
represent the cosine similarity between the topics. However, to our knowledge,
networks haven’t been used to generate visual overviews of literature reviews.

In this work, we propose a methodology based on NLP and network visualization
to summarize articles’ findings in a descriptive and interpretable manner. In
the rest of the article, first, we explain the articles database to which we
will apply our NLP-supported literature review. Second, we introduce our NLP-
supported literature review algorithm, and apply it to the articles database.
Third, we explain the logic behind the network visualization and use it to
visualize the results from the NLP-supported literature review. Fourth, we
wrap up with the conclusions. Finally, we discuss the limitations and future
work.

##  2 Application to Farmers Adaptation to Climate Change

This section introduces the articles database to which we will apply our NLP-
supported literature review. The articles’ database focuses on drivers of
farmers’ adaptation to climate change. In the literature, there are some
reviews on what the agricultural farmers’ adaptation measures are [2, 3, 21].
But, there is still a gap in the literature regarding what factors motivate
farmers to adapt. In this application, we aim to shed some light on the
factors that motivate agricultural farmers (hereafter called farmers) to
follow climate change adaptation measures.

###  2.1 Data

Our data was retrieved from Scopus [9] during the first week of August 2022.
We derived the data from articles retrieved from SCOPUS in an inclusive
manner. This means that initially, we did not constrain the search to only
farmers. Instead, we downloaded all the articles about human adaptation to
climate change belonging to the categories: Multidisciplinary, Social
Sciences, Arts and Humanities, and Environmental Science.

To derive the data from articles retrieved from SCOPUS, we followed six steps.
First, we searched for articles related to adaptation to climate change using
related search terms, such as climate change, social change, regime shift,
etc. This resulted in 30,000 unique articles. Second, we clustered the
articles into ten different categories using unsupervised learning. Third, we
categorized 5% of the articles in each category as relevant or irrelevant to
the study of climate change adaptation using ASReview [8]. This gave us a
labeled database of 1600 articles. Fourth, using the results from the 5%
categorization, we trained a classifier to categorize the rest of the articles
in the database. This resulted in 2438 articles classified as related to human
adaptation to climate change. Five, we validated the articles were about human
adaptation to climate change and that they talked about the possible drivers
by checking all the abstracts from the articles. Finally, we kept only those
related to farmers’ adaptation to climate change from the resulting articles.
This means we looked for those that contained the words “farmer” or
“agriculture” in either the title or the abstract. Our final database consists
of around 276 articles that were written between the years 2005 and 2022.
Figure 1 shows the articles’ distribution by year and Scopus area.

Figure 1: Articles’ distribution by year and Scopus area (total N=275). The
acronyms stand for: Arts and Humanities (ARTS); Environmental Science (ENVI);
Social Sciences (SOCI); Multidisciplinary (MULT).

##  3 Methodology: General Overview of the Algorithm

We build upon [5] extractive text summarization system to perform the NLP-
supported systematic literature review. In this work, we followed [5] first
six steps and, at the end, introduced three new ones (Figure 2): 1) PDF Text
Extraction; 2) Text Classification and Filtering; 3) Text Normalization; 4)
IMRAD Context Detection; 5) Sentence Segmentation; 6) Sentence Filtering; 7)
Identification of Verbs used to Report Findings; 8) Extraction of Words Based
on Verbs; and 9) Network Visualization. In the following paragraphs, we
further explain each step from Figure 2 while using the data from our
application as an example.

Figure 2: Flow chart diagram presenting our algorithm based on Bui et al.
(2016). Boxes with bolded words signal the steps we added.

First, PDF Text Extraction, as the name hints, transforms the pdf file into
plain text. For this, we used the Python package pdfminer.six [10]. The result
of this step is a data frame where the rows are the articles, and the columns
are the articles’ names and their cleaned plain text. Second, Text
Classification and Filtering categorizes the extracted information into title,
abstract, body text, semi structure, and metadata. For this, we transformed
the articles’ metadata from Scopus [9] into a data frame where the rows are
the articles and the columns are the articles’ metadata values per article.
Then, we merged this with the cleaned plain text database from the previous
step. This resulted in a single database with all the metadata and text by
article.

Third, Text Normalization means transforming the text into canonical form. For
this, we cleaned the data using regular expressions to remove: non-ASCII
characters, URLs, and in-text references, such as “(NAME, YEAR)” and “et al.
(YEAR).” We also used regular expressions to remove all the strings that
referred to the journal’s information and copy-rights. Fourth, IMRAD Context
Detection assigns text snippets into standard scientific-articles’ categories
named IMRAD. IMRAD stands for Introduction, Methods, Results, and Discussion
[24]. For this, we implemented an algorithm to split the articles’ cleaned
plain text into the IMRAD categories. Each of these sections was saved as new
independent columns in the database.

Fifth, Sentence Segmentation involves tokenizing the sentences’ elements.
Here, we used the python package spaCy [25] to tokenize the sentences and to
extract the Parts-Of-Speech of each of the articles’ sections. Sixth, Sentence
Filtering involves filtering out irrelevant information. In our work, this
translated into keeping the columns Findings, Conclusions, and Discussions,
from which we will extract all the sentences where the findings are referred
to, as explain in step seven.

Steps seven to nine comprise the contribution of this work. Step seven
corresponds to Verbs Findings Identification, for this we built a dictionary
of the verbs commonly used in describing findings. We created a sorted
database with all the verbs found in the sentences from step six. Afterward,
for each verb, we read a random sample of ten sentences to categorize the verb
as: positive, negative, neutral, depend, and none. Positive means that the
verb denotes a positive relation, e.g., increase, improve, enhance, etc.
Negative refers to a verb that denotes a negative association, e.g., reduce,
prevent, constrain, etc. Neutral denotes an undirected association, for
example, relate, link, associate, etc. Depend refers to those verbs that
require more information to indicate a possible association and direction. For
example, have could denote a positive association if the following words
contain the word positive (e.g. have a positive correlation) and a negative
association if the following words contain the word negative (e.g. have a
negative association). Other verbs in this category are: affect, influence,
show, etc. However, we assign the verb’s sign as positive or negative once
these verbs are identified based on the extra information. Finally, None
refers to those verbs that do not denote an association between the words,
such as adopt, cope, and implement.

Step eight corresponds to Words Extraction Based on Verbs; it detects the
nouns and adjectives connected by the verbs found in step seven (Figure 3(a)).
For this, we used the Parts-Of-Speech returned in step five. Afterward, we
kept unique sentences in the database, namely unique sentences where the verbs
connect words formed by adjectives and nouns.

Finally, step nine corresponds to Network Visualization. Our implicit
assumption is that the findings can be plotted as directed weighted graphs
easily interpreted as possible associations depending on the verb. For this,
we dropped out the sentences linked by verbs in the None category. Researchers
in the social sciences tend to use vocabulary very carefully; therefore, it is
advisable to interpret their results as associations instead of causations.
For example, Figure 3(a) can be translated into the graph plotted in Figure
3(b). In Figure 3(b), the sign corresponds to the relation between the words
based on the verb: +: positively associated; +/-: neutrally associated; and -:
negatively associated. In the following section, we further elaborate on the
network visualization.

(a) Example sentence of variables relations

(b) Variables translation into a directed weighted graph.

Figure 3: Example of sentence transformation into network.

##  4 Results: Network Visualization of NLP-supported Findings

In this section, we further explain the aesthetics of the network
visualization and provide three possible ways to synthesize the findings using
networks. This section aims to show how networks make the relationships
between the extracted concepts more interpretable by applying the
visualizations to the results obtained from the Scopus papers about drivers of
farmers’ adaptation to climate change.

