# Reddit in the Time of COVID

Veniamin Veselovsky, Ashton Anderson

###### Abstract

When the COVID-19 pandemic hit, much of life moved online. Platforms of all
types reported surges of activity, and people remarked on the various
important functions that online platforms suddenly fulfilled. However,
researchers lack a rigorous understanding of the pandemic’s impacts on social
platforms, and whether they were temporary or long-lasting. We present a
conceptual framework for studying the large-scale evolution of social
platforms and apply it to the study of Reddit’s history, with a particular
focus on the COVID-19 pandemic. We study platform evolution through two key
dimensions: _structure_ vs. _content_ and _macro_ \- vs. _micro_ -level
analysis. Structural signals help us quantify how much behavior changed, while
content analysis clarifies exactly how it changed. Applying these at the
macro-level illuminates platform-wide changes, while at the micro-level we
study impacts on individual users. We illustrate the value of this approach by
showing the extraordinary and ordinary changes Reddit went through during the
pandemic. First, we show that typically when rapid growth occurs, it is driven
by a few concentrated communities and within a narrow slice of language use.
However, Reddit’s growth throughout COVID-19 was spread across disparate
communities and languages. Second, all groups were equally affected in their
change of interest, but veteran users tended to invoke COVID-related language
more than newer users. Third, the new wave of users that arrived following
COVID-19 was fundamentally different from previous cohorts of new users in
terms of interests, activity, and likelihood of staying active on the
platform. These findings provide a more rigorous understanding of how an
online platform changed during the global pandemic.

## Introduction

A major impact of the COVID-19 pandemic was a dramatic surge in online social
platform usage. Virtually every major site reported large increases in
activity, with many previously-offline facets of life necessarily being
replaced by online analogs (Feldmann et al. 2021). Although examples of this
sea change are familiar from our own experience, we have yet to systematically
quantify the structure and content of how social media changed during the
course of the pandemic. This leaves significant gaps in our understanding of
social media. Did the processes governing online platforms before COVID-19
continue throughout the pandemic, or did a new organization emerge? Were the
new users that joined post-2020 fundamentally different from previous cohorts
of new users? What exactly were the changes that occurred, and did they last?

Answering these questions is important for both theoretical and pragmatic
reasons. From a theoretical perspective, rigorously measuring what happened on
social media during COVID-19 is a prerequisite to understanding _why_ it
happened. Although anecdotal experiences provide important early guidance in
developing understanding, only rigorous measurement of what occurred can
provide a sufficiently solid empirical foundation for further theoretical
inquiry. Furthermore, understanding how social media changed as a result of
COVID-19 is important not only for understanding the effect of crises on mass
communication but it is also a rare opportunity to understand the nature of
social media more generally. Much like neuroscientists advance their
understanding of the brain through analyses of unique traumatic brain injuries
and their repercussions, we can better understand the “hivemind” of social
media by measuring how it responded to the unique shock of a global pandemic.
From a pragmatic perspective, quantifying how social media has (and hasn’t)
changed during this turbulent period will provide a baseline comparison for
current and future analyses of social media activity in a post-pandemic world
and help inform the design of tomorrow’s digital environments.

Despite their importance, these questions have proven difficult to answer. In
this work, we present a general approach for studying platform evolution that
is applicable during either times of crisis or stability. Our approach
(illustrated in Figure 1) defines two key dimensions of analysis: 1) structure
versus content and 2) platform (macro-level) versus user (micro-level).
Structural signals help us quantify how much behavior changed, while content
analysis clarifies exactly how it changed. Applying these at the macro-level
illuminates platform-wide changes, while at the micro-level we discover more
nuanced user evolution. We apply this framework to the entire history of
Reddit, a large social media platform used by millions of people both prior to
and during the pandemic.

Figure 1: An illustration of our conceptual framework and our research
questions.

The Present Work. We showcase our general approach alongside the questions we
answer in Figure 1. Our first focus of analysis is on the structure of Reddit
at both the macro- and micro-levels. We find that Reddit experienced an
immediate and significant increase in activity at the start of COVID-19.
Overall activity jumped, new communities and topics of discussion emerged, and
a large influx of users arrived, often disrupting the experience of previous
cohorts. These new users that joined during the pandemic had different
activity patterns and areas of interest; their early experiences differed from
the traditional new user experience. In parallel, new language emerged, even
beyond expected COVID-related vocabulary; new clusters of communities became
popular; and the behaviors of older and newer users drifted. Government
stimulus inspired investing discussions, quarantine increased political and
world events discussions, and activity in NSFW communities doubled. These
changes were sometimes short-lived, but others persisted. Measuring these
dynamics is important for distinguishing between what aspects of online life
have returned to our previous “normal” versus which aspects have become a “new
normal.”

Two years on, many of these changes have persisted. For example, the boost in
monthly activity and the number of active subreddits did not wane and new
users still differ from their pre-COVID counterparts. COVID-era language has
also reached a natural background level on the platform, with words like
“coronavirus,” “stimulus,” “masks,” and “asymptomatic” becoming commonplace.
The most notable difference, however, lies in the types of communities that
are popular.

In summary, our main contributions are the following, and the rest of the
paper discusses each in detail:

  * •

A conceptual approach to studying social media analyses separating structure
from content and macro-level from micro-level analysis (Section Data and
Methodology).

  * •

An analysis of the structural changes Reddit went through during COVID-19
(Section Structural Evolution).

  * •

An analysis of the content-level shifts Reddit experienced by the platform and
user-levels (Section Content Evolution).

## Related Work

COVID-19 has affected our interactions with the digital world, augmenting
everything from the number of bits that were communicated across the Internet
to the infrastructures built on top of it. In this section, we touch on some
of the existing literature that has captured some of these effects.

Internet and social media. Anja Feldmann et al. utilized data from a series of
ISPs and IXPs to illustrate the mass influx in Internet use, which not only
spiked with lockdowns, but exhibited a lasting and unequal growth depending on
location and customer profile (Feldmann et al. 2021). A large swath of the
COVID-19 social research has centered on the social media giants that acted as
connectors and mediators for our at-home interactions. A cross-platform study
was conducted tracking the number of users on various platforms, as well as
the content they discussed, with a particular focus on potentially misleading
link-sharing (Cinelli et al. 2020). Other studies have similarly focused on
various questions of misinformation and fake news, quantifying its size,
detection, and user “susceptibility” (Evanega et al. 2020; Shang et al. 2022;
Khan et al. 2022; Teng et al. 2022; Weinzierl and Harabagiu 2022). Other work
has instead adopted a reactive approach by studying how to combat
misinformation through correcting user responses (Seo et al. 2022) and
examining how consumers should engage with COVID-related content (Raza et al.
2022).

User changes. User-level studies have instead tracked how the individual
experience has changed. This type of work typically examines how users or
cohorts of users changed before and after the pandemic. Zhunis et al. found
that shortly after the outbreak of COVID-19 users the amount of emotionally
negative posts increased, but several months after, users returned to their
pre-COVID emotion levels. In sum, users that were generally positive before
the pandemic returned to being positive, and users that were generally
negative returned to being negative (2022). A similar paper conducted a global
analysis of sentiment prior and throughout the pandemic examining how
lockdowns and COVID-19 cases affected people’s well-being, where they utilized
synthetic controls to study neighboring regions (Wang et al. 2022). Other
studies have instead focused on how the user composition has changed
throughout the pandemic, like Wikipedia, which experienced a large influx in
volunteer editing following COVID. (Ruprechter et al. 2021).

Social media during times of crisis. Social media and Web use in times of
crisis has been historically an important area of study. Research has
highlighted social media’s role as a connector and a disseminator of
information, giving people the power of sharing what occurs (Kogan, Palen, and
Anderson 2015). Another study examined Twitter’s evolution across a few
disasters by looking at the structural change and showing that users who
joined during a crisis tend to be longer-lasting users (Hughes and Palen
2009). The question of what “should” social media’s role be in times of crisis
and how to educate people about it has been presented (Dailey, Soden, and
LaLone 2018).

