10.1109/ACCESS.2023.DOI

Corresponding author: Isa Inuwa-Dutse (e-mail: i.inuwa-dutse@hud.ac.uk).

# Detection of Offensive and Threatening Online Content in a Low Resource
Language

FATIMA MUHAMMAD ADAM1  ABUBAKAR YAKUBU ZANDAM2  ISA INUWA-DUTSE3  Federal
University Dutse, Nigeria (e-mail<EMAIL_ADDRESS> Federal
University of Technology Babura, Nigeria (e-mail<EMAIL_ADDRESS>University of Huddersfield, UK (e-mail<EMAIL_ADDRESS>
###### Abstract

Hausa is a major Chadic language, spoken by over 100 million people in Africa.
However, from a computational linguistic perspective, it is considered a low-
resource language, with limited resources to support Natural Language
Processing (NLP) tasks. Online platforms often facilitate social interactions
that can lead to the use of offensive and threatening language, which can go
undetected due to the lack of detection systems designed for Hausa. This study
aimed to address this issue by (1) conducting two user studies ($n=308$) to
investigate cyberbullying-related issues, (2) collecting and annotating the
first set of offensive and threatening datasets to support relevant downstream
tasks in Hausa, (3) developing a detection system to flag offensive and
threatening content, and (4) evaluating the detection system and the efficacy
of the Google-based translation engine in detecting offensive and threatening
terms in Hausa. We found that offensive and threatening content is quite
common, particularly when discussing religion and politics. Our detection
system was able to detect more than 70% of offensive and threatening content,
although many of these were mistranslated by Google’s translation engine. We
attribute this to the subtle relationship between offensive and threatening
content and idiomatic expressions in the Hausa language. We recommend that
diverse stakeholders participate in understanding local conventions and
demographics in order to develop a more effective detection system. These
insights are essential for implementing targeted moderation strategies to
create a safe and inclusive online environment.

Trigger Warning: Readers may find some of the terms in this study distressing
or disturbing; all examples are for illustration only.

###### keywords:

Low-Resource Language, Hausa Language, Offensive and Threatening Content,
Cyberbullying, Online Social Media

=-15pt

##  1 Introduction

The modern world is highly interconnected with many people and devices
connected to the Internet. This has enabled research and interconnectivity
among various devices and has changed the world [kennedy2006beyond,
warf2018sage]. It has also led to the development of digital social networks,
which have created a wide range of large datasets for various research
purposes [miller2010data, kumar2014twitter, haffner2018spatial]. Social media
platforms such as Facebook111https://facebook.com and Twitter (now
X222https://twitter.com) allow users to stay connected, share and express
opinions, organise civil movements and more [kumar2014twitter, rane2012social,
olteanu2015characterizing, freelon2016beyond, puspitasari2017arab,
nemes2021information, dambo2022office, bello2023endsars]. These platforms have
a significant impact on politics, governance, the economy, business, and other
social issues [dollarhide2021social]. However, online participation often
involves the use of offensive, threatening, racist, and sexist terms, known as
cyberbullying [caselli2020feel, caselli2020hatebert]. Cyberbullying can have
many consequences, including psychological suffering and isolation
[bullystat]. It can also lead to depression and suicide among people
[magee2013teens]. Therefore, various strategies have been proposed to detect
and eliminate such content and combat cyberbullying [zhong2016content,
altay2018detection, zois2018optimal, rafiq2018scalable, yao2019cyberbullying,
amin2020various, plaza2021multi]. These detection methods and resources are
based on high-resource languages, making them ineffective on low-resource
languages such as Hausa. Although it has more than 100 million speakers, the
majority of whom reside in southern Niger and northern Nigeria, the Hausa
language is considered a low resource due to limited resources to effectively
build many downstream tasks in NLP [inuwa2021first]. The language is also used
by millions of online users for their daily engagement with online social
media platforms to spread both beneficial and harmful content. This study aims
to address two specific problems: collecting relevant data and developing a
detection system for offensive and threatening content in the Hausa language.
We will focus on the following areas:

  1. 1.

Investigating the connection between offensive and threatening online content
expressed in the Hausa language.

  2. 2.

Differentiating abusive terms associated with banter from more offensive
terms.

  3. 3.

Examine the interplay between idiomatic expressions with a subtle abusive or
threatening tone.

  4. 4.

Identify the topical issues that are more likely to attract offensive and
threatening content.

As one of the first sets of studies on the detection of offensive and
threatening online content in the Hausa language, the following are the main
contributions of the study:

  * -

To enrich and support downstream tasks, the study offers the first collection
of annotated datasets on offensive content (HOC) and threatening content (HTC)
in the Hausa language.

  * -

A useful detection model capable of detecting offensive and threatening online
posts in the Hausa language.

  * -

Through the two sets of user studies, we provide information on the prevalence
of content related to cyberbullying and pointers to some mitigation
strategies. Thus, we offer our recommendations for improving civilised
interactions and mitigating the risk of cyberbullying in the Hausa language
that could evade detection.

The remainder of the paper is structured as follows. Section 2 is about the
background and relevant studies, Section 3 describes our approach, and Section
4 presents the implementation details. Section 5 presents relevant results and
a discussion. Finally, Section 6 concludes the study and recommends future
work.

##  2 Background and Related Work

The Oxford dictionary defines offensive content as very unpleasant and
insulting; participating in or marked by persistent violence and cruelty
[dictionary_key]. According to the General Policy Recommendation of the ECRI,
hate speech encompasses any form of advocacy, promotion, or incitement, in any
form, of the disparagement, abhorrence, criticism, harassment, negative
labelling, stigmatisation, threat, insult or negative labelling directed at a
person or group of persons [ECRI]. Offensive language is closely related to
various linguistic and societal issues, such as abusive and violent tone,
cyberbullying, racism, extremism, radicalisation, toxicity, profanity,
flaming, discrimination, and hateful speech [caselli2020feel,
founta2018large]. With the massive increase in social media interactions, so
do offensive content and other forms of cyberbullying. In May 2016, the EU
Commission reached an agreement with the tech giants333Facebook, Microsoft,
Twitter, and YouTube on a Code of conduct on countering illegal hate speech
online [eucomm2016]. These measures are most effective in high-resource
languages such as English and French. However, in low-resource languages such
as Hausa, detecting cyberbullying-related content is challenging due to the
lack of annotated datasets and relevant resources. This work is aimed towards
addressing these challenges.

###  2.1 Offensive Content Detection

Cyberbullying involves abusive online behaviour directed at an individual or
group of individuals. This attitude has many consequences including
psychological suffering and isolation [bullystat]. As mentioned earlier,
cyberbullying can take many forms with varying detection strategies
[zhong2016content, altay2018detection, zois2018optimal, rafiq2018scalable,
yao2019cyberbullying]. Earlier approaches to detecting any form of
cyberbullying rely on classic machine learning models [zhong2016content,
al2016cybercrime, altay2018detection, van2018automatic]. For instance, the
work of [zhong2016content] is focused on identifying the form of cyberbullying
associated with image data. A strategy based on modelling posts written by
bullies, victims, and bystanders of online bullying to detect cyberbullying
was applied in [van2018automatic]. Other useful detection strategies tend to
revolve around two main areas: scalability and timeliness of the detection
system. Timeliness is placed at the centre of the detection pipeline to enable
efficient and effective support for the victim of cyberbullying. A multi-stage
cyberbullying detection involving a scheduling mechanism was proposed to make
the detection task more efficient [rafiq2018scalable]. An approach based on a
sequential hypothesis testing problem was proposed as a solution to
scalability and timeliness issues [zois2018optimal]. The study of
[yao2019cyberbullying] is focused on addressing false positives, scalability
and timeliness using data from Instagram. Because offensive or hateful content
and sentiment analysis are closely interrelated [schmidt2017survey,
oriola2020evaluating], sentiment analysis is used as the basis for detecting
offensive and inappropriate online content [plaza2021multi]. A feature shared
by the above studies is the focus on addressing cyberbullying challenges
associated with high-resource languages. There is an increasing interest in
the detection of offensive content and hate speech in low-resource languages
such as Tamil [rajalakshmi2023hottest, balakrishnan2023tamil], Pashto
[khan2023offensive], Urdu [saeed2023detection], Persian [kebriaei2023persian].
Similar studies have been focused on improving resources for tackling
offensive and hateful content detection [kovacs2022leveraging,
sinyangwe2023detecting, miao2023detecting, markov2023holistic].

Identifying threatening content is a major challenge in the online world.
Cyberbullying can take the form of threats and hate speech directed at
individuals or groups [ECRI]. To combat this, various automatic detection
methods have been proposed to moderate and contain the proliferation of
violent content [gavai2015detecting, wester2016threat, lobato2018adaptive,
aminanto2019combating, jiang2019warder, pappaterra2021bayesian]. For example,
[wester2016threat] investigated the effect of linguistic features (lexical,
syntactic and semantic) in detecting threats of violence using YouTube data.
[gavai2015detecting] focused on abnormal behaviour in enterprise social and
online activity data of employees to detect insider threats. [jiang2019warder]
proposed an online insider threat detection system using diverse feature
dimensions involving content and behavioural modalities. [lobato2018adaptive]
proposed an adaptive threat detection architecture for real-time training of
the detection models. [pappaterra2021bayesian] proposed an automated threat
detection to enable both offline and online threat assessment. Strategies and
resources have also been put forward for identifying threatening content in
low-resource languages [malik2023contextual, rehan2023fine]. Additionally,
comprehensive surveys on threat detection techniques and moderation policies
on tackling such content by online platforms have been conducted
[crothers2023machine, arora2023detecting]. Named entity recognition (NER) is a
Natural Language Processing (NLP) task concerned with identifying,
classifying, and categorising entities such as a name, person, place, and
organisation into some predefined groups [stavrianou2014nlp,
yenkar2021gazetteer, roy2021recent]. NER can be achieved either via a ruled-
based or machine learning (ML) approach. While the ruled-based relies on
searching for matched entities from predefined hard-coded categories, the ML
approach involves the use of a trained model to identify patterns and
relationships in a given collection [yi2020cybersecurity]. However, NER poses
some challenges when working with informal data, especially in languages with
limited resources. To address this, a multilingual NER model has been
developed as an alternative for languages lacking large corpora
[oyewusi2021naijaner]. Cross-lingual transfer learning methods have also been
used for low-resource languages to transfer knowledge from high to low-
resource language [enghoff2018low, mbouopda2020named, oyewusi2021naijaner].
Despite this, many languages still lack sufficient linguistic resources for
NLP-related tasks [tsvetkov2017opportunities]. This is especially true for the
Hausa language, which has a shortage of relevant linguistic corpus due to
insufficient annotated corpora, part-of-speech (POS) tagger, morphological
analyser, chunker, parser, and related resources. This study is aimed at
addressing the issue of annotated datasets to combat some forms of
cyberbullying in the Hausa language.