###  4.1 Aesthetics of the Network Visualization

For all the networks, we used the following aesthetics. First, the words are
arranged in concentric circles where the central words denote terms that have
the highest node degree. First, we measure the words’ degree to place the
nodes in concentric circles. For this, we only considered whether the word
appears in at least one finding sentence of each article, but we did not add
up the frequency it appears in the article (i.e., by article, its maximum
value is one). Suppose we assume there are: $n$ articles and $p$ words in the
database. Then the nodes’ degree ($D$) would be given by Equation 1. In
Equation 1, $I_{\\{.\\}}$ denotes the indicator function, which returns 1 when
the word $W_{j}$ appears in the findings sentences of article $A_{i}$ and 0 in
any other case.

$D(W_{j})=\sum_{i}I_{\\{A_{i}(W_{j})\\}}\quad\textrm{i=[1,n]; j=[1,p]}$ (1)

Second, the edge weight denotes the weighted frequency the words appeared
related in the findings. To measure the edge weight, we use the ratio of the
number of times the relationship between the words appeared in one article
divided by the number of times the same relation appeared in all the articles.
The edge weight ($EW$) would be given by Equation 2. In equation 2,
$A_{i}(W_{k},W_{l})$ is 1 when there is a findings’ sentence that associates
words $W_{k}$ and $W_{l}$ and 0 in any other case.

$EW(W_{k},W_{l})=\frac{\sum_{i}A_{i}(W_{k},W_{l})}{\sum_{i}\sum_{j1}\sum_{j2}A_{i}(W_{j1},W_{j2})}\quad\textrm{i=[1,n];
j1,j2=[1,p]}$ (2)

Third, the edge color shows the most frequent weighted verb sign linking the
words. This is measured by the ratio of the number of times the verb sign
appeared in the relation between the words in one article divided by the
number of times the verb sign appeared between the words in all the articles.
The verb sign ($VS$) would be given by Equation 3. In equation 3,
$A_{i}(s,W_{k},W_{l})$ is 1 when there is a findings’ sentence that associates
words $W_{k}$ and $W_{l}$ by the sign s and 0 in any other case.

$VS(s,W_{k},W_{l})=\frac{\sum_{i}A_{i}(s,W_{k},W_{l})}{\sum_{i}\sum_{j1}\sum_{j2}A_{i}(s,W_{j1},W_{j2})}\quad\textrm{i=[1,n];
j1,j2=[1,p]}$ (3)

Finally, the color assigned to the node denotes the cluster to which it
belongs. However, as shown in the next section, “Application of the Network
Visualization,” the color of the verb sign could be applied to the node, in
which case the visualization would not show the edges. The following section
shows how these aesthetics help with the interpretability of the NLP-supported
literature review.

###  4.2 Application of the Network Visualization

In this section, we show three possible ways to synthesize the information
using networks. First, we show a network summarizing the articles’ findings.
Second, we show a word cloud derived from the networks. Finally, we show a
network of the factors that drive famers’ adaptation to climate change.

The first network visualization summarizes the articles’ findings. For this,
just showing the connections is not enough. The network visualization has to
be done so that the reader can synthesize the knowledge immediately.
Therefore, we visualize the connections based on the cluster to which the
words belong according to the modularity algorithm [15]. Figure 4 shows the
first four clusters of a random sample of 10 articles (we do not show the
results for the entire database because of space constraints). Here we see
that the clusters’ central terms (i.e., the nodes with the highest degrees per
cluster) are: adaptation, climate, farmer, and information. The first three
central terms give us all the necessary information to understand the main
topic of the papers: Farmers’ Adaptation to Climate Change. The fourth central
term refers to one of the, possibly, most important drivers of adaptation:
information.

Figure 4: First four clusters based on modularity for a random sample of 10
articles’ findings in our database. Nodes colors correspond to the words’
cluster.

As information appears to be an important driver of farmers’ adaptation to
climate change, then we can choose to plot the networks of words related to
information. As this network uses all the available information in the
database, it is necessary to simplify it. Therefore, we split the
visualization in two: the words whose target is information (Fig. 5) and the
words whose source is information (Fig. 6). Moreover, as the number of edges
increases, removing the edges and coloring the nodes based on the edge sign is
better. Figure 5 shows the network of words whose target is information. From
here, we see that some variables that enable information are, among others:
community, context, finance, age, and (in the case of women) external
assistance.

Figure 5: Network where information is the target word. The node color
represents the most frequent sign between the nodes and information.

Figure 6 shows the network of words whose source is information. Figure 6
shows that information enables awareness, adaptive capacity, skill, adaptation
measure, and adaptation strategy. However, in the case of adaptation to
climate change, Figure 6 shows two interesting results. While information is
positively associated with the node “adaptation climate change,” information
is neutrally associated with the node “climate change adaptation.” These two
last results show two limitations of our methodology. First, the algorithm
does not detect that those nodes refer to the same term. Second, the network
visualization is based on categorical definitions of the sign, namely, the
sign that appeared most frequently connecting those nodes. Therefore, it does
not consider that some variables might have very close percentages of
positive, negative, and neutral associated verbs.

Figure 6: Network where information is the source. The node color represents
the most frequent sign between information and the nodes.

The final visualization corresponds to the network of factors that drive
farmers’ adaptation to climate change (Figure 7). For this visualization, we
used the findings from all the articles. Still, we only kept those nodes that
target at least one of the following nodes: farmer adaptation, climate change
adaptation, and adaptation climate change. The node color is based on the
cluster, and the edge color corresponds to the most frequent sign associated
with the verbs connecting the words. This visualization makes it easier to
read the findings. For example, age, gender (shown as woman), education,
knowledge, and information are frequently positively associated with farmers’
adaptation to climate change. This aligns well with the in-depth empirical and
theoretical literature on Climate Change Adaptation, relating socio-
demographics of individuals to their adaptive capacity and information to
individual awareness, risk perception, and response efficacy of a Climate
Change Adaptation measure [12, 17]. Government and social capital frequently
have a neutral association with farmers’ adaptation in our results, confirming
that the trust in government-led adaptation and social influence and norms are
indeed universal drivers as theorized and found in individual cases [1, 19].
For the negative associations, we could not connect them with current
literature on climate change adaptation because the nodes “friend”, “climate”,
and “farmer adaptation climate” do not convey enough information.

Figure 7: Clustered networks of words that target at least one of the
following terms: farmer adaptation, climate change adaptation, and adaptation
climate change. Node colors correspond to the words’ cluster.

These network visualizations also make it easier to identify and see what
variables are more often connected in the articles. This is because of their
network location: the closer to the center, the bigger the node degree. For
example, in Figure 7, factors like age, information, and market are more
likely to appear as farmers’ adaptation factors in the database. In contrast,
transformative social learning, access extension agent, and government appear
less often. So, the visualization gives not only the direction and the
frequency of the association between the words but also the frequency the word
appears in the database relative to others.