## Data and Methodology

Our analysis focuses on Reddit, one of the largest social media sites which is
composed of hundreds of thousands of communities (called subreddits) dedicated
to everything from politics (r/politics) to Japanese Woodworking
(r/JapaneseWoodworking). This wide collection of communities, alongside
Reddit’s minimal algorithmic curation, makes it a good platform to study the
various effects of COVID-19 across social media. Our data consists of all
comments and submissions on Reddit from its inception in December 2005 to June
2022 (Baumgartner et al. 2020). This includes almost 13 billion comments by 89
million users and 5 billion submissions by 48 million users.

Reddit’s size comes with the potential for noise and spam. To account for this
we took several pre-processing steps to limit bots and spam users. First, we
defined a bot-account as any user with “bot” or “mod” (both case insensitive)
in their username. Second, we de-duplicated comments by removing all comments
with more than 100 characters that have the same text as more than two other
comments (a variation of the approach adopted by Nobel et al. (2021)).
Finally, we also pre-processed the text of the comments for Section 5 by
removing stop words, punctuation, numbers, and URLs, as well as dropping all
comments with fewer than 3 characters.

Defining the start of the pandemic. Since this paper compares Reddit before
and during COVID-19, we need a point of reference to define the start of the
pandemic. Past work has relied largely on case timelines or mobility
restrictions data (Ribeiro et al. 2021; Yang et al. 2022). On Reddit, however,
matching mobility restrictions with location data is infeasible since users do
not publicly disclose their geographic location. Moreover, since Reddit is
used all over the world, we cannot simply use the mobility restrictions from
any one country. Even if we chose the United States, which is responsible for
most Reddit traffic, we still would have state-by-state variations in mobility
restrictions. For this reason, we rely on a simple platform-wide approximation
for the start of the pandemic as of February 1, 2020. This errs on the side of
caution by defining COVID-19 as being earlier than other estimates. Any
changes we notice on Reddit are likely to only be exaggerated if we define the
start of COVID-19 at a later date.

Analysis framework. We propose an analysis framework for capturing platform
changes over time using two axes: structure versus content (major axis) and
macro- versus micro-level (minor axis). The structural analysis examines the
meta-level factors associated with _how_ Reddit changes, including the size of
the platform and the dynamics of its user base. On the other hand, the content
analysis explores _what_ areas of attention users are focused on and the
language they use. Macro-level (platform-level) versus micro-level (user-
level) is a complementary axis that instead focuses on changes occurring at
different resolutions on the platform. Macro changes examine aggregated
metrics across various dimensions, whereas micro-level changes capture the
user-level mechanisms that may be driving some of the macro movements.

Effect estimation. Comparing metrics before and during COVID-19 risks picking
up on natural trends instead of COVID-related impacts. More generally, there
exist natural changes that drive certain metrics on the platform which will
confound the effects of COVID-19. To accommodate this possibility we will rely
on a well-established time series prediction tool called Prophet, which
captures non-linear trends, holiday effects, and ignores outliers (Taylor and
Letham 2018), and is often used in temporal analyses (Zunic et al. 2020; Usher
and Dondio 2020). Our general setup is to fit Prophet for a metric on Reddit’s
historic pre-COVID data (up to February 2020) and then predict the future.
Concretely, prior to February 2020, the predicted curve measures the
prediction of month $T$, given months $t<T$. On February 2020, we stop the
month-to-month prediction and estimate until June 2022. Then, we use the
difference between real-world data (where COVID-19 happened), and the
counterfactual forecast produced by Prophet to estimate the impact of COVID-19
on Reddit. This setup helps us disentangle the natural growth in Reddit from
that associated with COVID-19 and assess whether the change was long-lived.

Figure 2: Reddit’s growth in a series of metrics. The blue line represents
actual growth and the yellow line is a forecast fit with data until 1 February
2020.

## Structural Evolution

The first series of variables we examine are Reddit’s structural components.
While this can refer to many platform-level features, we restrict our study to
transformations in the user engagement patterns, which include everything from
raw activity counts to their habits on the platform. This form of meta-
analysis is important since it will lay evidence to some of the often-touted
claims surrounding the digital transformation which arose as a result of
COVID-19. How large was our increase in activity? What drove this spike in
activity? And most importantly, what should we expect a post-COVID social
media landscape to look like? Moreover, the structural patterns will act as a
solid basis for the proceeding content analysis, without which we can make
faulty conclusions about the changes we see. An example of why this is
important is shown in the work of Waller and Anderson where they attribute an
observable polarization on Reddit not to a radicalization of existing users,
but to a large influx of new users (2021) .

| Month | Loss | z-score
---|---|---|---
Submissions | 2020-04-01 | 0.13 | 0.74
2019-03-01 | 0.12 | 0.64
2016-11-01 | 0.10 | 0.35
2020-03-01 | 0.10 | 0.33
2019-05-01 | 0.10 | 0.30
New users | 2020-04-01 | 0.26 | 1.39
2020-05-01 | 0.25 | 1.31
2018-07-01 | 0.20 | 0.81
2021-01-01 | 0.16 | 0.51
2019-01-01 | 0.16 | 0.50
Subreddits | 2016-01-01 | 0.37 | 1.64
2016-02-01 | 0.34 | 1.44
2015-12-01 | 0.32 | 1.32
2015-10-01 | 0.21 | 0.54
2020-05-01 | 0.21 | 0.53

| Month | Loss | z-score
---|---|---|---
Comments | 2020-05-01 | 0.12 | 1.90
2020-04-01 | 0.09 | 1.13
2020-07-01 | 0.09 | 1.08
2020-06-01 | 0.08 | 0.96
2018-07-01 | 0.07 | 0.69
Active users | 2020-04-01 | 0.09 | 1.84
2020-05-01 | 0.09 | 1.81
2018-07-01 | 0.06 | 0.93
2019-03-01 | 0.05 | 0.67
2020-03-01 | 0.05 | 0.66
Large Subs | 2015-07-01 | 0.02 | 1.67
2021-11-01 | 0.01 | 0.94
2021-10-01 | 0.01 | 0.66
2020-04-01 | 0.01 | 0.35
2021-12-01 | 0.01 | 0.22

Table 1: For each metric (submissions, comments, new users, active users,
subreddits, and large subreddits) we present the five months where the Prophet
model performed the worst (Loss). We additionally show the z-score of the loss
as a signal for how wrong the prediction was in relation to all predictions.

### Macro-level structure

To gauge how Reddit changed as a whole, we track the evolution of several
platform-level metrics. Activity is measured through the number of submissions
and comments per month, the user base is measured by the number of new and
active users in a given month, and the number of communities is measured by
active and large (monthly comments $>$ 500) subreddits. To estimate these
changes we use the “effect estimation” design described in the methods
section, where for each metric we fit Prophet on Reddit’s historic pre-COVID
data and then use this line as an estimate for Reddit’s counterfactual growth
(no COVID-19).

The results from this setup are shown in Figure 2 where we contrast actual
(dark blue) vs. predicted (yellow) changes to these metrics. We observe large
jumps in each of the metrics with the most extreme changes occurring in the
number of new users and comments per month. In March 2020, the number of
submissions jumped by 49% year-over-year (YoY) from an average growth of 35%
YoY. Comments increased by 27% YoY, down from an average growth rate of 31%
YoY. This growth in activity was complemented by an influx of new users (52%
YoY) that came to the platform immediately after COVID. We similarly found a
large increase in the number of subreddits following COVID (47%), but a small
rise in the number of large subreddits. Unlike users, however, this increase
had a historical precedent, the period following the 2016 election (125%). A
deeper understanding of what leads to a rise in active subreddits warrants
further investigation.111 A possible explanation is that subreddit creation
can be used for karma-farming (increasing users’ reputation on the platform)

To further quantify these changes in relation to Reddit’s history, we report
the five most difficult months to predict for each metric in Table 1.
Specifically, we trained 89 Prophet models using data from months $t<T$, where
$T$ is a variable cutoff parameter encompassing each month between January
2015 and May 2022, and subsequently predicted the outcomes for month $T+1$. In
other words, instead of stopping the prediction on February 2020 and
predicting the future, we now stop the prediction right before each month.
This then gives a fair comparison across all-time on the magnitude of
different disruptions to the platform. We observe that for all metrics, the
start of COVID-19 is featured as one of the five largest shocks to the
platform; and occupies the top position for the number of comments, active
users, and new users. This indicates that the changes experienced on Reddit
during COVID-19 were truly historic in terms of scale and size.