###  2.2 Downstream Tasks in Hausa Language

The use of datasets from online social media platforms has been explored for a
variety of purposes, such as gauging people’s opinions [zishumba2019sentiment,
sanisentiment], identifying topics [kusumawardani2017topic,
antypas2022twitter], detecting sexist terms [aliyu2023hausanlp], recognizing
hateful speech [machuve2022herdphobia], classifying text [chaure2019text], and
recognizing entities [nie2020named]. Previous studies have also generated
relevant corpora in the Hausa language for various tasks [suleiman2019towards,
oyewusi2021naijaner, abubakar2021enhanced, inuwa2021first,
ibrahim2022development, muhammad2022naijasenti, zandam2023online]. A large
collection of tweets in Hausa, Igbo, Yoruba, and Nigerian Pidgin have been
compiled to improve sentiment lexicons in low-resource languages
[abubakar2021enhanced]. Additionally, a large collection of datasets
consisting of multilingual tweets annotated with sentiments in the major
languages of Nigeria has been created [muhammad2022naijasenti]. Despite the
potential of low-resource languages, there are still few studies involving
them, partly due to the lack of datasets and challenges such as the scarcity
of annotated datasets and associated lexicons. This study presents an approach
to detect offensive and threatening online content in one of the most widely
spoken low-resource languages, Hausa.

##  3 Methodology

In order to reach our primary objective, we must gather the pertinent data and
construct the suitable detection system. Therefore, our strategy is focused on
(1) carrying out user studies (2) collecting and labelling data (3)
constructing and assessing the detection system. Figure 1 illustrates the main
components of our approach.

Figure 1: An overview of the main aspects of the approach involving (1) a set
of user studies and (2) collection and curation of relevant datasets for
building detection systems for offensive and threatening content.

###  3.1 User Study

We conducted two sets of user studies to gain insight into the public
perception of offensive and threatening online content. The first study (HOC,
for offensive content) was conducted using
Questionpro444https://www.questionpro.com/ with 100 volunteers, and the second
study (HTC, for the threatening content) was conducted using Survey
Monkey555https://www.surveymonkey.com with 60 volunteers. The survey link was
shared on Facebook and X (formerly Twitter) to target social media users. All
participants gave their informed consent and no personally identifiable
information was collected. The study was conducted in accordance with the
institution’s ethical standards. Table 1 shows the relevant demographic
information on the research participants. The study focused on (1) the use of
Hausa language in sharing information among social media users, (2) the use
and effect of social media as a tool for disseminating offensive and
threatening online content, and (3) the utility of information extraction on
social media content to identify security threats that may lead to violence
before its occurrence. We sought to understand the following through
engagement with the public: (1) the public opinion on distinguishing abuse
terms used in an offensive manner or otherwise, (2) the different meanings
associated with offensive (and abusive) and threatening content, (3) how
threatening content could provoke physical violence, and (4) the public’s
definition of threatening content.

Table 1: The research involved 308 participants, and the following table shows
their demographics for both the HOC and HTC studies.

|  | User study ($n=201$) involving offensive content (HOC) |  |  |  |  |
---|---|---|---|---|---|---|---
Gender |  | Age |  | Education |  | Category |
Female 25.3% |  | 18-25 yrs. 28% |  | Sec. Edu. 0% |  | Student 46% |
Male 74.7% |  | 26-35 yrs. 63% |  | Undergraduate. 39% |  | Self-employed 23% |
|  | 36-45 yrs. 9% |  | Postgraduate 46% |  | Civil Servant 23% |
|  |  |  | Graduate 15% |  | Politician 1% |
|  |  |  |  |  | others 19% |
|  | User study ($n=107$) with threatening content (HTC) |  |  |  |  |
Female 29.8% |  | 18-25 yrs. 42.6% |  | Sec. Edu. 34% |  | Student 57.4% |
Male 70.2% |  | 26-35 yrs. 38.3% |  | Undergraduate 19.1% |  | Self-employed 14.9% |
|  | 36-45 yrs. 12.8% |  | Postgraduate 0% |  | Civil Servant 27.7% |
|  | 46-55 yrs. 6.4% |  | Graduate 46.8% |  | Politician 0% |
|  |  |  |  |  | others 8.5% |

###  3.2 Online Data Collection

Low-resource languages lack enough data to complete many natural language
processing (NLP) tasks. In this research, we used Twitter (now X) and Facebook
to gather posts related to various topics that are likely to contain offensive
and threatening content (see Table 2 for examples). We employed both an
automated method, using Twitter’s API, and a manual process for Facebook data.
Algorithm 1 outlines the steps we followed to search and retrieve relevant
data with the help of Twitter’s API.

Algorithm 1 _Text Collection_ : Using keywords to collect relevant data for
the study

1:Initialisation: $S=[S_{i}]_{i=1}^{n}$, list of search keywords, $n:$ number
of search keywords, $M:$ number of search iterations, $N:$ number of required
tweets

2:for each iteration $m=1$ to $M$ do

3: for each query in $S$ above $i=1$ to n do search tweets data with keyword
$S_{i}$ from $D=[D_{j}]^{k}_{j=1}$ where $D$ is a list of tweets

4: if $S_{i}$ matches $D_{j}$ then // get relevant fields

5: get posting date

6: get tweet ID

7: get retweet count

8: get favourite count

9: get full text

10: get screen-name

11: get urls

12: else

13: continue

14: end if

15: append $D_{j}$ to $T$

16: check for N, if satisfies maximum break

17: end for

18:end for

19:create a dataframe based on T

20:Output: a dataframe consisting of a list of tweets $[T_{t}]^{N}_{t=1}$

Table 2: Relevant keywords and sources used for data collection. Source |  Twitter (HOC)
---|---
Topics |  politics, strike action, banditry and social events
Keywords |  siyasa (politics), ASUU strike, yan bindiga (bandits), sallah celebration, love relationships, and zagi
Source |  Facebook (HOC)
Topics |  politics, sport, banter, and news reports
Keywords |  Dandalin Siyasar Jigawa, Hakika mabudin faira, legit.ng hausa, Ayi raha asha dariya, Alfijir Hausa, Rariya, Northern hibiscus, Dokin karfe TV
Source |  Twitter (HTC)
Topics |  politics, strike action, banditry and social events
Keywords |  zanga-zanga (protest), farmaki, mayaka (fighters), hari (attack), and yan bindiga (bandits)
Source |  Facebook (HTC)
Topics |  politics, sport, banter, and news reports (bbc hausa)
Keywords |  yan bindiga (bandits), hari (attack), farmaki, garkuwa da mutane, kudin fansa, and bindiga

####  3.2.1 HOC and HTC Datasets

We aim to maximise the chances of obtaining the desired data by concentrating
on areas likely to contain offensive or menacing language. Therefore, we
restrict our collection to topics such as politics, sports, banter,
relationships, abusive words, threatening terms, news reports, trending topics
and certain accounts. A few of these keywords and accounts are listed in Table
2. Since the collection is centred on offensive and intimidating material, the
gathered data is classified according to the following:

  * -

Hausa Offensive Content (HOC Dataset): This consists of an annotated corpus of
offensive posts and terms in the Hausa language.

  * -

Hausa Threatening Content (HTC Dataset): Similar to the HOC, this consists of
an annotated corpus of threats and terms in the Hausa language.

To strengthen the data collection aspect, we apply the following strategies in
defining offensive and threatening content in Hausa.

  * -

Dictionary definition: This is to ensure that the collection is consistent and
in agreement with the accepted meanings associated with abusive zagi and
threatening barazana content as defined in Kamus666Hausa dictionary
https://kamus.com.ng/. Moreover, drawing from anecdotal evidence, we label
each post according to the context in which terms conveying offensive or
threatening tones have been used.

  * -

Past data and user study: We use relevant data sets that contain abusive words
from previous studies [inuwa2021first] to inform our approach and enrich the
training data. Additionally, we take advantage of the examples given in the
user studies by the research participants.

####  3.2.2 Data Annotation

To achieve the desired goal of building a detection system, we manually
annotated the HOC and HTC datasets to train the prediction models. Each post
is annotated with the following additional information:

  * -

language: to denote the language of the post with a focus on posts in Hausa
language or Engausa777a post consisting of both Hausa and English terms, see
Figure 2(a).

  * -

sentiment: to indicate whether a post is positive, neutral, or negative (see
Figure 2(b) for a summary).

  * -

category: to denote the theme to which the post belongs (e.g. social,
political, religious, security, education, law, sport, health, agriculture,
security, and business). Figure 2(c) shows the distribution of topics with the
most common posts.

  * -

offensive: a column to indicate whether a post is offensive or not; abusive
words are considered offensive in this regard.

  * -

For the HTC data, it is crucial to identify additional details, such as named
entities in the collection. Thus, the following additional labels have been
added during the annotation:

    * -

location: to indicate and provide location information.

    * -

violence: to record any act of violence or conflict, e.g. zanga-zanga,
farmaki.

    * -

threat: to indicate and report any threatening term used in a post.

    * -

threat object: to record any weapon or dangerous tool in a post, e.g. bindiga,
makami.

    * -

class: to donate the post category, e.g. threat or no threat.

####  3.2.3 Data Cleaning

The collected dataset, especially the X (formerly Twitter) collection, is
noisy, containing many duplicates, retweets, hyperlinks, emoticons, numbers,
punctuation marks, non-Hausa posts, and other symbols that require cleaning.
To ensure accurate and meaningful analysis, our cleaning strategy entails the
following.

  * -

##### Stopwords removal:

Stopwords are words that do not add much meaning to texts. In addition to the
standard list of stopwords, we define a custom set of stopwords and characters
in Hausa such as ’a’, ’ni’, ’to’, ’su’,’.’, ’?’ and ’/’ which have been
flagged and removed accordingly.

  * -

##### Irrelevant terms:

This involves removing URLs, HTML tags, numbers, exclamation marks, question
marks, and punctuation marks. The hyperlinks, hashtags (#) and at (@) symbols
found in some tweets were removed by using a substring matching of the regular
expression. Also, to remove tweets that do not carry meaningful information,
the number of unique words per tweet was calculated and all tweets with
distinct words less than 8 were filtered out.

  * -

##### Duplicate removal and normalisation:

To avoid skewing the analysis, we identify and remove duplicate posts.
Normalisation of the data involves converting accented vowels and emojis and
correcting misspelt words using a dictionary containing the correct terms to
be used. After converting the tokens to lowercase, we applied a stemming
strategy to standardise the tokens.

##  4 Experiment

This section describes the development and evaluation of the prediction models
used in detection systems.

###  4.1 Preliminary Analysis

Figure 2 provides relevant statistics about the language, sentiment and major
themes in both HOC and HTC collections. There are many posts written by
combining Hausa and English (Engausa). The HOC attracts more diverse
discussion topics (2(c)) and the low proportion of negative sentiment (2(b))
suggests that many of the abusive terms have been used in a less offensive
context.

Figure 2: Statistics about language, sentiment and the major themes associated
with the HOC and HTC datasets.

We could not find a standard benchmark to compare how effective the detection
of offensive and threatening terms would be in the Hausa language. To maximise
the chances of developing an effective detection system, we explore how
existing translation engines perform in processing Hausa terms. Table 3
presents the most frequent terms found in the study data. Some of these terms
are quite offensive, but difficult to translate with a powerful translation
engine. To determine how effective Google’s translation engine performs in
translating offensive and threatening Hausa words into English, we took the
following top samples for comparison.