##  5 Conclusions

This article sought to propose an interpretable NLP algorithm and descriptive
visualizations to summarize articles’ findings when performing systematic
literature reviews. Our approach aimed to avoid using Black Box algorithms and
opaque machine learning methodologies to extract and synthesize information.
On the one hand, our method shows that the use of tailored verb dictionaries
help to interpret the results shown in the networks. On the other hand,
sensible visualizations of the results also facilitate synthesizing the
information.
In terms of our application, we found that information appears as the most
frequent term positively associated with farmers’ adaptation to climate
change. As, it might enable farmers’ awareness, adaptive capacity, skills,
adaptation measures, adaptation decisions, and adaptation strategies. Besides
information, other frequently positively associated variables are age, gender,
education, and knowledge. However, results derived from this methodology are
just intended to help researchers synthesize information from large amounts of
articles. Even though results interpretation should be made cautiously and
always based on previous knowledge, this methodology can help researchers keep
track of the findings from their areas of expertise.

##  6 Limitations and Future Work

Our work has some limitations that we would like to acknowledge. Compared to
machine learning, our methodology requires that researchers subjectively
classify the verbs found in sentences samples as conveying positive, negative,
or neutral associations between words. However, we do not believe this to be a
problem as long as researchers openly share the dictionary they used when
implementing this algorithm. Furthermore, as the results are descriptive,
everything can be tracked down to check why some words show specific
associations and to which article the relations belong. Another limitation is
that we based our network visualizations on categorical definitions of the
signs: the sign that appeared most frequently. This definition has the
limitation that it does not consider that some variables might have very close
percentages of associated positive, negative, and neutral verbs. Future work
can explore the possibility of showing signs’ colors in terms of continuous
variables.

##  7 Acknowledgements

This work was supported by the Netherlands Organization for Scientific
Research NWO VIDI grant number 191015. We want to thank the following people
for their help during different stages of this work. Professors Pradeep
Murukannaiah and Jazmin Zatarain-Salazar for their feedback during this
project’s early stages. For their help during the ASReview session, the Ph.D.
candidates: Alessandro Taberna, Asli Mutlu, Brayton Noll, Ignasi Cortés-
Arbués, Lauren Grimley, Liz Verbeek, and Thorid Wagenblast. Finally, Ryan Tan
Yi Wei, Joos Akkerman, and Thorid Wagenblast for their help validating and
downloading the articles.

##  8 Reproducibility

The code to replicate this work will be available at Gil-Clavel’s GitHub
repository (https://github.com/SofiaG1l) upon publication of the article.

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# Geometric frustration of hard-disk packings on cones

Jessica H. Sun  Harvard John A. Paulson School of Engineering and Applied
Sciences, Harvard University, Cambridge, MA 02138, USA  Abigail Plummer
Princeton Center for Complex Materials, Princeton University, Princeton, NJ
08540, USA  Grace H. Zhang  Department of Physics, Harvard University,
Cambridge, MA 02138, USA  David R. Nelson  Department of Physics, Harvard
University, Cambridge, MA 02138, USA  Vinothan N. Manoharan  Harvard John A.
Paulson School of Engineering and Applied Sciences, Harvard University,
Cambridge, MA 02138, USA  Department of Physics, Harvard University,
Cambridge, MA 02138, USA

###### Abstract

Conical surfaces pose an interesting challenge to crystal growth: a crystal
growing on a cone can wrap around and meet itself at different radii. We use a
disk-packing algorithm to investigate how this closure constraint can
geometrically frustrate the growth of single crystals on cones with small
opening angles. By varying the crystal seed orientation and cone angle, we
find that—except at special commensurate cone angles—crystals typically form a
seam that runs along the axial direction of the cone, while near the tip, a
disordered particle packing forms. We show that the onset of disorder results
from a finite-size effect that depends strongly on the circumference and not
on the seed orientation or cone angle. This finite-size effect occurs also on
cylinders, and we present evidence that on both cylinders and cones, the
defect density increases exponentially as circumference decreases. We
introduce a simple model for particle attachment at the seam that explains the
dependence on the circumference. Our findings suggest that the growth of
single crystals can become frustrated even very far from the tip when the cone
has a small opening angle. These results may provide insights into the
observed geometry of conical crystals in biological and materials
applications.

###### pacs:

Valid PACS appear here

##  I Introduction

The growth of a crystal can be frustrated by interactions with a curved
surface such as a spherical or hyperbolic substrate [1, 2, 3, 4]. When the
surface has nonzero Gaussian curvature, the frustration stems from variations
in the surface metric, which lead to stretching of the crystal lattice. This
type of geometrical frustration has been well studied, particularly in
colloidal systems [5, 6, 7, 8, 9, 10, 11, 12, 13].

Less well studied is frustration arising on surfaces with no Gaussian
curvature but on which crystals can form loops, such as cylinders. Although
such surfaces do not stretch the lattice, they can nonetheless frustrate a
crystal by imposing a closure constraint. As observed in experiments and
simulations on colloidal crystals on cylindrical fibers [14], crystals with
orientations that are incommensurate with the closure constraint form seams.
These seams, which are stable on cylinders but not on flat surfaces
unconstrained by periodic boundary conditions, break the translational
symmetry of the crystal.

Here we examine how the closure constraint affects crystallization on a cone,
which, unlike a cylinder, has a spatially varying circumference. As a
consequence, a seam must form with a width that varies in the axial direction
whenever the cone angle does not permit the crystal to wrap perfectly around
the cone (for a triangular lattice, such commensurate wrappings can be
achieved by placing, for example, a 60° disclination at the cone apex). The
seam is similar to a tilt grain boundary between two misoriented crystals on a
flat substrate [15, 16], except that it is a boundary between the misoriented
edges of a _single_ crystal that has wrapped around the cone. This seam can
break both the translational and rotational symmetry of the crystal. We seek
to understand how the closure constraint geometrically frustrates crystal
growth on a cone.

There are few previous studies of crystallization on a cone. Basin-hopping
simulations of colloidal crystals showed that interacting particles on a cone
form seams or scar-like defects [17, 18]. The aim of these simulations was to
understand the defect structure and how it changes with the cone geometry. An
experimental study of an atomic system, WS2, showed that crystals on a cone
form a distinct seam [19]. The cone in this work had a large opening angle,
and the size of the WS2 crystal was orders of magnitude larger than the size
of the atoms. The main aim of this study was to demonstrate the existence of
the seam, which the authors refer to as a tilt grain boundary.

Our aim is to examine the growth process, and in particular how conical
crystals grow after closure. In contrast to the basin-hopping simulations [17,
18], which reveal energy-minimizing crystal structures, we aim to determine
how an out-of-equilibrium growth process leads to disorder. Furthermore, we
focus on particles with short-ranged attractions, as seen in previous
experimental studies of crystallization on a flat surface [20], sphere [11],
and cylinder [14].

To isolate the consequences of geometrical frustration on the crystal
structure—and avoid complications associated with multiple nucleation sites
and kinetics—we use a greedy algorithm to simulate the idealized growth of a
crystal on a conical surface. Our approach, based on an algorithm developed by
Bennett to study metallic glasses [21], is inspired by previous work on
understanding the effects of geometrical frustration in metallic glasses, and
on spherical or hyperbolic surfaces [22, 23]. Because we aim to understand
effects in the quasi-two-dimensional colloidal systems realized
experimentally, we simulate disk packings rather than sphere packings to
simplify the computation.

Our simulation is designed to model the slow, reaction-limited growth of a
single crystal from a fixed nucleus. Briefly, we initialize the simulation
with three disks placed in a triangular configuration with a defined
orientation and position on the surface (Fig. 1A). At each subsequent step,
the algorithm places a single disk in a position that maximally reduces the
energy of the crystal interface. We do not allow the particles to rearrange
following placement. On a flat surface, this algorithm produces a perfect
crystal. Therefore any deviation from a perfect crystal on a conical surface
is the direct result of geometric frustration, due to a $\delta$-function of
Gaussian curvature at the cone apex [24]. Although this algorithm does not
account for the effects of temperature or kinetics, it is a simple and
effective way to model the effects of the closure constraint on crystal
growth. The details of the method are given in Sec. II.