Given that COVID-19 was a large momentary shock, we next examine how many of
these changes persisted throughout the two-year period following the initial
outbreak. For this analysis, we refer back to Figure 2 and measure how the
difference between the prediction (trained up to Feb. 2020) and the true data
for February 2022. We argue that if the metrics return to the pre-COVID growth
curve then COVID-19’s effects were a blip, and if they changed COVID-19 led to
a lasting change. Interestingly, we find that only the number of comments and
active subreddits experienced a lasting magnitude increase. On the other hand,
we observe that the number of submissions experienced a decline in activity
(-3.6%)222This drop in submissions may be due to some other structural changes
on Reddit. and the user metrics return back to their pre-COVID growth pattern,
even after experiencing a large temporary increase. This finding acts as a
word of caution for the arguments often stated (e.g. by (Qureshi and Woo
2022)) regarding the fundamental digital transformation COVID-19 accelerated.
On the contrary, we find that two years on, many of the structural social
media metrics returned back to their pre-COVID growth curve.

Figure 3: How activity on Reddit changed by different user activity levels and
cohorts. The left plot illustrates the proportion of activity in a given month
by activity buckets, and the right plot shows the relative activity of users
whose first comment were in different months. Figure 4: Proportion of users
active between the two given months of making their first post.

### Micro-level structure

In the previous subsection, we observed an immediate influx in commenting and
posting, alongside a rise in new users, but only a long-run increase in
comments and active subreddits. While this offers an interesting big-picture
understanding of what Reddit went through during COVID, it fails to capture
the mechanisms that could have been driving some of those changes. In this
section, we attempt to uncover what drove two of the changes: (1) commenting
behavior exhibited a lasting increase while (2) active users returned to their
pre-COVID growth trend. To understand both of these findings, we turn to the
micro-level and examine how different cohorts of users evolved, as well as
their propensity to stay active. In the first part, break down users by their
activity levels and measure the proportion of comments in a given month by
activity bucket and the proportion of a cohort in a given activity bucket. In
the second part, we show the fraction of users that remained active for a
significant period after their first comment.

Activity buckets. In Figure 3 we illustrate how the activity on Reddit changed
by different user activity levels. We determine activity level cutoffs by
manually examining user commenting distributions and defining intuitive
thresholds. More programmatic cutting techniques were attempted, like equal
bucket size cuts, but these results tended to be less intuitive and resulted
in similar patterns. The figure on the left illustrates the proportion of
activity in a given month by activity buckets (i.e. “Greater than 50” measure
the fraction of comments in a given month by users with more than 50 comments
that month, and “Less than 4” measures the fraction by users with less than 4
comments.) In general, more activity came from very active users. In fact,
users with more than 50 comments represent nearly 73% activity, a historic
high. In addition, there is a relative drop in low to moderate level active
users. Similarly, the figure on the right plots activity by first-comment
month. The x-axis represents users’ first commenting month, and the y-axis
indicates the fraction of activity that came from each activity level within
that group. Since each x-value is an aggregate over all months for that
cohort, we define their activity level as the mean number of comments during a
user’s active month. We observe that the user cohorts that joined following
the COVID-19 outbreak tended to be less active on average than previous
cohorts, with more and more of their activity coming from users with less than
4 comments a month on average.

Activity period. The new users that joined Reddit during the pandemic were
also less likely to be active for a longer period of time than users who
joined before the pandemic. We define a user as active for a longer stretch of
time if they comment in windows of $1$ to $3$, $3$ to $6$, $6$ to $9$, $9$ to
$12$ months after creating their account. This window technique allows us to
equally compare users across different periods of time. Figure 4 shows how the
proportion of active users changes over time. Here we observe a reversal in a
previous trend which saw users being active for longer periods of time.
Instead, the users that joined during COVID-19 users were less likely to stay
active for longer. Whereas in the above analysis, we noticed historic changes,
this change drop in activity pales in comparison to the drop in new-user
activity experienced during the 2016 US presidential elections.

These findings suggest that with the onset of COVID, Reddit experienced a
dramatic rise in new users that were not very active, as well as a surge in
activity by older cohorts that became more active. This raises important
questions about the needs that Reddit served for different cohorts of users
and may suggest that more established users became further enmeshed in the
Reddit ecosystem while newer users sought out new forms of entertainment.

Figure 5: Log odds ratio of language pre- and post-COVID language. The pre-
COVID vocabulary is defined as words that occurred twelve months prior to
COVID-19 and COVID-19 language is defined as language that emerged in the
first five months of the pandemic. Figure 6: Six LIWC categories that
experienced large change throughout COVID-19.

## Content Evolution

By examining the evolution in user activity we found multiple structural
transformations on the platform. This analysis, however, failed to capture how
the linguistic patterns and topics of interest changed. In this section, we
supplement the structure with an analysis of content, to gain insight into the
role Reddit served as well as a more nuanced understanding of the material it
supplied. We first examine the macro-level content changes on Reddit, with a
particular focus on language and topics. We explore what new language emerged
and how it differed from previous language, as well as the types of
communities that became more active. In the second half, we turn to the micro-
level by exploring disruptions to the user experience, again through the lens
of language (how different cohorts used COVID-related language?) and
consumption patterns (did the COVID-wave of users consume quantitatively
different content from previous waves of users?).

### Macro-level content

We begin by studying how content changed on the macro level; breaking down the
analysis into distinct, yet overlapping sections. First, we illustrate that
language underwent a transformation in terms of the raw occurrences and the
fundamental nature of use. Second, we turn to the subreddit-level changes, by
illuminating which topics became popular, and how they differed from previous
collections of topics.

Language. To study the language we defined a vocabulary of all words with more
than 99 uses in any month (an approach used related literature (Noble et al.
2021)). We then counted the number of times each of the words occurs for each
given month between January 2012 and June 2022.

In Figure 5, we compare the log probabilities of words that saw an increase or
decrease in usage before and during the COVID-19 pandemic. From this plot, we
can observe that the new language that emerged was largely COVID-related. For
instance, the increases were driven by words directly related to COVID-19,
such as “outbreak,” “distancing,” “antibody,” and “covid,” as well as
indirectly related words like “diy” (do-it-yourself). On the other hand, the
words that saw a decrease in usage tended to be related to politics or sports.
With the end of the impeachment of American President Donald Trump and the
formalization of Brexit in February 2020, we see a decline in the usage of
words frequently used during these events, such as “obstruction,”
“impeaching,” “Mueller,” and “Brexit.” Additionally, with the cancellation of
sporting events, there was a relative drop in words like “pass,” “fans,”
“defense,” and “team.” We also aggregate these words into LIWC categories
(Pennebaker et al. 2015) which includes a dictionary of words that capture
various psycholinguistic properties. In Figure 6 we present six categories
that went through dramatic changes throughout COVID-19.333A figure with all
LIWC categories is available in the code repository. We observe that pronouns
like ”She/He” and ”They”, words that ”negate” the meaning of a sentence, and
affect and positive emotion decline. On the other, there is a sharp temporary
spike in anxiety-related language.

To determine if this change was meaningful, we defined a simple unigram
language model over non-COVID-related language. The COVID-language vocabulary
was created using existing
datasets444https://www.btb.termiumplus.gc.ca/publications/covid19-eng.html and
a computational month-by-month comparison of word usage for COVID-related
words where we manually annotated the 1,000 words that changed the most from
November 2019 to June 2020 and classified them as COVID-related or not. Then,
for each month $t$, we measured the cross-entropy between the probability
distribution at month $t$ and $t^{\prime}$ to evaluate the similarity of the
two months in terms of word usage. We found that in March 2020 there was a
significant shift in language with a 0.032 increase in cross-entropy, the
largest increase between January 2018 and June 2022. This suggests that the
rate of language change accelerated, with certain words becoming more
prevalent and others falling out of usage.