##### Translation of Offensive Terms

Some examples of incorrectly translated offensive terms using Google’s
translation engine (’Hausa term’ $\longrightarrow$ meaning$\longrightarrow$
Google translation):

  * -

(’dan iska’ $\longrightarrow$ rascal $\longrightarrow$ and iska)

  * -

(’yan kutumar uba’ $\longrightarrow$ d***head $\longrightarrow$ father’s
cousins)

  * -

(’wawa jaki’ $\longrightarrow$ idiot, donkey $\longrightarrow$ wow what)

  * -

(’dan jaka’ $\longrightarrow$ jennet’s offspring $\longrightarrow$ and
jackets)

  * -

(’sakarai’ $\longrightarrow$ dull (male) $\longrightarrow$ boy

  * -

(’sakara’ $\longrightarrow$ dull (female) $\longrightarrow$ connection

##### Translation of Threat-Related Content

Some examples of incorrectly translated threat-related terms in the Hausa
language (’Hausa term’ $\longrightarrow$ meaning $\longrightarrow$ Google’s
translation):

  * -

(’zanga zanga’ $\longrightarrow$ protest $\longrightarrow$ my own)

  * -

(’hari’ $\longrightarrow$ attack $\longrightarrow$ day)

  * -

(’banga’ $\longrightarrow$ gang $\longrightarrow$ banga)

  * -

(’gumurzu’ $\longrightarrow$ tense contest $\longrightarrow$ tumblr)

  * -

(’idan maye ya manta’ $\longrightarrow$ if the perpetrator of a crime forgets
$\longrightarrow$ if the drunk forgets)

  * -

(’kare jini biri jini’ $\longrightarrow$ hotly contested tense situation
$\longrightarrow$ dog blood monkey blood)

Despite the common use of the terms mentioned (as seen in Table 3),
translation engines, like Google’s, which have access to abundant and varied
data, do not perform well. In the examples given, the best results are limited
to the literal meaning, and do not take into account the subtleties of the
language to detect offensive and threatening contexts. This highlights the
importance of enriching low-resource languages, especially in the current era
of large language models (LLMs).

Table 3: Some examples of the most frequent offensive and threatening terms
found in the HOC and HTC datasets.

Offensive (HOC) |  | Proportion |  | Subcategory |  | Threatening (HTC) |  | Proportion |  | Subcategory
---|---|---|---|---|---|---|---|---|---|---
dan iska |  | 27.7% |  | abusive |  | bindiga |  | 92.45% |  | threat-object
kutumar uba |  | 10.4% |  | abusive |  | kashe kashe |  | 31.97% |  | violence
shege/shegiya |  | 9.0% |  | abusive |  | tarzoma |  | 24.59% |  | violence
jaki |  | 8.7% |  | offensive |  | yan bindiga |  | 12.56% |  | threat
uwarka |  | 8.0% |  | abusive |  | farmaki |  | 9.02% |  | violence
dan kutumar uba |  | 6.9% |  | abusive |  | hari |  | 5.74% |  | violence
wawa |  | 6.9% |  | offensive |  | zanga zanga |  | 4.77% |  | threat
jahili |  | 6.90% |  | offensive |  | ta’addanci |  | 2.26% |  | threat
ubanka |  | 6.6% |  | abusive |  | rikici |  | 1.64 |  | violence

###  4.2 Detection System

The problem is modelled as a classification task consisting of two classes for
both the detection tasks - offensive/non-offensive involving the HOC data and
threat/non-threat for the HTC dataset. This section describes the development
and evaluation of applicable models.

####  4.2.1 Model Building

To determine the best detection model, we explore the following groups of
machine learning models:

  * -

##### Linear and kernel-based models:

This group consists of logistic regression, a collection of regression
algorithms that convey the relationship between variables (dependent and
independent) and support vector machines (SVM) that categorise information
separately using hyperplane to maximise the margin between them.

  * -

##### Naive Bayes

Naive Bayes is one of the popular models for classification tasks, especially
in NLP. It is based on Bayes’ theorem and assumes that features are
conditionally independent given the class label.

  * -

##### Ensemble models:

This group comprises random forest and XGBoost models. The random forest
classifier employs a set of decision trees. Closely related to the random
forest is the extreme gradient boosting algorithm (XGBoost), which is based on
the gradient tree boosting technique. These algorithms have been used for
their fast learning and performance scalability.

  * -

##### Neural networks:

This group consists of multilayer perceptron (MLP) and convolutional neural
network (CNN) models. The MLP is made up of an input layer, at least one
hidden layer of computational neurones, and an output layer. A CNN is a deep
neural network design consisting of convolutional and pooling or sub-sampling
layers that feed input to a fully connected network.

####  4.2.2 Feature Extraction and Training

The process of extracting relevant training features involves converting the
raw data (textual) into numerical vectors that can be fed into the learning
models for prediction. We apply both word embedding and the Term Frequency-
Inverse Document Frequency (TF-IDF) techniques in transforming the cleaned
version of the data. Word embedding maps words to a vector space for
representation. This technique enables the models to process the data and
capture the semantic relationships between words [mikolov2013efficient]. Using
the TFIDF technique, we tried unigrams, bigrams, and trigrams to maximise the
models’ prediction power.

####  4.2.3 Performance Metrics

Performance metrics are crucial to evaluating the quality and effectiveness of
machine learning models. We chose the following quantifiable metrics to assess
the efficacy of the models in building the detection system:

  * -

confusion matrix: is a composite metric that gives the prediction output and
quantification of how perplexed the model is. A true positive (TP) value
signifies that the positive value is correctly predicted, a false positive
(FP) means a positive value is falsely classified, a false negative (FN) means
a negative value is incorrectly predicted, and a true negative (TN) means the
negative value is correctly classified.

  * -

accuracy: is the number of successfully categorised instances divided by the
total number of instances. This metric can be derived from the confusion
matrix as the sum of TP and TN divided by the sum of TP, TN, FP and FN.

  * -

precision and recall: these are important metrics for binary classification
problems.

    * –

Precision is the proportion of true positive instances that are classified as
positive; it reflects the closeness of predicted values to one another given
by: $precision=\frac{TP}{TP+FP}$.

    * –

On the other hand, recall is the proportion of positive instances that are
correctly classified as positive, given by: $recall=\frac{TP}{TP+FN}$.

  * -

F1-score: This metric combines both precision and recall into a single metric,
giving equal weight to both.

In addition to the above objective metrics, we use the Google translation tool
service (see Section 4.1) to evaluate its efficacy in translating some
offensive and threatening content into the Hausa language.

##  5 Results and Discussion

This study explored the issues of offensive and threatening content detection
in the Hausa language through user studies and predictive analysis. In this
section, we present and discuss our main findings from user studies and the
detection task.

###  5.1 Public Engagement

In this section, we present and discuss the main takeaway from the set of user
studies. In total, we received responses from 308 participants who
participated in the surveys. We asked the participants to rate the degree of
offensiveness (Table 4) and threat (Table 5) on some selected posts.

####  5.1.1 Participants’ Take on Offensive Content

According to demographic data (Table 1), a typical participant was a male
between 25 and 35 years old who underwent a post-graduate study. In addition
to the student category, the primary work place is the public sector followed
by the self-employed. A large proportion ($49\%$) of the respondents lamented
the proliferation of offensive content, especially of abusive nature; $33\%$
reported encountering offensive content quite frequently. About 83% of the
respondents believe that young people tend to use abusive terms the most, and
21% reported the likelihood of commenting on abusive online posts. We
attribute this to the use of abusive terms in both jovial and offensive
contexts. Furthermore, about 41% of the respondents believe that abusive,
hateful, or offensive content hurts online engagements. A substantial
proportion (86%) of the respondents believe that political discourse is
responsible for many abusive and hateful online content. In Table 4, about 75%
rated example 1 as very offensive. There are mixed ratings for example 2;
about 38% rated the post as not offensive despite the use of a term (dan iska)
that is often considered offensive. Examples 3 and 4 also received mixed
ratings with 28% and 31% of the participants labelling them offensive. Further
detail is provided in the Remarks section of Table 4.

Table 4: Summary of the results from the second part of the HOC user study ($n=101$) about the public perceptions on the issue of offensive content in Hausa. Type: |  On the use of offensive abusive terms
---|---
Example 1: |  Dan gutsun uwarku watan maulid yaxo kuna bakin ciki don muna maulid amma duk tsinannan da yakarai mana happy new year sai munci uwarsa
Ratings: |  very abusive (74.5%); abusive (11.2%); somewhat abusive (7.1%); not abusive (2%)
Type: |  On the use of subtle abusive terms
Example 2: |  wallahi abokina basan dan iska bane sai da naga yayi 5mins yana dariya
Ratings: |  very abusive (5.1%); abusive (4.1%); somewhat abusive (17.3%); might be abusive (15.3%); not abusive (20.4%); not abusive at all (37.8%)
Type: |  On disparaging remark
Example 3: |  @user1 Shegiya me gudun dangi naga dae kema yar talakawa ce ko dan yanxu kin daena sae da awara da yaji iye
Ratings |  very abusive (27.8%); abusive (24.7%); somewhat abusive (27.8%); might be abusive (10.3%); not abusive (5.2%); not abusive at all (4.1%)
Type: |  On the use of offensive and idiomatic expression
Example 4: |  Bari ba shegiya bace ai, Azzamula kwai,Hakkinsa kuma yana nan Wallahi ranar da DSS basu da amfani bare wannan tsinannan Mulkin naku. @user2i gara dai kasa fir’auniyar matarka ta saki dan mutane dan Wallahi yanzu muka fara bayyana Zalincin da kukama Talakawa.
Ratings: |  very abusive (31.3%); abusive (24%); somewhat abusive (26%); might be abusive (9.4%); not abusive (5.2%); not abusive at all (4.2%)
Remarks: |  Example 1 is offensive and involves the use of strong abusive terms. Example 2 utilises abusive term but in a playful manner. Example 3 is derogatory and Example 4 combines subtle abusive terms and idiomatic expressions to lament on the state of leadership. Most of the participants’ ratings agree with these observations.

####  5.1.2 Participants’ Take on Threatening Content

For the user study involving threatening content, a typical participant was a
male between 18 and 25 years of age with a first degree. The primary work
place after the Student category is the public sector (see Table 1). About
$36\%$ believe that people use social networks to post threatening or violent
content; $54\%$ reported that threatening content is prevalent in discourse
related to politics, ethnicity, and religion. Similarly, $70\%$ of the
respondents believe that extracting relevant information such as name or
location details from a post will be a useful preventive measure against
security risks. Table 5 shows the participants’ ratings about how threatening
the sample posts are. The examples contain some terms with threatening tones
and subtle expressions of threats. Examples 1 and 2 involve reporting violent
incidents that could lead to backlash in the name of revenge. Example 3 is a
threat directed at an individual, but the individual terms show little
relationship with any form of threat. Example 4 reports on a planned protest
that could lead to violent clashes. Most of the participants’ ratings agree
with these observations.