By modeling crystal growth in this manner, we find a proliferation of defects
(defined here as particles with anomalous coordination numbers) for a crystal
growing towards the tip of a cone with a small opening angle, as shown in Sec.
III. As we shall show, this onset of disorder results from a finite-size
effect that depends strongly on the local circumference and is insensitive to
the seed orientation and cone angle. Intuitively, the disordered regions
appear when a significant fraction of the growing interface consists of the
seam of the crystal. We develop a theoretical model that explains the results
in Sec. IV.1 and discuss the influence of the crystal seed location in Sec.
IV.2 (the appendices provide additional context, including discussions about
corrections due to the three-dimensional (3D) nature of the particles,
commensurate packings on cylinders, and a particularly interesting alternative
seed composed of a ring of particles). We conclude by noting that the
transition to a disordered packing can occur surprisingly far from the tip,
which may give some insights into the morphology of crystals seen in biology
and materials (Sec. V).

##  II Methods

Figure 1: Geometric parameters in the simulation. (A) Diagram of a triangular
crystal seed. The angle $\theta$ describes the orientation of the lattice
vector relative to the red cone circumference curve $C$. (B) Rendering showing
results of a Bennett-type simulation [21] for a 2D packing of disks. We
initialize the crystal seed (marked in black) on a 3D cone of angle $\beta$.
The color of each particle indicates its coordination number $N_{j}$. (C)
Rendering showing a mapping of the 3D cone in B to a 2D unrolled sector of
angle $\phi=2\pi\sin(\beta/2)$, where the seed is at a sector radius $R$ from
the apex. In this example, $\theta=30^{\circ}$, $\phi=20^{\circ}$, $C=12a_{0}$
and $R=54.4a_{0}$. A seam consisting of particles with coordination number
$N_{j}<6$ runs in the axial direction.

Our algorithm is designed to simulate an idealized crystallization process in
which a crystal grows from a single nucleus in a reaction-limited fashion. To
create the nucleus, we place three close-packed particles with diameter
$a_{0}$ in a tangent plane at a radius $R=C/\phi$ from the sector vertex and
with an orientation $\theta$ with respect to the cone circumference $C$ (Fig.
1A). We then add particles one by one, such that each particle contacts the
greatest number of other particles, or, equivalently, forms the greatest
number of “bonds.” A bond is formed when the centers of two particles are less
than $a_{0}$ apart with a tolerance of $10^{-4}a_{0}$, representing an
interaction potential with a narrow attractive well, as is found in a number
of colloidal adsorption experiments [11, 14]. If there are several degenerate
options for placing the particle, we randomly select one option. We do not
place particles such that they form only one bond with the existing assembly
because the position of a dangling bond is not well-defined, and a rotation of
the dangling bond quickly leads to contact with two particles. We also do not
allow any previously placed particles to move, nor do we let the entire
crystal translate or rotate.

We choose 2D circular disks of diameter $a_{0}$ to represent the effective
shapes of 3D spherical particles adsorbed onto a conical surface, and we
neglect the anisotropic, position-dependent stretching of particle projections
onto a conical surface. This approximation allows us to map the three-
dimensional (3D) cone of cone angle $\beta$ into a flat two-dimensional (2D)
circular sector with a periodic boundary condition and a sector angle
$\phi=2\pi\sin(\beta/2)$. The mapping is one-to-one because the circumference
$C$ of a circular cross-section of the 3D cone (Fig. 1B) is equivalent to the
arc length $C$ of the 2D sector at a given $R$ (Fig. 1C). The resulting 2D
algorithm is computationally simpler, yet is still able to capture the effects
of the closure condition on frustration (see Appendix A for a discussion of
the limitations of this approach).

We truncate the sector at a radial distance of $20a_{0}$ from the seed
position to encourage growth into the sector tip, where we expect to see
interesting structures. A simulation terminates when no more particles can be
placed into the sector or 1000 particles have been placed. For comparison, the
maximum number of particles that can pack into a $\phi=5^{\circ}$ sector is
around 800.

Since our aim is to understand the effects of geometrical frustration, we
select conditions under which perfect triangular crystals cannot form. Perfect
crystals lack seams and can form only when the sector angle is a “magic” angle
$\phi=60^{\circ}P$, where $P$ is an integer [25, 26, 24]. For example, a
cylinder is a $P=0$ magic cone with $\phi=0^{\circ}$. We therefore restrict
our study to cones with $\phi\neq 60^{\circ}P$. A seam on such a cone is shown
in Fig. 1B and C. Particles at the crystal self-boundary have coordination
number $N_{j}<6$. For particles with Lennard-Jones-like pair potentials where
the range of attractive interaction is comparable to the defect hard-core
diameter, one might expect a grain boundary to form [5]. Our simulations,
however, have an interaction range of order $10^{-4}$ times the hard-core
diameter, and the resulting seam, like a stacking fault, maintains its
integrity.

We choose to study near-cylindrical cones with small opening angles of
$\phi\leq 30^{\circ}$ to facilitate comparison with results of previous
studies of crystals on cylinders [27, 14]. When simulating growth on a surface
with $\phi=0^{\circ}$, corresponding to a true cylinder, we select a seed
orientation $\theta$ such that seams are still geometrically required.

To characterize the structures, we calculate the bond orientational order
parameter $\psi_{6,j}$ for each particle $j$ with nearest neighbors indexed by
$k$ [28, 29]:

$\psi_{6,j}=\frac{1}{N_{j}}\sum_{k=1}^{N_{j}}e^{i6\theta_{jk}},$ (1)

where $\theta_{jk}$ is the angle between the circumferential axis and the
vector from particle $j$ to nearest neighbor $k$, and $N_{j}$ is the number of
nearest neighbors, or coordination number, of particle $j$. We consider only
particles that are separated by $a_{0}\pm 10^{-4}a_{0}$ as nearest neighbors.

We also calculate the defect density $\rho$ as a function of distance $R$ from
the vertex. We define defects as particles that have
$\left|\psi_{6,j}\right|<0.9$. This definition allows us to distinguish
defects, which disrupt the order of the crystal, from particles on the
boundary of the seam, which are part of an ordered crystal. Since the same
defect trends are preserved for different cutoff values of
$\left|\psi_{6,j}\right|$, we choose a high cutoff value to obtain a sensitive
measure of the defect density (see Appendix B). To calculate the defect
density, we first bin the particles by $R$. We then calculate the number of
defects per number of particles within each bin of width $a_{0}$, averaged
over 100 trials.

##  III Results

Figure 2: Variation of particle packings with seed orientation $\theta$ on
cones with sector angle $\phi$. (A) Rendering of simulation results at
unrolled cone angle $\phi=5^{\circ}$ show that particles in the vicinity of
the seed placed at $R=114.59a_{0}$ $(C=10a_{0})$ are ordered. As the crystal
approaches the vertex, the packing becomes disordered. (B) Plot of the defect
density $\rho$ as a function of radial distance $R$ from the tip, where $\rho$
is the number fraction of particles with $\left|\psi_{6,j}\right|<0.9$ in a
given $R$ bin averaged over 100 trials at each seed orientation $\theta$. The
defect density curves are similar for all $\theta$ at $\phi=5^{\circ}$. Note
that the defect density rises from the seed toward the tip and then drops to
zero at $R\approx 28.5a_{0}$ because the particles are squeezed out of the
tip.