One likely cause of this change is the emergence of new words. This is
evidenced by a marked increase in the popularity of certain words, as shown in
Figure 7, which displays a five-fold increase in the number of popular words.
We define a word as popular if it rises in popularity from below the 93rd
percentile to above the 94th percentile, which corresponds to an approximate
rise to 1000 times per month, from 500 times or less in February 2020. This
trend holds true across different percentile cutoffs. Furthermore, the new
language that emerged is more semantically diverse than before, as indicated
by the 1-d heatmap above the plot. We measure semantic similarity within a set
of new words by using a static word embedding. Explicitly, we map each word in
a set of new words onto its GloVe word embedding representation (Pennington,
Socher, and Manning 2014) and then calculate the centroid of all new words. We
then take the average cosine similarity between all the individual words and
the centroid and define this to be the “generalist-specialist score” (GS-
score) of the new language that emerged (Waller and Anderson 2019). In Figure
8, we notice that COVID-19 brought about the most diverse collection of new
languages since 2014. Language change is often a reflection of broader
societal phenomena, and the topics discussed online can provide insight into
what people are concerned about. The plot suggests that during the COVID-19
pandemic, users began discussing a wider range of dissimilar concepts. This
highlights the significant impact and global reach of COVID-19, as seen in
increased discussions of topics such as the economy, healthcare, connection,
and well-being. More interestingly, in other periods where we saw increases in
new language, (2016, 2019) the average cosine similarity between the words
maintained relatively high, indicating the shocks were less diffuse.

Figure 7: All words that increased in usage from less than the 93rd percentile
to above the 94th percentiler. The 1-d heatmap on the top illustrates the GS
score of the words that emerged. A lighter indicates a more general vocabulary
and a darker color indicates more specialized language. Figure 8: All
subreddits that increased in usage by 10 percentile points. The 1-d heatmap on
the top illustrates the GS score between the subreddits that emerged. A redder
color indicates a more general set of subreddits and a bluer color indicates
more specialized subreddits.

Interest Areas. We quantify the areas of the platform affected by increased
activity identified in the previous structural analysis by creating clusters
of subreddits to gain a high-level understanding of Reddit’s evolution. We
used the clustering method outlined in Waller and Anderson (2021). In the
work, the authors create and validate a behavioral embedding of Reddit, and
then cluster on the subreddit vectors. The embedding is created through
word2vec applied on subreddits (as words) and users (as contexts), which
results in a dense representation of subreddits. The proximity between two
subreddits indicates a similar set of users is active in both. We then cluster
on these dense representations, labeling the clusters based on the communities
they represent and their descriptions. A full list of the subreddits that make
up these topics is available in our online code repository and a sample of the
top five is included in the Appendix.555https://github.com/CSSLab/reddit-
covid-19

Figure 9: Evolution of topic interests over time. The blue line shows the
actual change in activity on a topic whereas the yellow line shows the
predicted change. The yellow area surrounding the yellow plot is the 95
confidence interval.

Figure 9 presents activity in these embedding-generated topics with the blue
line illustrating actual activity, and the yellow line representing predicted
activity. Similar to Figure 2, the prediction line is trained on the growth
trend for each topic until February 2020 and then predicts two years into the
future. Consequently, the yellow line represents the expected growth of a
subreddit topic prior to COVID. We find that there were some clear winners and
losers. To begin, the International cluster experienced a major growth in
activity. This cluster was composed of a series of global communities that saw
a dramatic rise in COVID-19 language and political discussion. These included:
r/thenetherlands, r/France, r/India, r/Germany, r/ontario, r/CasualUK, and
r/Philippines. As the COVID-19 pandemic continued, this topic saw no sign of
dying down and indicates that COVID-19 led to a rapid increase in Reddit’s
global adoption. Another two topics that experienced immediate and sharp
increases were (1) Advice: Dating and Life, and (2) Hobbies: 2. These topics
were composed of subreddits like r/NoFap, r/datingoverthirty, and
r/selfimprovement (1), and r/getdisciplined, r/selfimprovement, and
r/depression (2). The biggest winner, however, was NSFW (not safe for work)
content. This material in general had little mention of COVID-19-related
language but experienced large gains in activity. In fact, during the pandemic
NSFW content came to represent almost a two-fold increase in total Reddit
activity going up to 13% from 6.7% of all comments.

Interestingly, in addition to some of these major topics that experienced
increases, there were also a collection of subreddits that dropped in
activity. These subreddits represented traditionally popular content on Reddit
and included Animals, Memes, and Minecraft-like video games (Video Games 1).
Each one of these topics provides a rich lens into how our society evolved
throughout COVID-19 and a deep dive into all of these changes is beyond the
scope of the paper. However, these initial findings do reveal the complex and
ever-encompassing influence the COVID-19 period had on our lives.

We complement Figure 9 with the subreddit analog of the new words analysis
(illustrated in Figure 7) by looking at the patterns in subreddits that gained
a lot of attention. To do this we measured the subreddits that increased in
activity by 10 percentile points and used the subreddit embeddings to
calculate the GS-score (same process as described in the language section). We
plot this in Figure 8, and also observe not only a historic rise in the number
of subreddits but also that these new subreddits were largely dissimilar from
each other. This result is consistent with the change that we saw in language
and signals the diffuse impacts COVID-19 had on our lives. Additionally,
whereas prior new communities tended to be concentrated around one semantic
area, like investing, politics, or sports, the growth that took place during
COVID-19 touched on many distinct areas at once. Contrasting Reddit’s macro-
changes from a content and structural standpoint, we notice a stark
difference. While both have large immediate changes, content changes are far
more long-term. So while the habits of people on Reddit may not have changed,
the content they consumed did change dramatically. Consequently, it appears
that in the long-run COVID-19 accelerated a change in interest, but not a
change in activity.

Figure 10: How different cohorts of users were disrupted by COVID. Figure 11:
Divergence in user commenting patterns (over subreddits) between users whose
account formed in month $t$ and $t-12$ months commenting in month $t$.

### Micro-level content

The final section of this paper will complete our proposed 2x2 framework by
covering how content changed on the micro-level. Here we connect the various
findings we observed above through a study of three questions. (1) Did all
users’ activity get disrupted equally? (2) Was the large influx of new users
similar to that of new users in the past? (3) To what extent did various
cohorts turn to Reddit as a place to discuss COVID? The basic unit of analysis
in this section are cohorts. We define a user as being part of cohort $k_{m}$
if they commented for the first time in month $m$.

Cohort differences. We begin by examining how different cohorts of users were
disrupted by COVID-19. To measure this, we will compare cohort commenting
behavior between time $t$ and $t-12$ months and define a “deviation,” $D$, as
a disruption from a previously active set of communities. Explicitly, for
cohort $k_{m}$ we measure their commenting probability distribution over
subreddits for each month $t$, which we will denote $P_{m,t}$. Then we will
take the KL-divergence for the same cohort across two different periods, and
thus the deviation will be $D_{m}=D_{\mathrm{KL}}(P_{m,t}||P_{m,t-12})$ A high
kl-divergence indicates that the communities that the cohort was active in at
time $t$ are significantly different from that cohort’s activity in time
$t-12$; whereas a low kl-divergence suggests relative persistence in
interests. Figure 10 shows us that in March 2020 there is a historic rise in
divergence between all the cohorts; meaning that all users on the platform
experienced a shock to their consumption patterns. Older users also returned
to their pre-COVID consumption patterns quicker than newer users who typically
maintained a higher KL-divergence into the pandemic—indicating that their
active communities were more subject to change.