Table 5: Summary of the results from the second part of the HTC user study ($n=47$) about the public perceptions on the issue of threatening content. Type: |  On the use of threatening terms
---|---
Example 1: |  yan bindiga sun kashe matar sarkin fulani tare da binne gawar ta
Ratings: |  very threatening (40.4%); threatening (19.1%); somewhat threatening (19.1%); not threatening (21.3%)
Type: |  On reporting insecurity incidence
Example 2: |  yan bindiga sun kashe yan banga a jihar zamfara ranar zaben gwamna an ce yan bangan sun fita sintiri ne
Ratings: |  very threatening (32.6%); threatening (26.1%); somewhat threatening (13%); not threatening (28.3%)
Type: |  On the use of threatening content
Example 3: |  duk wanda ya isa ya kara magana yaga abin da zai faru
Ratings |  very threatening (28.9%); threatening (26.7%); somewhat threatening (15.6%); not threatening (28.9%)
Type: |  Protest report
Example 4: |  zuwa anjima za’a fara maka zanga zanga a cikin gari
Ratings: |  very threatening (21.3%); threatening (23.4%); somewhat threatening (23.4%); not threatening (31.9%)
Remarks: |  Examples 1 and 2 involve reporting violent incidences that could lead to backlash in the name of revenge. Example 3 is a threat directed at an individual, but the individual terms show little relationship with any form of threat. Example 4 reports on a planned protest that could lead to violent clashes. Most of the participants’ ratings agree with these observations.

###  5.2 Detection Task

In this section, we present and discuss the efficacy of the trained models in
the detection of offensive and threatening content.

####  5.2.1 Offensive and Threatening Content Detection

Table 6 and Figure 3 show the performance of trained models in the detection
of online offensive content in Hausa. XGBoost and CNN achieved the best result
(with f-score values of $0.86\%$ and $0.84\%$, respectively). This is followed
by the SVM, MLP, and Logistic Regression. The use of trigrams greatly improves
performance in all models. For detecting threatening content, Decision Trees
and XGBoost perform better with an accuracy of 75% and 73% (see Table 7 and
Figure 4), respectively. These two models also perform better on the remaining
metrics using the various n-grams. These results, especially those with higher
recall and precision (Tables 6 and 7), show a promising start in the task of
detecting offensive and threatening online content in the Hausa language.

Table 6: Performance of the models trained on the HOC datasets for the detection of offensive content. With the exception of the CNN, the trigrams strategy results in improved performance across all the models. | N-grams | Unigrams | Bigrams | Trigrams
---|---|---|---|---
| Metric | Accuracy | Recall | Precision | F-score | Accuracy | Recall | Precision | F-score | Accuracy | Recall | Precision | F-score
Models | Random Forest | 0.75 | 0.75 | 0.75 | 0.75 | 0.77 | 0.77 | 0.77 | 0.77 | 0.80 | 0.80 | 0.80 |  0.80$\uparrow$
SVM | 0.72 | 0.72 | 0.72 | 0.72 | 0.79 | 0.79 | 0.79 | 0.79 | 0.82 | 0.82 | 0.82 |  0.82$\uparrow$
Logistic Reg. | 0.72 | 0.72 | 0.72 | 0.72 | 0.77 | 0.77 | 0.77 | 0.77 | 0.80 | 0.80 | 0.80 |  0.80$\uparrow$
XGBoost | 0.75 | 0.75 | 0.75 | 0.75 | 0.81 | 0.81 | 0.81 | 0.81 | 0.86 | 0.86 | 0.86 | 0.86$\uparrow$
MLP | 0.73 | 0.73 | 0.73 | 0.73 | 0.73 | 0.73 | 0.73 | 0.73 | 0.80 | 0.80 | 0.79 |  0.80$\uparrow$
CNN | 0.84 | 0.84 | 0.84 | 0.84 | 0.84 | 0.84 | 0.84 | 0.84 | 0.84 | 0.84 | 0.84 | 0.84

N-grams are contiguous sequences of $n$ tokens in a given dataset, usually
text or speech. To build effective detection systems, we tried various n-grams
across the chosen models. We use bigrams and trigrams because of the
prevalence of two- and three-adjacent words in both offensive and threatening
terms. For example, dan iska is an abusive word that is used both offensively
and jovially. The word is made up of two independent words,  dan (son of or
affiliated with) and iska (air). The use of bigrams or trigrams will maximise
correct identification and prediction of the next word in a sentence. As
demonstrated in Tables 6, 7 and Figures 3 and 4, the strategy of using ngrams,
especially trigrams, culminated in a significant improvement in the prediction
task. This has the effect of identifying common phrases and expressions in
data collection.

Table 7: Performance of the models trained on the HTC datasets. The trigrams strategy results in improved performance across all the models. | N-grams | Unigrams | Bigrams | Trigrams
---|---|---|---|---
| Metric | Accuracy | Recall | Precision | F-score | Accuracy | Recall | Precision | F-score | Accuracy | Recall | Precision | F-score
Models | Random Forest | 0.72 | 0.60 | 0.74 | 0.60 | 0.81 | 0.71 | 0.87 | 0.73 | 0.84 | 0.76 | 0.89 |  0.78$\uparrow$
XGBoost | 0.73 | 0.64 | 0.69 | 0.65 | 0.84 | 0.77 | 0.86 | 0.79 | 0.86 | 0.80 | 0.87 |  0.82$\uparrow$
Decision Tree | 0.75 | 0.71 | 0.72 | 0.71 | 0.77 | 0.72 | 0.74 | 0.72 | 0.89 | 0.85 | 0.89 | 0.87$\uparrow$
Naive Bayes | 0.70 | 0.54 | 0.61 | 0.51 | 0.48 | 0.54 | 0.54 | 0.47 | 0.68 | 0.61 | 0.61 |  0.61$\uparrow$
Figure 3: Accuracy and F-score performance of the trained models on the task
of predicting offensive content (using the HOC dataset). Figure 4: Accuracy
and F-score performance of the trained models on the task of predicting
threatening content (using the HTC dataset).

###  5.3 Discussion

####  5.3.1 Relationship between offensive and threatening content

There are many similarities between both offensive and threatening Hausa terms
that are widely used by online users. It is not always a straightforward
approach to distinguish offensive and threatening terms in the context being
used. For example, in Tables 3 and 8, there are some overlaps in the meanings
associated with the terms. The distinguishing factor is the context and
sentiment or polarity of the text. A high proportion of abusive terms and
negative sentiment tend to point to threatening content. To establish an
effective distinction, manual moderation and a large collection of annotated
datasets are required to discern offensive content with a threatening tone.

##### Distribution of offensive and threatening content

To support our observation about the frequency or dominance of offensive
content in political and religious discourse, we identify the proportion of
each theme in the whole data (see Figure 2(c)). This prevalence can be due to
the following: political and religious discourse tends to be tense and often
results in physical violence or threats. Table 3 shows the most frequently
associated terms with offensive and threatening content in the distribution.

##### Topical issues vs offensive and threatening terms

Building on the insight from the two user studies, our thematic analysis
results show that political discourse, especially among young people, is at
the forefront of attracting a high proportion of abusive terms (see Figure 2
and Table 8). For threatening terms, discourse on religion, ethnicity, and
political issues top the category. Most of the threatening content is about
reports or incidents, not threats directed at an individual. However, some
reports such as yan bindiga sun kashe matar sarkin fulani tare da binne gawar
ta have the tendency to incite violence in the name of revenge. One area of
application where this insight could be useful is within the security sector
to better inform the strategy and resources to tackle the challenges.

Figure 5: A summary of the areas to focus towards improving the detection system. Due to limited resources, the areas highlighted need further studies to build more effective and comprehensive detection systems for offensive and threatening content in the Hausa language. Table 8: Sample comments from the participants. Type | Sample Post | Category
---|---|---
|  HOC Collection (Facebook and Twitter) |
abuse |  dan shegiya | offensive
abuse |  Dan jaka shege dakiki | offensive
abuse |  dan iska | banter
abuse |  Dan iska da fuska kamar ichen kabari taya bazatai block naka bah, malam you need madarar tamowa fah … | social
abuse |  @user1 @user2 @user3 Nace uwarka zai gyara anan, idan baka iya enigilishi ba kaje a fassara ma alaji. | social
abuse |  @user1 @user2 Na rantse da Allah ni nasan su si kuwa wlhi sae dae su Mutu yan kutumar uba Munafukai | political
other |  @user1 At all. Ai sai an dauki at least one and a half mins ana warm upðŸ….. | other
abuse (allegation) |  @user1 Thank you Dr. Jeffery. Atiku is corruption personified! You need to see how he made billions as chairman of National Council on Privatization during OBJ’s tenure.Barawo BANSA Atiku.I am OBIDATTI. | political
|  HTC Collection (Facebook and Twitter) |
threat (news) |  dakarun mnjtf sun yi nasarar kama iyalai da masu taimaka wa yan ta’adda an kama mutanen ne a dajin sambisa | security
– |  wayan da suke koyawa yaran mu daukan makami saboda siyasa su kashe wasu allah ya isa tsakanin mu dasu | security
threat |  dan majalisa ya sha da kyar an kusa kashe shi an kona motar shi kurmus | political
threat |  yan bindiga sun kashe malamin coci ranar da aka fara azumi watau alhamis a kaduna shugaban can yace bayan halaka malami | security
threat (news) |  dakarun jamhuriyar nijar sun kashe yan taadda | security
threat (prayer) |  Allah Ya kawo Mana Karshen Ta’addadci da yan ta’adda da Masu daukar Nauyinsu, Albarkacin Rasulullahi | security
threat (news) |  Buhari ya farga.. yace bazai sassautawa ’yan ta’adda ba. | security
threat |  Manoman sun tafka asarar ne lokacin da wasu ’yan bindiga suka kai masu hari a kasuwar sayar da albasa da ke ?aramar Hukumar Mbaise da ke Jihar Imo. | security
threat (prayer) |  Tsinuwa Da Fushin Allah Su Tabbata Akan Masu Daukar Nauyin Ta’addanci A Najeriya | security
threat |  Jami’in Tsaro Sun Dakile Harin Yan Bindiga a Hanyar Katsina Zuwa Jibia, Sannan Sun Kama Yan Bindiga Masu Yawa | security

##### Distinguishing jovial and offensive abusive terms

Ambiguity represents a semantic condition in which words or phrases can
seemingly carry multiple semantic interpretations or meanings. Discerning
context is crucial since the meaning derived from a given text requires many
factors to be considered. The problem of ambiguity involving the Hausa
language has been analysed in [ishakucontrastive]. The context in which an
ambiguous statement is presented is crucial in deciphering its intended
meaning. This is required since some of the abusive terms are often used in
informal and jovial conversations among friends. To make the context clear and
explain the distinction, we compute the sentiment associated with each post.
By extracting the abusive terms and corresponding proportions in negative and
positive (and neutral) cases, the distinction is made clear. Abusive terms
associated with negative sentiment often point to offensive content. Although
offensive in tone, abusive terms are commonly used in banter and playful
engagement (see Tables 4, 5 and 8 for examples).