We first explore the effects of seed orientation on the structure of the
system. We find that the crystal initially grows from the seed as an ordered
packing of particles, represented by grey particles with $N_{j}=6$ in Fig. 2A.
As the crystal wraps around the cone, it meets itself to form a seam
consisting of particles with $N_{j}<6$. However, as the seam approaches
smaller circumferences, these defects begin to dominate the growth interface,
leading to the formation of a disordered region consisting primarily of defect
particles with $N_{j}<6$ (see Fig. 2A, regions near tips of cones).

For all seed orientations, the defect density follows a similar curve (Fig.
2B), tending to increase as $R$ decreases. The local minimum at approximately
$R=40a_{0}$ corresponds to the location where a crystalline cluster three
particles wide can form. The defect density then increases again for smaller
$R$, until it reaches a maximum value and falls rapidly to $\rho=0$ at
approximately $R=28.5a_{0}$. Note that $R\approx 23a_{0}$ represents the limit
for packing two particles side by side on unrolled cones of angle
$\phi=5^{\circ}$. Overall, the tendency of the defect density to increase as
the cone narrows suggests that finite-size effects are responsible for the
disorder near the tip.

When we initialize crystals on cones with different sector angles with fixed
seed orientation $\theta=40^{\circ}$, we find that the position at which the
disordered region emerges also varies, as seen for unrolled cones in Fig. 3A.
For small sector angles, we find more disorder farther from the tip, while for
larger sector angles, the disordered region is found nearer the tip (Fig. 3B).
These results show that cones with small angles can have long disordered
regions.

By rescaling $R$ by $R\phi=C$, we find that the defect density collapses as a
function of cone circumference (Fig. 3C). Defect proliferation therefore
depends on the circumference and not strongly on the sector angle or seed
orientation. Although the precise sector angle and seed orientation might
alter the details of the closure constraint at the single-particle scale, the
circumference at which closure occurs has a greater effect on crystal growth.

The density curves do not perfectly collapse because of our discrete binning
procedure, which results in a systematic variation with the sector angle. As
$\phi$ increases, the gradient of C also increases. An annular bin of fixed
width centered at C will not only access larger circumferences but also have
more particles at the larger circumference, which are less likely to be
defects. Therefore, at a given C, the defect density for the annular bin is
biased towards a lower $\rho$ as $\phi$ increases. It is possible that the
gradient of $C$ could also affect the defect density in other ways.
Nonetheless, the near-collapse of the density curves upon rescaling shows that
the circumference, rather than the gradient, is the most important parameter
to consider.

To show that defect proliferation can be understood as a function of
circumference, we examine crystal growth on cylinders, which have a constant
circumference. We find that cylinders with large circumferences have lower
defect densities than cylinders with small circumferences (Fig. 4A). For large
cylinders, the crystal forms a seam, as expected, and is ordered with few
defects. As the circumference decreases, however, the crystal becomes
increasingly fragmented as more defects are incorporated into the packing. For
thin cylinders, the packing is predominantly disordered.

We find that the defect densities on cylinders as a function of circumference
follow the same trend as the defect densities of cones (Fig. 4B), provided the
orientation $\theta$ of the triangular seed cluster is not tuned to the
special phyllotactic value that allows a commensurate tiling by a triangular
crystal [30, 31, 32]. Because our focus here is on seeds with random
orientations, we neglect the interesting regular tilings that occur for
commensurate seed orientations at fixed cylinder radius, or commensurate
cylinder radius at fixed seed orientation (Appendix C). For seed orientations
$\theta=40^{\circ}$, we find that the defect density distributions appear
exponential for small $R$, regardless of whether the surfaces are conical or
cylindrical (Fig. 4C). Crystals on cylinders therefore reproduce the finite-
size effects seen in crystals on cones, if special commensurate tilings are
ignored [30, 31].

Figure 3: Variation of particle packings with unrolled sector angle $\phi$ at
fixed seed orientation $\theta$. (A) Renderings of simulation results show a
long region of $N_{j}<6$ particles at $\phi=5^{\circ}$, while the disordered
regions are concentrated closer to the tip at $\phi=10^{\circ}$ and
$\phi=15^{\circ}$. The seed crystals are at $C=10a_{0}$ with
$\theta=40^{\circ}$. (B) Plot of the defect density $\rho$ as a function of
$R$, averaged over 100 trials. The distribution is broad for $\phi=5^{\circ}$
and becomes narrower for increasing sector angles. (C) For each $\phi$, $\rho$
is mapped to $C$ by $C=R\phi$. $\rho$ collapses as a function of $C$.  Figure
4: Particle packings on 2D cylinders of different circumferences, measured in
units of the hard core diameter $a_{0}$. (A) Rendering of representative
results for simulations. At $C=10a_{0}$, the crystal consists of primarily
$N_{j}=6$ particles. As $C$ decreases, the crystal becomes fragmented and
disordered. The seed orientation is fixed at $\theta=40^{\circ}$. (B) Plot of
defect density of cylinders (circles) superimposed on the plot for cones from
Fig. 3C (lines). The cylinder defect densities are calculated as the number
fraction of defects relative to the total number of particles, averaged over
100 trials at each circumference. (C) As the circumference decreases, the
defect density grows exponentially at small circumferences for both cones and
cylinders. The chosen circumferential values $C/a_{0}=$2, 4, 6, 8, and 10 in
this figure do not include any of the special values that would result in
perfect packings for $\theta=40^{\circ}$ (Appendix C) [32].

##  IV Discussion

###  IV.1 Circumference determines defect density

To provide some intuition for exponential defect proliferation observed in our
simulations at small cone circumferences, we introduce a simple theoretical
model for slow, reaction-limited growth on a _cylindrical_ substrate. We
expect our algorithm to simulate slow, reaction-limited growth because
particles attach one-by-one to the growing crystal, and each added particle
minimizes the local energy of the crystal interface. We use a cylindrical
substrate to simplify our theoretical arguments because, as shown above,
cylinders reproduce the finite-size effects seen on cones.

Given a crystal with a seam and smooth facets as in Fig. 5, we consider the
types of lattice sites that an additional particle can diffuse to in the
context of a 2D terrace-ledge-kink model, used to describe ideal surface
crystal growth [33]. Particles sitting at the crystalline edges form the
ledges where new particles can adsorb. Kinks describe missing particles along
the ledge. In our simulation of slow ideal growth, kinks, which have three or
more dangling bonds by definition, are higher-energy sites than the smooth
ledges, which have two dangling bonds. Therefore, a particle diffusing to the
crystal will adsorb to kinks first.

Once the kink sites have been filled, a particle can attach to two types of
energetically equivalent lattice sites. The first, a ledge site, is continuous
with the preexisting crystal (Fig. 5A, dark blue circle). The second, a seam
site, is incompatible with the preexisting crystal (Fig. 5B, dark red circle).

Figure 5: Illustration of the defect growth process at a seam. Particles in
the preexisting crystal are shown as filled gray circles. When all candidate
sites result in the formation of two bonds, a particle can attach randomly at
(A) a ledge site that initiates a new crystal row or (B) a seam site that
creates more sites that break the symmetry of the preexisting crystal. Dark
open circles show new particle locations, with lighter open circles
emphasizing candidate sites created by attachment of the new particle. Note
that this schematic depicts growth on a cylinder—the crystal rows on either
side of the seam are parallel.