Differences in new users. In the structure-macro section we observed an
unprecedented rise in new users. In this section we explore who these new
users were—would they have joined the platform either way or were they a
fundamentally new segment of the population? Typically when new users join a
platform they have a distribution similar to previous cohorts of users. In
other words, their activity doesn’t deviate from what’s considered the norm
during a specific time-frame. However, if this deviation begins to grow, then
it might indicate that the new users that join the platform exhibit patterns
that are divergent from previous cohorts of users, and thus might not
represent the “typical Reddit user.” We operationalize this approach by
running a similar setup to the prior analysis only now comparing the
distribution in month $t$ between users who were part of cohorts $k_{m}$ and
$k_{m-12}$. In a similar notation to above, we’ll define new user deviation as
$D^{*}_{t}=D_{\mathrm{KL}}(P_{m,t}||P_{m-12,t})$. We illustrate these
divergences in a regression discontinuity in time design in Figure 11. In our
context, we consider the treatment at the start of the pandemic with cutoff 1
February 2020 and fit the curve $\hat{y}=\beta_{0}+\beta_{1}X+\beta_{2}t$,
where $X$ is an indicator variable equal to $1$ for $t\geq$ “1 February 2020”
and $0$ otherwise, $t$ is the date and $\hat{y}$ is an estimate of
$D^{*}_{t}$. The vertical displacement between the two curves estimates the
effect of COVID-19 on the new user activity divergence. The jump is equal to
$0.1437$ (95% CI: $[0.08,0.21]$) and indicates that the mass influx of new
users that joined alongside COVID-19 represented a user base whose interests
diverged from previous cohorts of users.

COVID related language. To better understand the emergence of the new COVID-19
lexicon we ran a cohort-level analysis on a sample of user comments. Each user
was bucketed into the year they first commented, and then using a 10% sample
of comments from January 2019 to June 2021 we measured the proportion of each
cohort’s language that was COVID-related. We present these proportions by
cohort in Figure 12. Immediately, we notice a large jump in COVID-language at
the onset of COVID-19. It spiked in March and April 2020 and then saw a
leveling off following May 2020. Interestingly, while all cohorts spoke about
COVID-19 more, we observe that newer cohorts tended to invoke COVID language
less than older cohorts. This may indicate that Reddit served different
purposes for well-established users and newer users, and further analysis of
this asymmetric pattern may be warranted.

Figure 12: Differences in the fraction of language in a given month that is
COVID-19 related across cohorts.

Finally, we’ll answer the three questions we prompted this discussion with.
First, we find that all cohorts on Reddit were affected by COVID-19, but the
older cohorts were able to return to their pre-COVID patterns quicker. Second,
the new users that joined Reddit around the time of COVID-19 were
fundamentally different from previous sets of new users that joined the
platform. As COVID-19 progressed this divergence between new users and users
that were one year old continued to increase. Finally, we found a stark
difference in how different cohorts started to use COVID-related language. We
notice almost a perfect gradient with newer users consistently less likely to
use COVID-related language than older cohorts.

## Discussion

In this paper, we complement previous studies of social media change by
presenting a systematic approach to studying platform change and apply it
during a focal event. Our framework relies on two axes (structure versus
content and macro-level versus micro-level). Structure captures the meta-
levels like activity and user composition, whereas content analyzes the actual
topics of discussion and language. The macro- versus micro-level change
instead examines platform versus user changes. This lets us pinpoint whether
macro-level changes are driven by new users entering the platform, or whether
there are new behaviors emerging across the whole platform.

At the beginning of this paper, we defined a series of questions that our
framework was able to solve. Now in Figure 13 we summarize the answers to each
of these questions. In review, Reddit experienced a truly historic increase in
activity in relation to its predicted growth. We find that across all six
structural-macro changes, COVID-19’s emergence led to some of the hardest
months to forecast. It was one of the five worst months for all metrics
covered (comments, submissions, new users, active users, active subreddits,
and large subreddits), and the worst month for 4/6 metrics: submissions,
comments, new users, and active users. In the long run, some of these rises
were short-lived. For instance, the number of new users and active users
returned to its previous growth curve two years on. But for the number of
subreddits and comments per month, Reddit still experiences increases from
COVID-19. On the user level, we notice that these increases were due to older
cohorts that became more active; whereas the experience of the new users on
the platform was relatively shorter-lived than previous cohorts of new users.

Figure 13: An illustration of the 2x2 (structure vs. content, macro vs. micro)
framework as well as the answers to questions we use initially posed.

We then complement the structure analysis by examining how the content of
discussions evolved. Content is examined both through the linguistic changes
and evolution of areas of discussion (i.e. clusters of subreddits). We first
notice that following the pandemic, Reddit experiences an unprecedented change
in the language that was used on the platform. By defining a language model
for each month and predicting language use in other months, we find that the
cross-entropy rose after March 2020. This was in part due to a rise in new
COVID-related language. This new language that emerged, tended to remain
popular for a longer period than the new language that emerged prior to
COVID-19 which was usually short-lived and driven by memes and political
language. Instead, words like “coronavirus,” “masks,” “virus,” and “zoom” all
became commonplace and part of our lives.

We also analyzed content through which people commented by using a custom-made
topic classification of subreddits on Reddit. The classification was built
through a neural embedding of communities and similarity between communities
was defined by similar users commenting in both communities. We found that the
topical landscape of which communities became salient also rapidly changed. In
general, there was a rapid rise in NSFW, political communities, and investing;
and a relative drop in “traditional” Reddit content. We connected the
linguistic changes with topical changes in activity to track which subreddits
experienced a rise in activity by becoming the centers for COVID-19
discussions and which subreddits increased in activity due to people simply
staying at home. Afterward, aimed to better understand these changes by
shifting our attention to the user level to study the topical and linguistic
changes. Here we have two key takeaways. First, the older cohorts tended to
view Reddit as a place to discuss COVID-19 whereas newer cohorts in general
tended to use less COVID-19-related language. Secondly, everyone was disrupted
equally by COVID-19, but older cohorts were quicker to shift their interests
back.

There are a few limitations in the current analysis that need to be discussed.
To begin, this paper presents a description of the changes that occurred on
Reddit during the pandemic. This structure, however, does not allow us to make
causal claims of COVID’s impacts. Instead, we have rigorously documented what
Reddit experienced during the pandemic. For this reason, additional causal
structures would have to be added to the analysis to understand the causal
mechanisms at play. With social media occupying an increasingly important role
in our lives, it is becoming more important than ever to study and audit these
platforms and how they change, since understanding them will help us
understand ourselves. We believe that the methodology and framework we propose
are general enough that they can be applied to different platforms during
times of crisis and calm. Additionally, one limitation of using a unigram
language model is that it misses how the semantics of language change. In
general, our approach assumes that over a short period of time, a change in a
word is likely driven by an increase in the use of that word and that a
significant change in meaning will be accounted for by the use of that word.

### Conclusion, Broader Impacts, and Ethical Concerns

This work acts as an empirical foundation for future analyses of social media.
We focused on the “what changed” question; we believe that this is a necessary
precursor to the next set of analyses on “why” they changed. In particular, we
report a series of interesting, and sometimes surprising, changes that we
experienced over the course of COVID-19. Additionally, our approach is general
so it can be used in future analyses to study Reddit in times of calm and
change.

There are a couple of possible ethical concerns in our study that we attempt
to mitigate. To begin, the Pushift Reddit dump potentially contains deleted
comments and posts. But given our aggregated analysis, these harms are
minimal. Additionally, no authors were paid or associated with Reddit during
the course of the analysis limiting any potential conflicts of interest.

With social media occupying an increasingly important role in our lives, it is
becoming more important than ever to study and audit these platforms and how
they change, since understanding them will help us understand ourselves. We
believe that the methodology and framework we propose are general enough that
they can be applied to different platforms during times of crisis and calm.

Acknowledgements. Thank you to Manoel Horta Ribeiro for his helpful comments.
As well as NSERC, CFI and ORF for funding the project.

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## Appendix

### Subreddits in Topics

In Table 2 we illustrate the top five subreddits in each cluster.