##### Idiomatic expressions vs offensive/threatening content

Idiomatic expressions are phrases that have a figurative, non-literal meaning.
They often convey a specific idea, emotion, or concept that may not be
immediately apparent to someone unfamiliar with the language. Because they are
often metaphorical and difficult to understand or translate for nonnative
speakers, the set of words in idiomatic expressions has meanings that differ
from the individual words. Some of these expressions can conceal offensive and
threatening content in a more cryptic tone that will elude automated
detection. To examine the interplay between idiomatic expressions conveying
offensive or threatening tones, we focus on identifying the commonly used
expressions and their meaning vis-a-vis offensive or threatening tones found
in the data. Some examples of these expressions include: (1) idan maye ya
manta (2) kare jini biri jini, (3) bari ba shegiya bace and (4) abokin barawo
barawo ne. The above expressions can be used in different contexts and are
subject to various interpretations. As shown earlier, Google’s translation
engine failed to correctly translate expressions (2). The bottom line is to
collect a huge collection of rich data and explore various training
strategies.

####  5.3.2 Impact of Offensive and Threatening Terms

A tirade of online abuse and threat is likely to incite the public and
precipitate physical violence. This is more pronounced in religious and
political discourse. Unchecked cyberbullying could lead to continuous
harassment both online and in physical space. When violence erupts, the use of
abusive and threatening terms proliferate, thereby creating some sort of
vicious cycle. At this juncture, it is pertinent to seek to understand the
potential consequences of the forms of cyberbullying often encountered by
online users. In the survey we conducted, some of the respondents lament how
the use of certain words on social media platforms influences and instigates
violence within the community. For instance, a certain post alludes to the
killing of a public figure’s wife within a particular ethnic group, quoted as:
yan bindiga sun kashe matar sarkin fulani tare da binne gawar ta. These
insights could be leveraged for preventive measures in combating security
risks and other forms of cyberbullying that are likely to go unnoticed if
expressed in low-resource languages such as Hausa.

##  6 Conclusion

Online social media platforms provide users with the opportunity to express
their opinions, share messages, and socialise. Unfortunately, offensive and
threatening content can make the online space toxic and unwelcoming. Although
many detection strategies have been proposed, identifying cyberbullying-
related content in low-resource languages is still a challenge. This study
focused on the Hausa language, one of the most widely spoken Chadic languages,
and contributed the following:

  * •

The first annotated datasets consisting of offensive and threatening content
to support downstream tasks in the Hausa language.

  * •

Detection systems to identify offensive and threatening posts in the Hausa
language.

  * •

Insights from two sets of user studies about the impact of offensive and
threatening online content

We found that some offensive and threatening content have subtle relationships
with idiomatic expressions that are expressed in the local tone, making them
difficult to detect. This underscores the need to engage native speakers and
resources to better understand local conventions and demographics in order to
effectively combat cyberbullying in the Hausa language.

##### Limitations and Future Work

Limitations of the present work include (1) lack of larger annotated datasets
(2) overlapping abusive terms with ambiguous meanings, and (3) the prevalence
of idiomatic expressions with subtle offensive and threatening tones. Future
work will focus on enriching the data to convey nuances and better
contextualisation in the language, curating relevant idiomatic expressions,
and using more powerful pre-trained language models (LLMs) for fine-tuning.
Engagement with diverse stakeholders, such as the general public, experts
(linguistics) and major news organisations, will be necessary to improve the
training data. Workshops, interactive sessions and crowdsourcing initiatives
can be used to encourage the public to contribute to a repository that
documents cyberbullying-related content in the Hausa language. Additionally,
curating relevant idiomatic expressions and using pre-trained language models
for fine-tuning will be essential. Figure 5 outlines the focus areas that need
to be improved. We hope this will be a starting point to motivate further
research into detecting various forms of cyberbullying in low-resource
languages, particularly Hausa.

[]Fatima Muhammad Adam completed her Master’s degree in Computer Science from
Federal University Dutse, as well as a Bachelor’s degree in Computer Science
from the same university. She holds a National Diploma in Statistics from
Jigawa State Polytechnic, Dutse. Her research interests span the areas of
artificial intelligence, NLP, and social networks analysis, with a focus on
low-resource languages. Several of her research papers have been featured in
prominent academic conferences and scholarly publications. She is currently a
full-time lecturer at the Federal University Dutse, Nigeria, and a part-time
lecturer at the Rawafid International College of Health Science and
Technology, Dutse. She is a data science enthusiast and an active member of
many data science communities, including the Arewa Data Science Fellowship,
DSN-Ladies in AI, DSN, and the Global AI Hub. []Abubakar Yakubu Zandam is an
educator and researcher working at the Federal University of Technology,
Babura, as a full-time lecturer in the Department of Computer Science.
Abubakar completed an MSc degree in Computer Science from Federal University
Dutse and a Bachelor’s degree in Computer Science. He is a passionate advocate
for the transformative potential of AI and Machine Learning. His research
interest lies within AI and NLP with a focus on low-resource languages to
support and empower underserved linguistic communities. He previously held the
position of Assistant Lecturer at Jigawa State College of Education, Gumel.
Abubakar’s research work has been published in reputable academic journals and
conferences. []Isa Inuwa-Dutse is a lecturer at the University of
Huddersfield. Isa holds a doctoral degree in Computer Science from Edge Hill
University, UK, an M.Sc. from The University of Manchester and the first
degree from Bayero University Kano. His research interests span areas of
artificial intelligence, text mining, and social network analysis and
modelling. Some of his research publications appeared in major academic
conferences and journals. He contributes to various conferences and journals
as a reviewer. His previous roles include a graduate teaching assistant at
Edge Hill University, a full-time lecturer in the Department of Computer
Science, at Federal University Dutse, and the Department of Statistics and
Computer Science, at Kano State Polytechnic. He holds full membership of the
British Computer Society, Computer Professional Council of Nigeria, Royal
Statistical Society, UK, and a fellow of Advance Higher Education, UK
(formerly HEA).


††thanks: Equal contribution††thanks: Equal contribution

# Birth and Death of One-dimensional Domains in Cylindrically Confined Liquid
Crystals

Madina Almukambetova  Department of Physics, Ulsan National Institute of
Science and Technology, Ulsan, Republic of Korea  Arman Javadi  Department of
Physics, Ulsan National Institute of Science and Technology, Ulsan, Republic
of Korea  Jonghee Eun  Department of Physics, Ulsan National Institute of
Science and Technology, Ulsan, Republic of Korea  Juneil Jang  Department of
Biomedical Engineering, Ulsan National Institute of Science and Technology,
Ulsan, Republic of Korea  Cheol-Min Ghim  Department of Physics, Ulsan
National Institute of Science and Technology, Ulsan, Republic of Korea
Joonwoo Jeong<EMAIL_ADDRESS>Department of Physics, Ulsan National
Institute of Science and Technology, Ulsan, Republic of Korea

###### Abstract

Nematic liquid crystal (LC) is a partially ordered matter that has been a
popular model system for studying a variety of topological behaviors in
condensed matter. In this work, utilizing a spontaneously twisting achiral LC,
we introduce a one-dimensional (1D) model system to investigate how domains
and topological defects arise and annihilate, reminiscing the Kibble-Zurek
mechanism. Because of the unusual elastic properties, lyotropic chromonic LCs
form a double-twist structure in a cylindrical capillary with degenerate
planar anchoring, exhibiting chiral symmetry breaking despite the absence of
intrinsic chirality. Consequently, the domains of different handedness coexist
with equal probabilities, forming the topological defects between them. We
experimentally measure the domain-length distribution and its time evolution,
best fitted by a three-parameter log-normal distribution. We propose that the
coalescence within a train of 1D domains having the normal length distribution
and randomly assigned handedness, may lead to the domains of the log-normal-
like length distribution. Our cylindrically confined LC provides a practical
model system to study the formation and annihilation of domains and defects in
1D.

††preprint: APS/123-QED

##  I Introduction

Symmetry-breaking phase transition is of great interest in various fields,
ranging from cosmology to condensed matter Kibble (1976); Zurek (1985); Hendry
_et al._ (1994); Weiler  _et al._ (2008); Monaco  _et al._ (2009); del Campo
_et al._ (2010); Ulm  _et al._ (2013); Pyka  _et al._ (2013); Deutschländer
_et al._ (2015); Fowler and Dierking (2017); Bowick  _et al._ (1994); Chuang
_et al._ (1991a, b). When a system undergoes a phase transition from higher to
lower symmetry phases, different regions of broken symmetries appear. If the
symmetries of adjacent domains are incompatible, topological defects may form
at their boundaries. Kibble Kibble (1976) and Zurek Zurek (1985) proposed a
theoretical model to statistically understand the dynamics of defect formation
and annihilation, utilizing the examples of cosmic string formation in the
universe and superfluid phase transition in helium, respectively. The
resulting scaling laws connect the density of defects and phase transition
rate. Regardless of a system’s nature, various experimental systems have
validated the Kibble-Zurek mechanism and scaling laws: superfluids Hendry  _et
al._ (1994) and superconductors Monaco  _et al._ (2009) to laser-cooled ion
crystals del Campo  _et al._ (2010); Ulm  _et al._ (2013); Pyka  _et al._
(2013), trapped atom gas Weiler  _et al._ (2008), and two-dimensional
colloidal systems Deutschländer  _et al._ (2015).

Nematic liquid crystals (LCs) have been a popular system for studying the
formation of domains and accompanying defects involved in phase transitions.
Nematic LC with no long-range positional ordering, but the orientational order
is arguably the simplest partially ordered matter and allows a facile
observation of topological defects under optical microscopes. Thus, it is one
of the earliest experimental systems verifying the Kibble-Zurek mechanism
Chuang  _et al._ (1991a, b); Bowick  _et al._ (1994). Specifically, the
formation and evolution of strings, monopoles, and textures in nematic LC
films have been observed, and the Kibble-Zurek scaling laws successfully
describe the relation between the defects density and the cooling rate Chuang
_et al._ (1991a); Bowick  _et al._ (1994); Bradač  _et al._ (2002); Fowler and
Dierking (2017). In the same vein, water-based lyotropic chromonic liquid
crystals (LCLCs) with a broad isotropic-nematic coexistence regime have been
employed to investigate the morphogenesis of nematic tactoids and defects
formation by coalescence of the tactoids Kim  _et al._ (2013); Wang  _et al._
(2018); Golovaty, D  _et al._ (2020).

In this work, we introduce a one-dimensional (1D) model system to investigate
how domains and defects form, evolve, and annihilate. We utilize a
cylindrically confined LCLC exhibiting the double-twist director
configuration. A train of right- or left-handed domains is formed with
topological defects between two adjacent domains of different handedness. Upon
quenching from the isotropic to nematic phase, we observe nematic tactoids’
nucleation, growth, and coalescence to form the train of domains with
alternating handedness, reminiscing the Kibble-Zurek mechanismKim  _et al._
(2013) and 1D Ising model. Taking advantage of the 1D system, we investigate
how the domains get coarsened by lowering the system’s elastic free energy.
Lastly, to justify an early-stage domain-size distribution, we propose a
statistical distribution best describing our experimentally observed domain-
size distribution and a theoretical model considering a stochastic merging of
pre-domains.