Particle attachment to a ledge site results in on-lattice growth, meaning that
the symmetry of the preexisting crystal is preserved. Particle attachment to a
seam site results in off-lattice growth, meaning that the symmetry of the
preexisting crystal is broken. We expect the density of disordered defects to
be related to the proportion of seam sites to ledge sites. Crucially, off-
lattice growth increases the probability of further off-lattice growth because
adsorption of a seam particle increases the number of off-lattice candidate
states overall (light red circles in Fig. 5). Thus, if off-lattice growth is
likely, this argument predicts that it will only become more likely as growth
proceeds, initiating the formation of a disordered region as we see in our
simulations.

To further develop this simplified picture, we estimate the probability of
disordered growth. A perfect crystal with rows composed of
$N_{\textnormal{row}}$ particles and a single seam as in Fig. 5 has of order
$N_{\textnormal{row}}$ candidate sites that lead to on-lattice growth, and
only order one candidate sites that lead to off-lattice growth. Therefore, the
probability of on-lattice growth occurring initially is
$P_{1}(t=0)\sim(1-1/N_{\textnormal{row}})$, and the probability of off-lattice
growth is $P_{2}(t=0)\sim 1/N_{\textnormal{row}}$. If on-lattice growth
occurs, $P_{1}$ and $P_{2}$ do not change. If off-lattice growth occurs, we
expect that $P_{2}$ increases by an amount that scales with
$1/N_{\textnormal{row}}$. If $P_{2}$ reaches some threshold value—say
$P_{2}=1/2$, at which half of the candidate sites lead to off-lattice
growth—runaway off-lattice growth results, leading to the formation of a
disordered region.

What is the probability of $P_{2}(t)$ increasing to this threshold value? If
on-lattice growth occurs following off-lattice growth, there may be some
healing of the disordered seam region, and $P_{2}$ may decrease. We therefore
make the simplifying approximation that $P_{2}$ increases only when off-
lattice growth occurs many times in a row, and $P_{2}$ increases by
$1/N_{\textnormal{row}}$ every time off-lattice growth occurs consecutively.
With these assumptions, the probability of off-lattice growth occurring $s$
consecutive times scales as

$\displaystyle P_{s}={}$ $\displaystyle P_{2}(t=0)\,P_{2}(t=1)\ldots
P_{2}(t=s)$ (2)
$\displaystyle\sim\frac{1}{N_{\textnormal{row}}}\times\frac{2}{N_{\textnormal{row}}}\times\ldots\times\frac{s}{N_{\textnormal{row}}}=\frac{s!}{\left(N_{\textnormal{row}}\right)^{s}}.$

To find the probability that $P_{2}$ reaches $1/2$, we let
$s=N_{\textnormal{row}}/2$ and make Stirling’s approximation:

$P_{N_{\textnormal{row}}/2}=\frac{(N_{\textnormal{row}}/2)!}{\left(N_{\textnormal{row}}\right)^{N_{\textnormal{row}}/2}}\sim\frac{\sqrt{N_{\textnormal{row}}}}{2^{N_{\textnormal{row}}/2}}e^{-N_{\textnormal{row}}/2}.$
(3)

Therefore, this simple model predicts that the probability of a disordered
region initiating increases exponentially as the number of particles in a
crystal row encircling the cylinder decreases. The predicted exponential
scaling is consistent with the simulation results, which show that the defect
density increases exponentially with decreasing circumference (Fig. 4C). The
same argument can explain defect formation on a cone, albeit with some
subtleties, which we discuss in Appendix D.

###  IV.2 Seeds close to the tip

Figure 6: Particle packings on sectors initialized at different seed
positions. (A) Rendering of simulation results for $\phi=5^{\circ}$ with seed
positions at $C=5a_{0}$, $C=7a_{0}$, and $C=9a_{0}$ and a seed orientation of
$\theta=40^{\circ}$. (B) Plot of the defect density $\rho$ as a function of
$C$, averaged over 100 trials. Crystals seeded at small circumferences have
dips in the defect density that correspond to the seed location.

These results, all of which concern seeds placed far from the tip of the cone,
raise the question of whether placing seeds closer to the tip of the cone
might help prevent disorder near the tip. We therefore examine simulations of
crystallization with seeds placed at different circumferences.

We find that while crystals seeded far from the tip (at $C=9a_{0}$) are able
to grow normally towards the tip until the onset of disorder, crystals that
are seeded closer to the tip (at $C=7a_{0}$ and $C=5a_{0}$) first form a small
crystal before disorder emerges (Fig. 6A). The formation of these small
crystals is reflected in the defect density, which shows a dip at the
circumference corresponding to the seed location for all seeds placed at
$C<9a_{0}$ (Fig. 6B).

We can explain these results using the model from Section IV.1. A crystal
grows until the closure constraint demands formation of a seam. But when the
crystal is seeded near the tip, the circumference is small, and hence
$N_{\textnormal{row}}$ is small. Therefore off-lattice growth is more probable
at these small circumferences, and the crystal becomes frustrated a short
distance from the seed.

Interestingly, in some cases we see that new crystals can form at the wider
part of the cone, as shown in the $C=7a_{0}$ example. However, these new
crystals quickly become frustrated again as new seams and grain boundaries
form. Consequently, the defect density dips at the circumference of the seed
and then rises with increasing circumference, until it exceeds the defect
density for a seed placed at $C=10a_{0}$ (Fig. 6B). We conclude that crystals
that are seeded near the tip can temporarily escape the finite-size effect
leading to disorder near the tip, though at the expense of increased disorder
farther from the seed site.

##  V Conclusion

We have shown that crystal growth on a cone is geometrically frustrated. For
any non-magic cone angle, a seam is required. A disordered region forms near
the tip because defects tend to appear at the seam, and the probability of
these defects proliferating increases exponentially as the circumference
decreases.

This type of frustration has implications for slow, reaction-limited crystal
growth on cones. Near-cylindrical cones have long sections in which defects
form with high probability, resulting in large areas of potentially disrupted
crystallization. In wider cones, the increase in defect probability is
concentrated at the tip and can block tip closure, leading to holes at the
tips of conical shells.

These results may help explain tip-closure problems observed in experimental
systems. For example, the conical capsids of HIV often exhibit large holes at
the tip [26]. Also, crystals of WS2 have been found to terminate unexpectedly
far from the tip [19]. Future experiments on colloidal systems, such as the
system described in Chapter 6 of Ref. 34, might shed light on whether tip-
closure failures are the result of the frustration mechanism revealed by our
simulations.

Our results also show that control over nucleation may be crucial to
fabricating conical crystals for applications. Disordered regions form near
the tip of the crystal, regardless of whether the seed is far or close to the
tip. But a crystal seeded near the tip can temporarily bypass the finite-size
effect, resulting in a locally reduced defect density. Therefore, if the
surface or interactions can be controlled such that nucleation occurs close to
the tip, at least small crystals can be formed in this region. Furthermore, if
the geometry of nucleation can be controlled, new crystalline structures might
be realized experimentally. In Appendix E, we discuss how a nucleus consisting
of a ring of particles might grow, and how the size of the resulting crystal
depends on the elastic modulus.

Our simulations used a greedy algorithm because our aim was to reveal the
geometric frustration faced by crystals growing on a conical surface. Our
simulations do not account for kinetics, thermal fluctuations, or vibrational
entropy. Future simulations and experiments are therefore needed to develop a
more complete physical understanding of conical crystal growth. Nonetheless,
our results show that, apart from the special case of magic-angle cones, any
conical crystal is subject to geometrical frustration that promotes disorder
at small circumferences.