Topic name | Top 5 subreddits
---|---
Academics and Politics | politics, conspiracy, worldnews, news, unpopularopinion
Advice: Dating and Life | Tinder, relationship_advice, RedditSessions,
distantsocializing, sex
Animals | Eyebleach, rarepuppers, AnimalsBeingDerps, aww, cats
Anime | OnePiece, pokemon, anime, Animemes, Genshin_Impact
Art | whatisthisthing, fountainpens, Aquariums, gardening, Art
Board Games and Movies | marvelstudios, magicTCG, DnD, StarWars, PrequelMemes
Cryptocurrency | CryptoCurrency, Bitcoin, wallstreetbets, Superstonk, dogecoin
Hobbies 2 | AskWomen, relationships, AmItheAsshole,
TwoXChromosomes, Random_Acts_Of_Amazon
Hobbies: 1 | personalfinance, Fitness, guns, cars, AskMen
International | de, Cricket, soccer, FIFA, formula1
Memes 1 | AskOuija, wholesomememes, starterpacks,
insanepeoplefacebook, me_irl
NSFW 1 | Celebs, selfie, amihot, feetpics, gentlemanboners
NSFW 2 | electronic_cigarette, TrueFMK, FreeKarma4U, Drugs, gonewild
Popular | AskReddit, funny, pics, todayilearned, videos
Programming | sysadmin, linux, cscareerquestions, ProgrammerHumor, programming
Reactions | trashy, interestingasfuck, facepalm, PublicFreakout, nextfuckinglevel
Sports and Music | hockey, SquaredCircle, CFB, nba, nfl
Technology | apple, Android, pcmasterrace, MechanicalKeyboards, buildapc
Video Games 1 | dankmemes, PewdiepieSubmissions, Minecraft, teenagers, memes
Video Games 2 | leagueoflegends, FortNiteBR, DotA2, gaming, DestinyTheGame
Table 2: Top 5 subreddits in each cluster.


# Reorientation dynamics of microswimmers at fluid-fluid interfaces

Harinadha Gidituri, Zaiyi Shen, Alois Würger, Juho S. Lintuvuori  Univ.
Bordeaux $\&$ CNRS, LOMA (UMR 5798), 33405 Talence, France

(Received: date / Accepted: date)

###### Abstract

We study the orientational and translational dynamics of spherical
microswimmers trapped at fluid interfaces, in terms of the force dipole and
source dipole components of their flow field. Using numerical simulations and
analytical calculations, we show that the force dipole exerts a torque,
orienting pushers parallel to the interface, and pullers in normal direction.
The source dipole results in particle rotation only for a finite viscosity
contrast between the two fluids, in agreement with previous studies. The
superposition of these two contributions leads to an rotational dynamics with
a steady-state orientation that depends on the relative magnitudes of the
force and source dipoles. In the general case, swimmers with weak force
dipoles and strong pullers are observed to align perpendicular to the
interface and become stationary, while strong pushers have a finite
inclination angle towards the lower viscosity fluid and swim along the
interface.

Introduction.– Microscopic active particles, such as bacteria (eg. E. coli)
and algae (eg. Chlamydomonas) often found in confined environments near solid-
fluid and fluid-fluid interfaces [1], and surface interactions can affect
their dynamics. For example, E. coli swims in clockwise circular motion near a
no-slip surface [2] and anti-clockwise close to air-liquid interface [3]. In
the last two decades, researchers have synthesized artificial microswimmers
(eg. active Janus particles) [4]. These artificial swimmers have potential for
applications, such as drug delivery [5] and environmental remediation [6, 7,
8, 9]. For example, active Janus particles have been used to collect oil
droplets using sea water as a fuel [7]. Understanding the dynamics of these
particles at interfaces, is crucial for designing such applications.

The dynamics of microwsimmers in the vicinity of an interface is governed by
the hydrodynamic stresses created by the particles [10, 11], and has been
studied both experimentally [3, 12, 13, 14, 15, 16, 17, 18, 19] and
theoretically [20, 21, 22, 23, 22, 24]. Previous investigations mainly
considered microswimmers near interfaces. Less is known of particles
straddling a clean interface between two fluids, where they can be
thermodynamically trapped due to a Pickering effect [25]. The hydrodynamic
boundary condition at the contact line between the two fluids and the particle
surface, can play a role both in the linear and orientational dynamics of the
swimmer [26]. Simulations of self-diffusiophoretic colloids at fluid-fluid
interface predicted an emergence of an aligning torque on a particle at an
interface between two fluids with equal viscosities [27].

In general, the two fluids have different viscosities, characterised by the
ratio $\lambda=\eta_{2}/\eta_{1}$. The effects of viscosity stratification on
the dynamics (translational and rotational) of microswimmers have been studied
both experimentally and numerically [28, 29, 30, 31, 32, 33, 34, 35], where
reorientation towards negative gradients (towards the less viscous fluid) is
typically observed in fluids with the viscosity changing over a length scale
considerably larger than the particle size. This negative viscotaxis has also
been predicted by simulations of catalytic swimmers [36] and observed in
recent experiments of bacteria [33, 34] in sharp viscosity gradients.

In this work, we use lattice Boltzmann simulations to study the dynamics of a
spherical swimmer trapped at a clean interface separating two fluids. We
consider the experimentally most relevant case of low Reynolds and capillary
numbers, where inertial effects and interface deformation are small. Both the
viscosity contrast and the hydrodynamic boundary condition are described in
terms of a Ginzburg-Landau-functional for an interface of finite thickness.
The swimmer is described by the squirmer model [37], the flow field of which
consists of two components: a source dipole resulting from a sink and a source
flow, and a force dipole corresponding to the rotational flow of a pair of
opposite point forces (Fig. 1).

Figure 1: (a,b) Slip flows at the particle surface corresponding to (a) source
and (b) force dipoles, inclined at an angle $\phi$ with respect to the
interface.

We find that both flow field components contribute to the reorientation
dynamics of the swimmer, albeit in a rather different manner. We demonstrate
that the torque exerted by the force dipole, depends crucially on the squirmer
characteristics (pusher or puller). On the other hand, the torque arising from
the source dipole term (neutral squirmer) has been predicted to be
proportional to the viscosity difference [33, 36]. Our simulations agree with
this. We show that these two contributions are independent of each other.

We take into account both the source (Fig. 1a) and the force dipole
contributions (Fig. 1b) as well as the viscosity ratio $\lambda$, and
construct a state diagram for the steady state orientation. We observe that in
the steady state, weak swimmers, dominated by the source-dipole contribution,
become stationary and orient perpendicular to the interface pointing towards
the lower viscosity fluid. Further, we show that the force dipoles are
insensitive to the viscosity contrast but the hydrodynamic boundary condition
at interface leads to a reorienting torque. Strong pullers are observed to
turn perpendicular to the interface, and become immobile, while strong pushers
swim along the interface, pointing towards the lower viscosity fluid with a
finite angle respect to the interface normal.

Computational model and parameters.–

To simulate the finite size squirmers [38, 39], we impose a slip velocity at
the particle surface [40, 41]

$v_{s}=B_{1}\sin\theta+B_{2}\sin\theta\cos\theta,$ (1)

where $\theta$ is the polar angle with respect to the particle axis
$\mathbf{e}$. The first term corresponds to a source dipole, which is
responsible for the bulk swimming speed $U_{0}=\frac{2}{3}B_{1}$ along the
axis $\mathbf{e}$, and generates a far-field varying as $B_{1}r^{-3}$ with the
distance $r$ from the particle center [42]. The second term arises from a
force dipole, with a far-field component $\propto B_{2}r^{-2}$ that is at the
origin of long-range hydrodynamic interactions. The ratio of their amplitudes
defines a squirming parameter $\beta=B_{2}/B_{1}$, which characterises the
swimmer type: pullers (pushers) correspond to $\beta>0$ ($\beta<0$) while
$\beta=0$ is a neutral swimmer.