##  II Results and Discussion

Figure 1:  Defects formation in cylindrically confined nematic DSCG. (a) and
(b) Time-lapse polarized optical microscopy (POM) images of the 14.0% (wt/wt)
DSCG between crossed polarizers with a full-wave plate during the isotropic-
to-nematic phase transition at different cooling rates: (a) 2∘C/min, (b)
20∘C/min. Yellow arrows represent pass axes of the polarizers, and a tilted
blue arrow corresponds to the slow axis of the full-wave plate of 550-nm
retardance, inserted before the analyzer. All scale bars are 100 $\mu$m. White
texts correspond to the time elapsed after cooling. (c) POM images of three
types of defects after relaxation: point (Top), domain wall (Middle), and
disclination ring (Bottom). (d) Schematic diagram of the DT director
configuration with the disclination ring. Rods representing the nematic
director are color-coded according to the angle between the capillary axis and
the director. Here red- and blue-shaded regions correspond to right- and left-
handed domains, respectively, according to the right-hand rule. The black
circle represents the disclination-ring defect.

###  II.1 Formation of Double-twist Domains and Time Evolution of Their Size
Distribution

Our model system is a cylindrically confined nematic LCLC having the double-
twist (DT) director configuration. Nematic LCLCs have unusual elastic
properties of LCLC, such as giant elastic anisotropy and large saddle-splay
modulus Zhou  _et al._ (2012, 2014); Nayani  _et al._ (2015); Davidson  _et
al._ (2015); Long and Selinger (2022), which often give rise to chiral
symmetry breaking of the director configurations under confinement despite
their achiral building blocks Tortora and Lavrentovich (2011); Jeong  _et al._
(2014, 2015); Davidson  _et al._ (2015); Nayani  _et al._ (2015, 2017); Fu
_et al._ (2017); Javadi  _et al._ (2018); Eun  _et al._ (2019); Srinivasarao
(2021). Specifically, the nematic disodium cromoglycate (DSCG)—a
representative LCLC—exhibits spontaneous chiral symmetry breaking to form the
DT structure when confined in cylindrical capillaries with degenerate planar
anchoring Nayani  _et al._ (2015); Eun  _et al._ (2019). The directors of the
DT configuration twist along the capillary radius while the directors at the
capillary center lie along the capillary axis Davidson  _et al._ (2015);
Nayani  _et al._ (2015); Eun  _et al._ (2019). The large saddle-splay modulus
lowers the total elastic free energy by making the surface directors at the
inner capillary wall align along the circumferential direction Koning  _et
al._ (2014); Selinger (2018). The resultant DT configurations with the right-
and left-handed twist have the same energy. Thus, domains of different
handednesses coexist in a capillary with the same appearance probability $P$
($P_{\mathrm{Right}}=P_{\mathrm{Left}}=1/2$). There is a topological defect
between two domains of different handedness.

Figure 2:  Comparison of the appearance probability of each defect type
according to the cooling rate and the capillary size. We investigate the types
and number of defect in capillaries of four different diameters, ranging from
50 to 200 $\mu$m. The numbers above each column indicate the total number of
defects investigated; “n/a” means that the number is not available because we
did not count them. “Ambient” data are from the capillaries after injecting
room-temperature nematic DSCG without heating and cooling procedure, and these
capillaries exhibit only “wall” defect. See Fig. 1(c) for the types of
defects. The fast and slow cooling correspond to 2 and 20∘C/min, respectively.
Figure 3:  Statistical characterization of the time-evolving distribution of
domain sizes. The specimen is 14.0% (wt/wt) DSCG in 100 $\mu$m-diameter
cylindrical capillary after a fast cooling at the rate of 20∘C/min. (a) Time-
lapse POM images of the DT configuration with ring defects separating the
domains of opposite handedness. The scale bar is 200 $\mu$m, and polarizer
axes are represented by double arrows. (b) The experimentally measured
cumulative distribution function of non-dimensionalized domain sizes
$\tilde{l}=l/D$ (colored solid lines). Each black dashed line corresponds to
the best fit to each data with the three-parameter log-normal distribution
function. An inset shows the probability distribution function of
representative data at 3 hours. The data is taken from five independent
experiments, and each experiment investigates six capillaries of approximately
30 mm in length at the designated time interval. (c) The total number of
domains in a single experiment with six capillaries as a function of time. The
symbols show the average values of the five independent experiments, and the
error bar is the standard deviation. (d)-(f) The fitting parameters of the
three-parameter log-normal fittings in (b): (d) mean $\tilde{l}_{c}$, (e)
standard deviation $\tilde{l}_{w}$, and (f) cutoff $\tilde{l}_{0}$ of the
logarithm of non-dimensionalized domain size $\tilde{l}$. See Eq. (2) for the
distribution function.

The domain formation process and resulting defect density strongly depend on
the rate of isotropic-nematic phase transition before the nematic DT
configuration forms. Fig. 1 (a) and (b) compare two experiments of phase
transition from the isotropic (45∘C) to nematic (21.5∘C) phase with different
cooling rates, 2 and 20∘C/min, respectively. Fig. 1(a) shows how the nematic
DT domains form at the cooling rate of 2∘C/min. Entering the isotropic-nematic
coexistence phase upon cooling from the isotropic phase, a nematic shell grows
from the capillary wall. However, as shown in Fig. 1(b) with the cooling rate
of 20∘C/min, the nematic shell grows from the capillary wall, and also nematic
droplets appear and grow in bulk, simultaneously Javadi  _et al._ (2018). The
nematic texture relaxes over time to form one-dimensional (1D) DT domains
eventually, while nematic domains merge and annihilate. In our experimental
conditions with the glass capillary of 100-$\mu$m inner diameter, it takes
approximately three hours to clearly identify defect-separated 1D domains of
different twist-handedness. The linear defect density in the slow-cooling
case, i.e., the number of defects per unit capillary length, is over ten times
smaller than the density in the fast-cooling case.

This observation that slow cooling leads to larger domains than fast cooling
reminisces the scaling laws in the Kibble-Zurek mechanism, but the systematic
study of the Kibble-Zurek scaling law under this confinement can be pointless.
As shown in Fig. 1 (a) and (b), the nematic droplets formed only in the fast-
cooling case make the texture more disordered than in the slow-cooling case,
presumably resulting in the shorter domains. On the other hand, during the
droplet-free slow cooling, the Rayleigh-Plateau instability of the
confinement-induced cylindrical core-shell interface seems to play a major
role in the determination of the domain sizes Javadi  _et al._ (2018). This
indicates that the mechanism of the domain formation can differ by the cooling
rate, thus making it difficult to expect a single governing scaling.
Furthermore, because acquiring a statistically meaningful number of domains by
slow cooling is impractical, we mainly study the fast-cooling case.

In DSCG’s nematic DT configuration, we identify three different types of
defects between neighboring domains. A previous study presents that energetics
chooses the type of the defect separating the adjacent domains of opposite
handednesses: either a point defect or a domain wall Davidson  _et al._
(2015). For instance, only point defects have been observed in another
representative LCLC, Sunset Yellow (SSY) Davidson  _et al._ (2015). However,
we find that three distinct types can appear in our 14.0% (wt/wt) DSCG: point,
domain wall, and disclination ring as shown in Fig. 1(c), and their appearance
probability depends on factors such as capillary size and cooling rate. Fig. 2
compares the experimentally measured appearance probability of each defect
type. In this work which focuses on 100 $\mu$m-diameter capillaries filled
with 14.0% (wt/wt) DSCG quenched at the rate of 20∘C/min, the disclination
ring shown at the bottom row of Fig. 1(c) is the dominant type of defect with
the appearance probability greater than 95% Eun  _et al._ (2019). This type of
defect is not observed in SSY, and Fig. 1(d) sketches its proposed director
configuration. Note that we sometimes observe the domain-wall defect (the
middle row of Fig. 1(c)), which is not observed in SSY, and the DSCG’s point
defect (the top row of Fig. 1(c)) is similar but different from the one in SSY
because of their broken symmetry. The defect energetics in DSCG requires
further investigation in the future.

Utilizing the 1D model domains, we investigate the domain-size distribution of
the fast-cooling case and its time evolution during which the domains coarsen
by merging of defects as shown in Fig. 3(a). For better statistics, we
simultaneously observe six different capillaries after quenching them on the
same temperature-controlled slide glass, and domains in all six capillaries
are analyzed as one data set. The domains from the six capillaries span
$\sim$165 mm in the capillary length, and there are approximately five hundred
defects after three hours of relaxation from the quenching. We observe that
the domain and defect arrangements before and after the heating-and-cooling
process are independent, meaning that we can erase the information via the
temperature cycle and conduct independent experiments multiple times with the
same specimen. Therefore, repeating the same temperature experiments, we
acquire five independent data sets. Note that we study only the fast-cooling
data for the size-distribution study. It is because the defect density in the
slow-cooling case is too low to analyze the domain-size distribution
statistically, i.e., ten times smaller than the fast-cooling case.

We estimate the distribution of the dimensionless domain length $\tilde{l}$ at
different times after the quenching. The length $l$ of a single domain is
obtained by measuring the distance between two neighboring defects as
described in Materials and Methods and is non-dimensionalized in units of the
capillary diameter ($D$ = 100 $\mu$m), i.e., $\tilde{l}=l/D$. Fig. 3(b) shows
the cumulative distribution function (CDF) of $\tilde{l}$ in all combined five
data sets; The inset shows a probability distribution function (PDF) after
three hours from the cooling, which is right-skewed. The coarsening shown in
Fig. 3(a) is evident in the time evolution of CDF, i.e., the curve shifting
toward the longer $\tilde{l}$ as time goes by. Fig. 3(c) shows how the number
of defects, $N$, decreases during the coarsening as a function of time. The
driving force of this coarsening process is the defects’ elastic energy
penalty; defects annihilation by merging them can lower the total energy of
the system Jeong  _et al._ (2015). In this work, we focus on understanding the
domain-size distribution from the statistical point of view, not the
energetics of defect annihilation.

###  II.2 Theoretical Understanding of Domain-Size Distribution

To understand the observed domain-size distribution, we propose a theoretical
model based on a stochastic merging of neighboring domains. As shown in Fig. 1
(b), when the system enters the isotropic-nematic coexistence phase from the
isotropic phase by quenching, the nematic droplets nucleate, grow, and
coalesce to form the train of 1D domains shown in Fig. 3(a). We assume that
the droplets have twisted bipolar director configuration, and their
probability of having either right or left-handedness is $1/2$ because of the
achiral nature of the LCLC Jeong  _et al._ (2014). If adjacent droplets as
‘pre-domains’ have the same handedness, they merge to form a longer domain.
Otherwise, they become two separate domains with a defect between them. Thus,
the sizes of the pre-domains and their stochastic merging determine the number
and sizes of the domains.