## Data Availability Statement

Data for the simulations are openly available on the Harvard Dataverse [35].
Code for the simulations is available under the GNU General Public License v3
at https://github.com/manoharan-lab/cone-disk-packings.

###### Acknowledgements.

We thank Lara Braverman for insightful conversations. This research was
primarily supported by the National Science Foundation through the Harvard
University Materials Research Science and Engineering Center under grant
number DMR-2011754. Additional support was provided by the National Science
Foundation Graduate Research Fellowship Program under grant numbers
DGE-2140743 and DGE-1745303.

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##  Appendix A 3D corrections to effective particle shape

Using a constant particle shape simplifies our analysis considerably. However,
a position-dependent particle shape would more accurately model 3D spheres
assembling onto a cone. Though we do not consider these corrections in this
work, we describe them briefly here.

To provide intuition, we first consider two circular particles tangent to each
other and to the surface of a circle, as shown in Fig. S1.

Fig. S1: Diagram showing that the distance between particles along a surface
differs from $2a_{0}$ when the surface is curved.

The distance between particle centers is $d_{\textnormal{ctr}}=2a_{0}$.
However, measured along the surface, the distance between the two points at
which the particles touch the substrate is

$\displaystyle d_{\textnormal{surf}}$
$\displaystyle=2R\sin^{-1}\left(\frac{a_{0}}{R+a_{0}}\right)$ (4)
$\displaystyle\approx
2a_{0}-\frac{2a_{0}^{2}}{R}+\mathcal{O}\left(\frac{a_{0}^{3}}{R^{2}}\right).$

In the limit $a_{0}/R\to 0$, $d_{\textnormal{ctr}}=d_{\textnormal{surf}}$,
while for finite $R$, $d_{\textnormal{surf}}<d_{\textnormal{ctr}}.$ This
difference can be significant: for $R=3a_{0}$, $d_{\textnormal{surf}}$ is
approximately 75% of $d_{\textnormal{ctr}}$, for example. We can represent
this 2D assembly in 1D by “unwrapping” the circle to form a line segment of
length $2\pi R$ with periodic boundaries, upon which we place two adjacent
line segments of length $d_{\textnormal{surf}}$. The dimensional reduction
creates an effective particle size that depends on the surface curvature.

This idea can be straightforwardly extended to a cylindrical substrate with
radius $R$, as described in Memet  _et al._ [36]. Two adjacent particles
aligned with the axial direction are separated by a distance
$d_{\textnormal{ctr}}$, while two adjacent particles aligned with the
circumferential direction are separated by a distance $d_{\textnormal{surf}}$
measured along the surface. Upon unwrapping the cylinder to form a flat 2D
domain with periodic boundaries, we find that the effective particle shape is
an extended, ellipse-like shape. When centered at the origin of the
$xz$-plane, this effective shape is described by the equation

$(a_{0}+R)^{2}\sin^{2}(x/R)+z^{2}=a_{0}^{2}.$ (5)

Note that this is the equation of a circle in the limit $R\gg a_{0}$, as
expected. Thus, we can simulate crystallization on a cylinder using a flat 2D
domain while taking into account the 3D shape of the particles if we
generalize our algorithm to use the oblong shapes given by Eq. 5 instead of
circles.

For a cone, the effective particle shape is more complicated. Far from the
cone tip, curvature effects will be small, and the effective particle shape
will be approximately circular. As we move closer to the tip, curvature along
the circumferential direction increases, and the effective particle shape will
become more elongated.

We can solve for this position-dependent effective particle shape by
considering horizontal slices through the cone and a tangent spherical
particle. Effective particle shapes for different locations on the domain are
shown in Fig. S2.

Fig. S2: Diagram showing that the effective particle shape on a cone is a
tapered ellipse-like shape, pointing towards the sector center. The correction
is much more pronounced close to the tip.

Using a position-dependent effective particle shape will likely further
disrupt crystallization close to the sector center/apex (where the particle
shape is not approximately circular). These 3D effects should be considered by
simulations that seek to more accurately model experiments. Though solving
explicitly for the effective particle shape in Fig. S2 is useful for
estimating the corrections to our model, simulating the assembly process
directly in three dimensions would be a more straightforward way to
incorporate this effect.

##  Appendix B Defect definition

We choose $\left|\psi_{6,j}\right|<0.9$ to define a defect. We use the bond
orientational order instead of the more commonly used definition based on
coordination number so that we can filter out dislocations that form along the
seam, which have $\left|\psi_{6,j}\right|\geq 0.9$. These dislocations would
otherwise contribute a background defect density. Changing the cutoff value
from 0.9 does not alter the shape of the defect density distribution, as seen
in Fig. S3.

Fig. S3: Plots of defect density for various choices of cutoff for
$\left|\psi_{6,j}\right|$. The simulations were carried out for sectors with
$\phi=5^{\circ}$ and for seeds initiated at $C=10a_{0}$ with orientation
$\theta=40^{\circ}$. Each curve is an average over 100 trials.

##  Appendix C Commensurate packings on cylinders

Fig. S4: Commensurate chiral and achiral particle packings on cylinders. (A)
Plot of Eq. 6 which represents the circumferences $C$ and seed orientations
$\theta$ that should result in commensurate packings, where $\theta$ is
related to $(m,n)$ by Eq. 7. The blue vertical line represents a seed
orientation of $\theta=40^{\circ}$ as shown in Fig. 4 of the main text, while
the red vertical line represents a seed orientation of $\theta=59^{\circ}$, as
shown in Fig. S5. Note that the vertical lines do not intersect any
commensurate points for $C<20a_{0}$. Here we show commensurate points for
$C<12a_{0}$, which is the circumference range studied. (B) Renderings of
simulation results on an unrolled cylinder with crystal seed parameters set at
commensurate points. We observe perfect packings even for chiral seed
orientations and small circumferences. Fig. S5: Nonmonotonic defect density
behavior of near-commensurate cylinders. (A) Rendering of representative
results for simulations on an unrolled cylinder for seed orientation fixed at
$\theta=59^{\circ}$. As $C$ decreases, the crystal becomes fragmented and
disordered. (B) Plot of the defect density of cylinders (circles) superimposed
on a plot for cones (lines) for simulations fixed at $\theta=59^{\circ}$. Note
that the cylinder defect densities are nonmonotonic, with local minima at
integer $C$ values.

On a cone, perfect crystals can form only when the sector angle is a “magic”
angle $\phi=60^{\circ}P$, where $P$ is an integer [25, 26, 24]. The only magic
angle in the range $\phi\leq 30^{\circ}$ studied here is $\phi=0^{\circ}$,
corresponding to a cylinder. On a cylinder, there are many commensurate
packings, depending on the circumference and angle of the crystal. These
commensurate packings are determined by a pair of phyllotactic indices $(m,n)$
where [32]

$\frac{1}{2\pi}\left|C\right|=\frac{a_{0}}{2\pi}\sqrt{m^{2}+n^{2}-mn},$ (6)

and $a_{0}$ is the lattice spacing. The integers $m$ and $n$ are related to
the crystal orientation by [32]

$\tan{\left(\frac{\pi}{2}-\theta\right)}\approx\frac{2}{\sqrt{3}}\left(\frac{m}{n}-\frac{1}{2}\right),$
(7)

where $\theta$ is the angle between the crystal lattice and circumferential
axis.