Figure 2: (a) Schematic representation of an active squirmer radius $R$ and
swimming speed $U_{0}$ at an interface separating two fluids with viscosities
$\eta_{1}$ and $\eta_{2}$. The orientation angle $\phi$ is defined as the
angle measured between the squirmer orientation and fluid-fluid interface and
$\theta$ is the polar angle. (b) Temporal evolution of the orientation
$\phi(t)$) for $B_{2}<0$ and $B_{2}>0$ for $\eta_{1}=\eta_{2}$. (c) The
measured angular velocity $\Omega(\phi)$ as a function of the angle $\phi$.
The blue (red) corresponds to $B_{2}>0$ ($B_{2}<0$) and dotted lines are fits
to $\hat{\Omega}_{F}\sin(2\phi)$. (d) The prefactor $\hat{\Omega}_{F}$ as a
function of the force dipole strength $B_{2}$.

The fluid-fluid interface is realised in terms of a Ginzburg-Landau free
energy functional [43],

$F[c]=\int dV\left({-\frac{A}{2}c^{2}+\frac{B}{4}c^{4}+\frac{\kappa}{2}|\nabla
c|^{2}}\right)$ (2)

where $-A=B>0$ and $\kappa$ are constants, and $c$ is the phase composition,
where $c^{*}=\pm 1$ are the equilibrium compositions. The temporal evolution
of the phase field variable $c$ is governed by a Cahn-Hilliard advection-
diffusion equation, and the fluid velocity is obtained by solving the
incompressible Navier-Stokes equation (for more details of the model see e.g.
[43, 44]). The coupled equations are solved using a hybrid finite difference
lattice Boltzmann scheme detailed in [43, 44]. The phase dependent viscosities
are implemented through the relation [45],

$\eta(r)=\eta_{1}^{\frac{1+c}{2}}\eta_{2}^{\frac{1-c}{2}}$ (3)

where the viscosity takes the values $\eta_{1,2}$ well above or below the
interface where $c=\pm 1$. This law expresses the fact that the viscosity of
liquid mixtures varies exponentially with the concentrations of their
components, $\ln\eta=c_{1}\ln\eta_{1}+c_{2}\ln\eta_{2}$, as first proposed by
Arrhenius in 1887 [46]. In physical terms, it is related to the fact that in
many liquids the viscous motion arises from activated jumps, such that
$\ln\eta_{i}$ is a measure of the free enthalpy barrier of molecular component
$i$.

Unless otherwise mentioned we fix $U_{0}=\frac{2}{3}B_{1}=10^{-3}$, and vary
$B_{2}$ to study the relative contributions between source and force dipole
flows. We consider a particle with radius $R=12$ in a simulation domain
$160\times 160\times 160$ with periodic boundary conditions. The lattice
spacing $\Delta x$, time step $\Delta t$ and density $\rho$ are set to unity.
The binary fluid parameter are chosen as $B=-A=0.0258$, surface penalty
$\kappa=0.04$ and mobility $M$ = 0.5. This leads to a flat interface at $c=0$
with an interfacial width $\chi_{0}=\sqrt{2\kappa/|A|}\approx 1.76$ and
interfacial tension
$\sigma=\sqrt{8\kappa|\mathcal{A}|^{3}/9\mathcal{B}^{2}}\approx 0.03$ [43].

The relevant non-dimensional quantities are the capillary number
$\mathrm{Ca}=\eta_{1}U_{0}/\sigma$ which compares viscous stresses with
interfacial tension, and the Reynolds number $\mathrm{Re}=\rho
RU_{0}/\eta_{1}$ which is the ratio of inertial and viscous forces. With the
above parameters, we find $\mathrm{Re}\sim 10^{-2}$ and $\mathrm{Ca}\sim
10^{-2}$, which means that inertial and interfacial deformation effects are
negligibly small.

Using physical parameters of water, and a particle radius $\sim 1\mu$m,
$\mathrm{Re}\sim 10^{-2}$ corresponds to $U_{0}\sim 10^{-2}$m/s. We can map a
single length and time unit as $\Delta x\sim 0.1\mu$m, and $\Delta t\sim
10^{-5}$s.

Results.– A neutrally wetting particle placed at the interface, adopts a
symmetrical position (Fig. 2a) [44]. We start with the case $B_{1}=0$, where
the particle has zero linear velocity yet is subject to a force dipole $B_{2}$
(Fig. 1b). In the bulk, such a “shaker particle” does not move yet produces
long-range flow field components proportional to $r^{-2}$ and $r^{-4}$ [42].
When trapped at an interface, however, the particle shows rotational motion in
respect to the interface (Fig. 2). The final stable orientation depends on its
squirmer characteristics: A puller ($B_{2}>0$) turns its axis $\mathbf{e}$
towards the interface normal, and a pusher ($B_{2}<0$) parallel to the
interface (Fig. 2b).

We have measured the reorientation velocity $\Omega_{F}$ resulting from a
force dipole, as a function of the angle $\phi$ between particle axis and
interface, and find a sinusoidal dependence

$\Omega_{F}=\hat{\Omega}_{F}\sin(2\phi),$ (4)

as shown by the symbols in Fig. 2c. The prefactor is proportional to the
squirmer parameter, $\hat{\Omega}_{F}\propto B_{2}$ (Fig. 2d).

Figure 3: a) Decomposition of the force dipole at an angle $\phi$ with respect
to a rigid interface with slip boundary conditions. The second term exerts a
torque on the particle. The boundary conditions at the particle surface and
fluid interface require additional higher multipoles [47]. b) Slip velocity
$v_{s}$ at the contact line. c) The interface imposes a discontinuity of
$v_{s}$ and an intricate flow profile in the vicinity of the contact line.

The source dipole slip-flow is symmetric at the interface (see e.g. Fig. 1a).
Thus, the rotational motion can only result from the coupling of the interface
to the force dipole (Fig. 1b),
$\mathbf{f_{\pm}}=\pm\mathbf{e}f_{0}\delta(\mathbf{r}\mp a\mathbf{e})$, with
the particle axis $\mathbf{e}$ and where the squirmer coefficient is defined
as $B_{2}=f_{0}a/2\pi\eta R^{2}$. In Fig. 3a the force dipole is decomposed in
analogy to Blake’s treatment at a solid surface [47]. This gives rise to two
terms compatible with a highly rigid fluid interface. From their symmetry it
is clear that the first one does not affect the particle motion, whereas the
second one results in the torque

$\mathbf{T}=4\pi\eta R^{2}\,\mathbf{e\cdot n(n\times e)}=2\pi\eta
R^{2}\sin(2\phi)\mathbf{\tau},$ (5)

where the unit vector $\mathbf{\tau}$ is perpendicular on the particle axis
$\mathbf{e}$ and the interface normal $\mathbf{n}$.

This rationalizes the dependency $\Omega_{F}\propto\sin 2\phi$ observed in the
simulations, yet it does not provide the drag coefficient $T/\Omega_{F}$,
which is determined by the additional contributions to the velocity field.
Fig. 3b shows the slip velocity (1) at the contact line, which is clearly
incompatible with the presence of the interface. Starting from the force-
dipole flow and satisfying the boundary conditions both at the interface and
at the particle surface, results in a series of source and force multipoles,
similar to that of an interfacial particle driven by a self-generated
Marangoni flow [48]. The resulting flow profile in the vicinity of the contact
line, is illustrated in Fig. 3c.

Figure 4:  (a) Angular velocity $\Omega_{S}(\phi)$ observed for a neutral
squirmer for different values of the viscosity ratio $\lambda$. The arrow
indicates an increase from $\lambda=0.1$ to 0.7, with a step size of 0.1. The
dashed lines are given by $\Omega_{S}=\hat{\Omega}_{S}\cos\phi$. (b) The
prefactor $\hat{\Omega}_{S}$ obtained from the simulations (full circles),
whereas the solid line is calculated from by eq. (7) (solid line) with the
constant $-0.225U_{0}/R$.

Source dipole contribution.– Now we turn to self-propelling microswimmers with
a finite $B_{1}$, which have been shown to be sensitive to variations of the
viscosity of the swimming medium. In previous experimental and theoretical
investigations, a negative viscotaxis has been observed [28, 29, 30, 31, 32,
33, 34, 35, 36]. To study the effect of the source dipole $B_{1}$ we consider
a neutral squirmer ($B_{2}=0$) and introduce a viscosity ratio
$\lambda=\eta_{2}/\eta_{1}$.