Although the actual merging and relaxation process seems complicated, as shown
in Fig. 1(b), we assume that stochastic merging of 1D pre-domains having a
certain length distribution determines the DT domain-size distribution at the
early stage shown in the top panel of Fig. 3(a). The 1D pre-domain’s non-
dimensionalized size $\tilde{x}_{\mathrm{pre}}=x_{\mathrm{pre}}/D$ as a random
variable has a probability distribution function (PDF)
$f(\tilde{x}_{\mathrm{pre}})$; $x_{\mathrm{pre}}$ and $D$ are the length and
diameter of a cylindrical pre-domain, respectively. Since the size $\tilde{x}$
as a dimensionless length is linearly additive, the dimensionless size of a
‘domain’ by the merging of $n$ pre-domains is a random variable
$\tilde{l}(n)=\sum_{i=1}^{n}\tilde{x}_{i,\mathrm{pre}}$, where the size of
$i$th pre-domain is $\tilde{x}_{i,\mathrm{pre}}$ as an independent random
variable. Then, the domain-size PDF $f(\tilde{l})$ is a mixture distribution
$f(\tilde{l})=\sum_{n=1}^{\infty}P(n)~{}f(\tilde{l}(n))$ with a PDF
$f(\tilde{l}(n))$ of the random variable $\tilde{l}(n)$ and the probability
$P(n)$ of having a merged domain by $n$ consecutive pre-domains. The
probability $P(n)$ as a weighting factor should satisfy
$\sum_{n=1}^{\infty}P(n)=1$. Given that the probability of having a left-
handed pre-domain is $p$ and $1-p$ for the right-handed one,
$P_{\mathrm{Left}}(n)$ is the probability of having $(n-1)$ consecutive left-
handed pre-domains that follow a given left-handed pre-domain as a starting
point of the domain. Additionally, the $(n-1)$ pre-domains should be followed
by a right-handed pre-domain as a starting point of a new right-handed
adjacent domain. This gives $P_{\mathrm{Left}}(n)=1\times p^{n-1}\times(1-p)$,
and $P(n)=(\frac{1}{2})^{n}$ for the equally probable left- and right-
handedness.

Figure 4:  Comparison of experimental (a) CDF and (b) PDF of non-
dimensionalized domain sizes (gray solid fill) with the three-parameter log-
normal fittings (red dashed line) and our theoretical model based on a
stochastic merging (blue dotted line). The experimental data is the 3-hour
data in Fig. 3(b).

We find that pre-domains having the normal size distribution can simulate
approximately the experimentally measured domain-size distribution. Inspired
by our experimental observation that the nucleated nematic droplets have a
quite uniform size with some variation as shown in Fig. 1(b), we try
$\tilde{x}_{\mathrm{pre}}\sim\mathcal{N}(\mu,\,\sigma^{2})$, where $\mu$ and
$\sigma$ are mean and standard deviation of the normal distribution
$\mathcal{N}$, respectively. Then, we have
$\tilde{l}(n)=\sum_{i=1}^{n}\tilde{x}_{i,\mathrm{pre}}\sim\mathcal{N}(n\mu,\,n\sigma^{2})$
and

$F(\tilde{l})=\sum_{n=1}^{\infty}P(n)~{}F_{\mathcal{N}}(\tilde{l}(n))=\sum_{n=1}^{\infty}\frac{\left[1+\textrm{erf}\left(\frac{\tilde{x}-\mu
n}{\sigma\sqrt{2n}}\right)\right]}{2^{n+1}},$ (1)

where $F_{\mathcal{N}}(\tilde{l}(n))$ is a cumulative distribution function
(CDF) of $\mathcal{N}(n\mu,\,n\sigma^{2})$ and $F(\tilde{l})$ is the CDF of
the domain-size distribution with $P(n)=(\frac{1}{2})^{n}$. In Fig. 4, we
compare our experimental data and Eq. 1 to find $\mu$ and $\sigma$
numerically. With the cutoff $n_{\mathrm{max}}=10$, $\mu=1.693~{}\pm~{}0.004$
and $\sigma=0.518~{}\pm~{}0.011$ best describe the experimental data in the
CDF plot shown in Fig. 4(a); Fig. 4(b) shows the fitting in the PDF plot.
Instead of $\tilde{x}_{\mathrm{pre}}\sim\mathcal{N}(\mu,\,\sigma^{2})$, one
may try other distributions guaranteeing positive domain size, such as Weibull
or truncated normal distribution, we still find numerical similarity between
the resulting CDF and experimental data (data not shown here). Note also that
this comparison is only for the domain-size distribution at the early stage,
i.e., the first snapshot after the relaxation into the DT configuration,
because the time evolution of the distribution afterwards will be determined
by the elasticity-driven interaction between the defects and resultant domain
coarsening, not by the merging of pre-domains.

We find that all experimental CDFs can be approximated to the log-normal
distribution’s CDF (see Statistical Tests section under Materials and
Methods). The dashed lines in Fig. 3(b) are the log-normal distribution fits
to the time-varying experimental CDFs, and the fitting parameters are shown in
in Fig. 3(d-f). The log-normal distribution is a continuous probability
distribution of a random variable, whose logarithm follows the normal
distribution. This log-normal nature is frequently observed in various systems
such as aerosol droplets, bacteria division, avalanche processes in cells
Sangal and Biswas (1970); Hosoda  _et al._ (2011); Polizzi  _et al._ (2021).
We adopt the CDF of the 3-parameter log-normal distribution

$F_{\mathrm{log-
normal}}(\tilde{l})=\int_{\tilde{l}_{0}}^{\tilde{l}}\frac{1}{\sqrt{2\pi}\tilde{l}_{\mathrm{w}}t}\exp\left(\frac{(\ln(t-\tilde{l}_{0})-\tilde{l}_{\mathrm{c}})^{2}}{2\tilde{l}_{\mathrm{w}}^{2}}\right)dt,$
(2)

where $\tilde{l}_{\mathrm{c}}$ and $\tilde{l}_{\mathrm{w}}$ correspond to the
mean and standard deviation of the logarithm of non-dimensionalized domain
size $\tilde{l}$ after subtracting the offset $\tilde{l}_{0}$. As shown in
Fig. 3 (d-f), both $\tilde{l}_{\mathrm{c}}$ and $\tilde{l}_{\mathrm{w}}$
according to Eq. 2 monotonically increase as time passes.
$\tilde{l}_{\mathrm{c}}$ increases because the average domain size gets larger
while the number of defects decreases as they annihilate by merging (See Fig.
3(a)). Note that, even at the later stage with the larger
$\tilde{l}_{\mathrm{c}}$, there still exist short domains since the coarsening
keeps occurring, which results in the large variance in the domain size
$\tilde{l}_{\mathrm{w}}$. On the other hand, we see no consistent trend in
change of $\tilde{l}_{0}$. Theoretically, domains shorter than $\tilde{l}_{0}$
should not exist. The experimental interpretation of $\tilde{l}_{0}$ is that
domains shorter than $\tilde{l}_{0}$ is hardly observed because the
neighboring defects at a short separation quickly annihilate by merging. This
cutoff is determined by the energetics, which does not depend on time.
Therefore, $\tilde{l}_{0}$ remains almost constant throughout the experiments.

Figure 5:  Two statistical tests to find the best fit to the experimental
domain-size distribution shown in Fig. 3(b). Using eight right-skewed trial
probability distribution functions, we evaluate (a) Coefficient of
determination $R^{2}$ with (b) the enlarged view of the region around
$R^{2}=1$. (b) Kolmogorov-Smirnov test statistics $KS$ of the eight trial
functions. (c) Quantile-Quantile plot of the three-parameter log-normal
distribution in the logarithmic scale. LN, LL, and W stand for the log-normal,
log-logistic, and Weibull distributions, respectively. The following number,
i.e., 2 or 3, indicates the number of parameters. F is for the Fréchet
distribution with three parameters and, G for the gamma distribution with two
parameters.

##  III Materials and Methods

###  III.1 Sample Preparation

Disodium cromoglycate (DSCG, purity of 95%), acquired from Sigma-Aldrich is
used without further purification. By dissolving DSCG into deionized water
(18.2 M$\Omega~{}$cm), we prepare the 14.0% (wt/wt) DSCG solution. We fill
this solution into a cylindrical glass capillary (CV1017, Vitrocom) of
100-$\mu$m inner diameter with $\pm$10% tolerance, at the room temperature of
22$\pm$2∘C with no surface treatment of the capillary. The DSCG-filled
capillaries are put on a slide glass and capillaries’ both ends are sealed
with epoxy to minimize water evaporation during experiments. We place a
coverslip on top of the capillaries and fill the space between the slide glass
and coverslip with an index-matching oil (n = 1.474 at 589.3 nm, Cargille
Labs).

###  III.2 Optical Microscopy

We observe the slide on a temperature controller (T95-PE120, Linkam) using a
polarized optical microscope (BX53-P, Olympus). First, the sample is heated to
45.0∘C at the rate of 20∘C/min, and nematic DSCG melts into the complete
isotropic phase. After incubating the sample for ten minutes at 45.0∘C, we
decrease the sample’s temperature to 21.5∘C while controlling the cooling
rate: 20∘C/min for the fast cooling process and 2∘C/min for the slow one.
Unless specified otherwise, all observations of domains are conducted at the
final temperature, 21.5∘C, where the 14.0% (wt/wt) DSCG solution is in the
fully nematic phase.

The polarized optical microscopy (POM) images with crossed linear polarizers
are taken using a CCD camera (Infinity3-6UR, Lumenera) under illumination
derived from LED lamp (LED4D067, Thorlabs). As shown in the first row of Fig.
3(a), the first set of images is taken three hours after reaching 21.5∘C, and
the time evolution is recorded typically at 3-hour intervals for 24 hours. To
facilitate the measurement of the domain-size distribution in a wide field of
view, we use a 4$\times$ objective lens to take and stitch multiple images
covering the whole sample area; See Data Analysis for how to stitch multiple
images together. For the observation of domain formation shown in Fig. 1(a)
and 1(b), we use a 20$\times$ objective lens to take images at the rate of one
frame per five seconds.

###  III.3 Data Analysis

To measure the domain-size distribution in a whole capillary, we stitch
multiple images using ImageJ’s ‘Grid/Collection Stitching’ plugin Preibisch
_et al._ (2009). Then, from the intensity profile along the stitched capillary
images, we identify the positions of defects and measure the distances between
all neighboring defects, i.e., domain lengths. Specifically, as shown in the
POM images under crossed polarizers such as Fig. 3(a), the defects appear as
dark lines between two bright regions. A location where the second derivative
of the intensity profile along the capillary is greater than a threshold
value, corresponds to the defect position. Collecting these positions and
distances between the neighboring defects, we estimate the domain-size
distribution of $\sim$500 domains per each experiment, as shown in Fig. 3(b)
and (c), and repeat the experiments five times. Note that we present the data
as the cumulative distribution function with no need to set the bin size.

###  III.4 Statistical Tests

Testing eight right-skewed probability distributions to fit our experimentally
measured distributions, we find that the log-normal distribution with three
parameters best describes the experimental data. As shown in Fig. 4(b), the
experimental probability distribution function(PDF) is right-skewed. We choose
eight distributions of a similar shape to fit our experimental data from 3 to
24 hours: log-normal, log-logistic, Weibull, Fréchet, and gamma distributions.
For log-normal, log-logistic, and Weibull distributions, we test both the
typical two-parameter distribution and the three-parameter distribution with
an additional offset parameter. Avoiding the binning issue, we use CDFs for
fitting and test the goodness of the fits by two methods: the coefficient of
determination, $R^{2}$ (Fig. 5(a)), and the Kolmogorov-Smirnov test (Fig.
5(b)) Young (1977); Pobočíková  _et al._ (2017).