The commensurate points in the $(C,m/n)$ plane are plotted in Fig. S4A
according to Eq. 6. The blue vertical line represents the crystal orientation
$\theta=40^{\circ}$. Along this line there are no commensurate packings, at
least for $C<20a_{0}$. The red vertical line represents crystal orientations
$\theta=59^{\circ}$. Along this line there are also no commensurate packings
for $C<20a_{0}$. The circled points are visualized in Fig. S4B. Our
simulations are able to reproduce the commensurate packings at the expected
phyllotactic values.

When the seed orientation is fixed at a near-commensurate value of
$\theta=59^{\circ}$, our simulations reveal interesting non-monotonic behavior
in the defect density plot (Fig. S5). While the overall boundary of the
sampled defect densities follows the same profile as cones, we observe local
dips at integer $C$ values, corresponding to values close to commensurate
points (Fig. S4A). Therefore, while cylinders of different crystal
orientations manage to capture the general defect density behavior of cones,
we choose to analyze seed orientations $\theta=40^{\circ}$, for which we
encounter fewer nearby commensurate points as cylinder circumference increases
(see blue line in Fig. S4).

##  Appendix D Circumference argument applied to cones

We now seek to extend the growth probability argument presented in Section
IV.1 to a conical substrate. On a cone, crystals grow along conical geodesics.
The theory for cylinders might need to be modified if the number of particles
along a geodesic interface were to differ significantly from the number of
particles that can fit around the cone circumference at a given height, or if
the interface/geodesic length were to depend strongly on the crystal
orientation.

However, we find that if we restrict our attention to geodesics that represent
realistic crystal boundaries (for example, the geodesics that start at a
particular point on the cone, circle the cone once, and returns to within
$a_{0}$ of the starting point) on cones with small angles, the length of a
geodesic is well-approximated by the circumference of a circular cross-
section. The simplest case is a geodesic at an angle $\theta=0$. The geodesic
length is $2R\sin(\phi/2)$, which is the same as the circumference at that
point, $R\phi$, in the small-angle limit. Perhaps more surprisingly, varying
$\theta$ leads to only a minor correction. Only a narrow range of $\theta$
values produce curves that circle the apex one time, returning to within
$a_{0}$ of their starting point, and these admissible curves are also well-
approximated by the circumference, as we show graphically in Fig. S6. It is
therefore a reasonable approximation to set
$N_{\textnormal{row}}\approx\text{circumference}/a_{0}$ and to explain the
dependence on circumference (and independence on seed orientation and cone
angle) in our cone simulations with our simplified growth model developed for
cylinders.

Fig. S6: Diagram showing examples of geodesics on the circular sector that
make permissible crystal boundaries. Three possible starting points are shown
(blue points) from which geodesics depart toward the bottom at a range of
angles, circle the cone once, and return within a distance $a_{0}$ of the
initial starting point (light blue disk). The lengths of the geodesics that
meet these conditions are well-approximated by the circumference of a circular
cross-section of the cone at the initial points.

##  Appendix E Initialization with a ring of particles

In the main text, we investigate crystal growth initiated by a seed of three
particles in contact with one another (Fig. 1A). In this appendix, we provide
a brief overview of the interesting effects that occur when a crystal instead
nucleates on a ring of particles oriented parallel to the base of the cone
(Fig. S7). Such a ring-shaped crystal could be realized experimentally by, for
example, milling a shallow trench or placing a metallic ring flush against the
conical surface. Reliably creating ring-shaped initial conditions in
experiments is the subject of ongoing work.

Fig. S7: Schematic of ring initial condition, defining the ring circumference
$C_{0}$ and the banded crystal width $w$.

First, we consider a ring of particles initiating crystal growth on a
cylindrical substrate. A crystalline band of any width is not strained on a
cylinder provided that the initial seeded ring is commensurate with the
circumference of the cylinder. The energy of a crystal of circumference
$C_{0}$ and width $w$ is given by

$E_{\mathrm{cylinder}}=E_{\mathrm{bound}}+E_{\mathrm{bulk}},$ (8)

where

$\displaystyle E_{\mathrm{bound}}$ $\displaystyle=$ $\displaystyle 2\sigma
C_{0}$ (9) $\displaystyle E_{\mathrm{bulk}}$ $\displaystyle=$
$\displaystyle-\Delta fC_{0}w,$ (10)

where $\sigma$ is the energy per unit length required to form a free boundary
in a crystalline bulk by breaking interparticle bonds, and $\Delta f$ is the
change in free energy of the non-crystalline phase relative to the crystalline
phase [37]. Note that $\Delta f>0$ is positive, as the disordered phase has
higher energy than the ordered crystalline phase. Since the energy in Eq. 8
decreases with increasing crystal width $w$ for any $w>0$, crystal growth
always lowers the energy of the system. Therefore, the seeded crystal would
nucleate and grow to cover the entire cylindrical surface. Note that the lack
of a nucleation barrier here is specific to our ring seeding configuration. An
energy barrier is typically present for nucleation from a more isotropic seed
on a two-dimensional surface.

In contrast, on the surface of a cone, crystal growth from an initially
unstrained ring of particles can generate nonzero elastic strain. In the limit
of a small cone angle, a crystalline band of width $w$ centered about an
initial commensurate ring of circumference $C_{0}$ has energy [38]

$E_{\mathrm{cone}}=E_{\mathrm{bound}}+E_{\mathrm{bulk}}+E_{\mathrm{strain}},$
(11)

where

$\displaystyle E_{\mathrm{bound}}$ $\displaystyle=$ $\displaystyle 2\sigma
C_{0}.$ (12) $\displaystyle E_{\mathrm{bulk}}$ $\displaystyle=$
$\displaystyle-\Delta fC_{0}w$ (13) $\displaystyle E_{\mathrm{strain}}$
$\displaystyle=$ $\displaystyle\frac{Y\tan^{2}(\pi\sin\beta/2)}{6C_{0}}w^{3},$
(14)

where $\beta$ is the cone angle defined in Figs. 1 and S7 and $Y$ is the two-
dimensional Young’s modulus of the colloidal crystal. Owing to the $\sim
w^{3}$ scaling of the strain energy in Eq. 14, the derivative of the total
energy $E$ with respect to $w$ becomes positive above a critical width $w_{c}$
given by

$\displaystyle w_{c}=\frac{2C_{0}}{\pi\beta}\sqrt{\frac{2\Delta f}{Y}},$ (15)

indicating the width at which colloidal crystallization is arrested. Note that
the nonzero Gaussian curvature of the conic surface, which manifests as the
varying circumference of the cone cross-section, is an essential ingredient
for this growth arrest.

Interestingly, we can also leverage this crystallization arrest phenomenon to
propose an alternative method for measuring the Young’s modulus of the
colloidal crystal, a quantity that has typically been extracted through
various inference methods [39, 11]. On inverting Eq. 15, we have

$\displaystyle Y=8\Delta f\left(\frac{C_{0}}{\pi\beta w_{c}}\right)^{2}.$ (16)

Since the critical width of a crystalline band on a cone is a directly
measurable quantity, we can use it to infer the Young’s modulus $Y$ through
Eq. 16.

In this calculation, and throughout the paper, we have assumed that the
triangular lattice is the most stable configuration. For longer-ranged
interactions, however, it is possible that the crystal could restructure
itself to accommodate the elastic strain. Future work might examine whether
other lattices, such as the rhombic lattice studied by Mughal and Weaire [31]
in the context of disk packings on cylinders, might represent stable
structures on the cone.