Our simulation data for the source-dipole driven angular velocity $\Omega_{S}$
are shown in Fig. 4, as a function of the particle orientation $\phi$ and the
viscosity ratio $\lambda$. For $\lambda\leq 0.7$ the variation with the angle
$\phi$ (Fig. 4a) obeys the relation

$\Omega_{S}=\hat{\Omega}_{S}\cos\phi,$ (6)

as expected from previous work [31, 36]. For larger values of $\lambda$, the
hydrodynamic torques are weak and a slow drift of $\phi(t)$ is observed, which
likely arises from numerical errors due to the finite capillary number used in
the simulations. In Fig. 4b we plot the dependence of the prefactor
$\hat{\Omega}_{S}$ on the viscosity ratio $\lambda$, and find a good agreement
with the theoretical prediction [36]

$\hat{\Omega}_{S}=\mathrm{const.}\times\frac{(2-\lambda)(1-\lambda)}{1-\frac{3}{32}(1-\lambda)^{2}}$
(7)

with the constant $-0.225U_{0}/R$.

General squirmer.– Finally, we consider the general case of a microswimmer
with both force and source dipoles. At low Reynolds number the reorientation
rate can be written as the sum of their respective contributions,
$\Omega=\Omega_{S}+\Omega_{F}$, where the former is proportional to $B_{1}$
and the latter to $B_{2}$. In the following we assume a constant self-
propulsion velocity $U_{0}=\frac{2}{3}B_{1}$, and discuss the dependencies on
the viscosity ratio $\lambda$ and the squirmer parameter $\beta=B_{2}/B_{1}$,

$\Omega(\lambda,\beta)=\hat{\Omega}_{S}(\lambda)\cos\phi+\hat{\Omega}_{F}(\beta)\sin
2\phi.$ (8)

The prefactor $\hat{\Omega}_{S}$ is a complex function of $\lambda$, whereas
$\hat{\Omega}_{F}$ is proportional to $\beta$. The stationary points
$\phi^{*}(\lambda,\beta)$ of the orientational dynamics are given by a zero
angular velocity, $\Omega=0$, and can be visualized by an effective
orientational potential $\Psi=-\int{\Omega d\phi}$ (Fig. 5). From eq. (8) it
is clear that its minima and maxima are given by
$\phi^{*}=\arcsin(\hat{\Omega}_{S}/2\hat{\Omega}_{F})$ for
$|\hat{\Omega}_{S}|<|2\hat{\Omega}_{F}|$, corresponding to strong pushers and
pullers, respectively. For weak force dipoles,
$|\hat{\Omega}_{S}|\geq|2\hat{\Omega}_{F}|$, only two stationary points
$\phi^{*}=\pm 90^{\circ}$ are observed (Fig. 5).

Figure 5: Orientational potential
$\Psi(\phi)=-\int_{0}^{\phi}{\Omega(\phi^{\prime})d\phi^{\prime}}$ for pushers
($\beta<0$) and pullers ($\beta>0$) with $\lambda=0.2$, calculated from eq.
(8).

Using (8), we construct a steady state diagram in terms of stable and unstable
fixed point in the $\lambda-\beta$-space (Fig. 6). To test these theoretical
predictions, we carried out simulations where the squirmer parameter $\beta$
and the viscosity ratio $\lambda$ were varied. Both theory and simulations
show three different ranges.

The first one (I) is observed for pushers ($\beta<0$) and is characterized by
a stable fixed point between $-90^{\circ}$ and $0^{\circ}$ (Fig. 6),
corresponding to minimum of $\Psi$ (see e.g. $\beta=-3$ and $\beta=-5$ curves
in Fig. 5). This arises from the competition between the force dipole
contribution turning the particle towards the interface and the source dipole,
which turns the particle towards the lower viscosity fluid. The resulting
steady-state orientation varies from parallel to the interface ($0^{\circ}$,
upper left corner) to normal orientation ($-90^{\circ}$, dashed line). The
$\phi^{*}$ observed from the simulations are given by circles, with a filling
color according to the color bar at the right in Fig. 6. The background color
corresponds to the theoretical expression
$\phi^{*}=\arcsin(\hat{\Omega}_{S}/2\hat{\Omega}_{F})$, where the constant
prefactors of $\hat{\Omega}_{F}$ and $\hat{\Omega}_{S}$ are taken from the
fits in Figs. 2d and 4b, respectively.

In range (II) the stationary orientation corresponds to a minimum at
$\phi=-90^{\circ}$, which occurs for sufficiently small $|\beta|$, where the
source dipole term $\Omega_{S}$ is dominant and orients the swimmers orient
towards the lower viscosity fluid. The dashed lines give the theoretical
boundaries of range II, defined by $\Omega_{S}=\pm 2\Omega_{F}$. The upward
orientation $\phi=90^{\circ}$ corresponds to an unstable fixed point (see e.g.
$\beta=\pm 1$ curves in Fig. 5).

Range (III) describes strong pullers. The force dipole contribution
$\Omega_{F}$ dominates and turns the swimmer towards one of the minima at
$\phi=\pm 90^{\circ}$. The steady state orientation is decided depending on
whether the initial orientation is below or above the unstable fixed point at
$\phi=\arcsin(\hat{\Omega}_{S}/2\hat{\Omega}_{F})$, as illustrated by the
curves for $\beta=+3$ and +5 in Fig. 5.

Figure 6: A state diagram for the stationary angle $\phi^{*}$ in a $\beta$ \-
$\lambda$ space. Simulation results (symbols) and theory (background) are
color-coded according to the color bar at the right. Range I (circles)
indicates a stable fixed point in the range $-90^{\circ}<\phi\leq 0$ for
pushers, and range II (diamonds) at $-90^{\circ}$ for sufficiently weak pusher
and pullers, where the source dipole contribution dominates. Range III
(squares) indicates the stationary states $\phi^{*}=\pm 90^{\circ}$ for strong
pullers. The dashed lines give the boundaries between these states and are
calculated from $\hat{\Omega}_{S}=\pm 2\hat{\Omega}_{F}$. (See text for more
details)

We observe a good agreement between the simulations and the theory based on
the superposition principle of the two interactions $\eqref{eq:omega_total}$.
The deviations observed for $\beta<0$ at strong viscosity contrast,
$\lambda<0.2$, are possibly due to numerical artefacts overestimating the
source dipole contribution.

Conclusions.– We have investigated the reorientation dynamics of spherical
microswimmers trapped at a clean fluid-fluid interface. In rather good
agreement with the theoretical models, our numerical simulations demonstrate
that the reorientation has two, independent, components: The force dipoles
give a rise to a torque, which drives a parallel steady-state orientation for
pushers and a perpendicular one for pullers. When a viscosity difference is
introduced, our simulations show that neutral swimmers orient towards the
lower viscosity fluid, in agreement with simulations of catalytic particles
[36] and bacterial experiments [33, 34] in sharp viscosity profiles. Our
results show moreover, that these two contributions are independent of each
other, and that their interplay defines the steady-state orientation.

In summary, in the case of a weak force dipole and strong viscosity contrast,
the swimmers align on the interface normal toward the less viscous fluid, such
that their translational velocity is zero. Similarly, strong pullers adopt a
normal orientation yet may be trapped in an upward or downward state and
become stationary. A different behavior is predicted for strong pushers, which
in the steady state reach a finite inclination angle, toward the lower-
viscosity fluid, and thus move at a finite velocity $U=U_{0}\cos\phi^{*}$
along the interface.

Acknowledgements.– HG would like to thank Sotiris Samatas for engaging and
helpful discussions. HG is grateful for the computational resources provided
at LOMA (University of Bordeaux) and cluster Curta at MCIA. HG, ZS and JSL
acknowledge the French National Research Agency (ANR) through Contract No.
ANR-19-CE06-0012-01, IdEx Bordeaux and la région Nouvelle-Aquitaine for
funding.

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