Both tests suggest that the three-parameter log-normal distribution (LN3) best
describes our experimental data. We calculate the coefficient of determination
$R^{2}=1-\frac{\mathrm{SS}_{\mathrm{R}}}{\mathrm{SS}_{\mathrm{T}}}$ where the
residual sum of squares
$\mathrm{SS}_{\mathrm{R}}=\sum_{i=1}^{N}\left[F(\tilde{l}_{i})-F_{\mathcal{D}}(\tilde{l}_{i})\right]^{2}$
and total sum of squares
$\mathrm{SS}_{\mathrm{T}}=\sum_{i=1}^{N}\left[F(\tilde{l}_{i})-\bar{F}(\tilde{l})\right]^{2}$.
Here, the experimental CDF is $F(\tilde{l}_{i})$ when $\tilde{l}_{i}$ is the
$i$th nondimensionalized domain length, with its mean
$\bar{F}(\tilde{l})=\frac{1}{N}\sum_{i=1}^{N}F(\tilde{l}_{i})$. The CDF of a
certain model distribution $\mathcal{D}$ is $F_{\mathcal{D}}(\tilde{l})$. Fig.
5(a) shows $R^{2}$ of experimental CDFs at different times to eight trial
distribution functions. Except for the two-parameter Weibull and gamma
distributions, the values of $R^{2}$ are close to one; $R^{2}=1$ means the
distribution model matches the data. The LN3 gives the highest $R^{2}$.

The Kolmogorov-Smirnov test determines the distribution best describing the
experimental data based on the largest absolute difference between the
experimental and model CDFs, i.e.,
${KS}=\sup_{\tilde{l}}\left|F_{\mathcal{D}}(\tilde{l})-F(\tilde{l})\right|$.
Here $\sup_{\tilde{l}}$ denotes the supremum of the set of absolute CDF
differences; the supremum stands for the value greater or equal to all the
other values in a set. If the test statistic ${KS}$ is smaller than a critical
value, the null hypothesis that the data comes from the particular
distribution is not rejected. In our test, the critical value for the test
statistic is not determined conventionally since the parameters of the fitting
distributions are estimated from the data itself. Therefore, we consider the
distribution with the smallest ${KS}$ to be the best model. Fig. 5(b) shows
${KS}$ for all trial distributions as a function of time, which reconfirms
that the LN3 fits our data best. The Quantile-Quantile (Q-Q) plots in Fig.
5(c) compare the experimental and LN3 CDFs. Except for the regions of very
short and long domains where the log-normal distribution expects more domains,
the experimental data points lie mostly on a 45-degree reference line,
indicating a fair agreement between the experimental distribution and LN3.

##  IV Conclusion

We study how nematic DT domains in a cylindrical capillary form and grow by
coarsening, focusing on their size distribution. Quenching from the isotropic
phase leads to the domains of different handedness separated by a singular
defect. The domains get coarsened by the annihilation of defects by merging
and exhibit a time-evolving length distribution. We propose a model describing
how stochastic merging within a train of nematic droplets as pre-domains with
normally distributed sizes and randomly assigned handedness, determines the
initial size distribution after the isotropic-nematic phase transition. Our
statistical tests propose that the three-parameter log-normal distribution
best approximates our experimentally measured size distributions of the
initial and coarsened domains. Our results manifest that the proposed 1D
system allowing facile optical observation and size characterization, provides
a useful model platform to study symmetry-breaking phase transition in
condensed matter, accompanying domains and topological defects formation.

Looking to future works, we believe understanding the energetics of DT domains
and their defects, would deepen our understanding of how the domain-size
distribution is determined and evolve as time goes by. Specifically, the
topological defects where the handedness of the ground-state DT domain changes
have an energy penalty, thus exhibiting effective attraction with each other
to lower the elastic free energy. Investigating the energy landscape may give
us hints at why the coarsening dynamics maintains the log-normal size
distribution over time evolution. It would also be intriguing to tune the
energy landscape by adding chiral dopants or applying external fields and see
how the coarsening dynamics would change Eun  _et al._ (2019); Dietrich  _et
al._ (2017). Then, this 1D model system with known and tunable energetics may
shed light on domain formation and coarsening dynamics in complicated systems
of higher dimensions.

## References

  * Kibble (1976) T. W. B. Kibble, J. Phys. A: Math. Gen. 9, 1387 (1976).
  * Zurek (1985) W. H. Zurek, Nature 317, 505 (1985).
  * Hendry  _et al._ (1994) P. C. Hendry, N. S. Lawson, R. A. M. Lee, P. V. E. McClintock, and C. D. H. Williams, Nature 368, 315 (1994).
  * Weiler  _et al._ (2008) C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley, M. J. Davis, and B. P. Anderson, Nature 455, 948 (2008).
  * Monaco  _et al._ (2009) R. Monaco, J. Mygind, R. J. Rivers, and V. P. Koshelets, Phys. Rev. B 80, 180501 (2009).
  * del Campo  _et al._ (2010) A. del Campo, G. De Chiara, G. Morigi, M. B. Plenio, and A. Retzker, Phys. Rev. Lett. 105, 075701 (2010).
  * Ulm  _et al._ (2013) S. Ulm, J. Roßnagel, G. Jacob, C. Degünther, S. T. Dawkins, U. G. Poschinger, R. Nigmatullin, A. Retzker, M. B. Plenio, F. Schmidt-Kaler, and K. Singer, Nat. Commun. 4, 2290 (2013).
  * Pyka  _et al._ (2013) K. Pyka, J. Keller, H. L. Partner, R. Nigmatullin, T. Burgermeister, D. M. Meier, K. Kuhlmann, A. Retzker, M. B. Plenio, W. H. Zurek, A. del Campo, and T. E. Mehlstäubler, Nat. Commun. 4, 2291 (2013).
  * Deutschländer  _et al._ (2015) S. Deutschländer, P. Dillmann, G. Maret, and P. Keim, Proc. Natl. Acad. Sci. U.S.A. 112, 6925 (2015).
  * Fowler and Dierking (2017) N. Fowler and D. I. Dierking, ChemPhysChem 18, 812 (2017).
  * Bowick  _et al._ (1994) M. J. Bowick, L. Chandar, E. A. Schiff, and A. M. Srivastava, Science 263, 943 (1994).
  * Chuang  _et al._ (1991a) I. Chuang, R. Durrer, N. Turok, and B. Yurke, Science 251, 1336 (1991a).
  * Chuang  _et al._ (1991b) I. Chuang, N. Turok, and B. Yurke, Phys. Rev. Lett. 66, 2472 (1991b).
  * Bradač  _et al._ (2002) Z. Bradač, S. Kralj, and S. Žumer, Phys. Rev. E 65, 021705 (2002).
  * Kim  _et al._ (2013) Y.-K. Kim, S. V. Shiyanovskii, and O. D. Lavrentovich, J. Phys.: Condens. Matter 25, 404202 (2013).
  * Wang  _et al._ (2018) P.-X. Wang, W. Y. Hamad, and M. J. MacLachlan, Philos. Trans. Royal Soc. A 57, 3360 (2018).
  * Golovaty, D  _et al._ (2020) Golovaty, D, Kim, Y-K, Lavrentovich, O. D., Novack, M, and Sternberg, P, Math. Model. Nat. Phenom. 15, 8 (2020).
  * Zhou  _et al._ (2012) S. Zhou, Y. A. Nastishin, M. M. Omelchenko, L. Tortora, V. G. Nazarenko, O. P. Boiko, T. Ostapenko, T. Hu, C. C. Almasan, S. N. Sprunt, J. T. Gleeson, and O. D. Lavrentovich, Phys. Rev. Lett. 109, 037801 (2012).
  * Zhou  _et al._ (2014) S. Zhou, K. Neupane, Y. A. Nastishin, A. R. Baldwin, S. V. Shiyanovskii, O. D. Lavrentovich, and S. Sprunt, Soft Matter 10, 6571 (2014).
  * Nayani  _et al._ (2015) K. Nayani, R. Chang, J. Fu, P. W. Ellis, A. Fernandez-Nieves, J. O. Park, and M. Srinivasarao, Nat. Commun. 6, 8067 (2015).
  * Davidson  _et al._ (2015) Z. S. Davidson, L. Kang, J. Jeong, T. Still, P. J. Collings, T. C. Lubensky, and A. G. Yodh, Phys. Rev. E 91, 050501 (2015).
  * Long and Selinger (2022) C. Long and J. V. Selinger, Journal of Elasticity , 1 (2022).
  * Tortora and Lavrentovich (2011) L. Tortora and O. D. Lavrentovich, Proc. Natl. Acad. Sci. U.S.A. 108, 5163 (2011).
  * Jeong  _et al._ (2014) J. Jeong, Z. S. Davidson, P. J. Collings, T. C. Lubensky, and A. G. Yodh, Proc. Natl. Acad. Sci. U.S.A. 111, 1742 (2014).
  * Jeong  _et al._ (2015) J. Jeong, L. Kang, Z. S. Davidson, P. J. Collings, T. C. Lubensky, and A. G. Yodh, Proc. Natl. Acad. Sci. U.S.A. 112, E1837 (2015).
  * Nayani  _et al._ (2017) K. Nayani, J. Fu, R. Chang, J. O. Park, and M. Srinivasarao, Proceedings of the National Academy of Sciences 114, 3826 (2017).
  * Fu  _et al._ (2017) J. Fu, K. Nayani, J. O. Park, and M. Srinivasarao, NPG Asia Materials 9, e393 (2017).
  * Javadi  _et al._ (2018) A. Javadi, J. Eun, and J. Jeong, Soft Matter 14, 9005 (2018).
  * Eun  _et al._ (2019) J. Eun, S.-J. Kim, and J. Jeong, Phys. Rev. E 100, 012702 (2019).
  * Srinivasarao (2021) M. Srinivasarao, Liquid Crystals: New Perspectives , 311 (2021).
  * Koning  _et al._ (2014) V. Koning, B. C. van Zuiden, R. D. Kamien, and V. Vitelli, Soft Matter 10, 4192 (2014).
  * Selinger (2018) J. V. Selinger, Liquid Crystals Reviews 6, 129 (2018).
  * Sangal and Biswas (1970) B. P. Sangal and A. K. Biswas, Water Resour. Res. 6, 505 (1970).
  * Hosoda  _et al._ (2011) K. Hosoda, T. Matsuura, H. Suzuki, and T. Yomo, Phys. Rev. E 83, 031118 (2011).
  * Polizzi  _et al._ (2021) S. Polizzi, A. Arneodo, F.-J. Pérez-Reche, and F. Argoul, Front. Appl. Math. Stat. 6 (2021).
  * Preibisch  _et al._ (2009) S. Preibisch, S. Saalfeld, and P. Tomancak, Bioinformatics 25, 1463 (2009).
  * Young (1977) I. T. Young, J. Histochem. Cytochem. 25, 935 (1977).
  * Pobočíková  _et al._ (2017) I. Pobočíková, Z. Sedliačková, and M. Michalková, Procedia Eng. 192, 713 (2017).
  * Dietrich  _et al._ (2017) C. F. Dietrich, P. Rudquist, K. Lorenz, and F. Giesselmann, Langmuir 33, 5852 (2017).