††thanks: These authors contributed equally††thanks: These authors contributed
equally

#  Gender disparities in the dissemination and acquisition of scientific
knowledge

Chiara Zappalà  Center for Collective Learning, Corvinus Institute for
Advanced Studies (CIAS), Corvinus University, 1093 Budapest, Hungary  Luca
Gallo  Department of Network and Data Science, Central European University,
1100 Vienna, Austria  ANETI Lab, Corvinus Institute for Advanced Studies
(CIAS), Corvinus University, 1093 Budapest, Hungary  Jan Bachmann  Complexity
Science Hub, A-1080 Vienna, Austria  Department of Network and Data Science,
Central European University, 1100 Vienna, Austria  Federico Battiston
Department of Network and Data Science, Central European University, 1100
Vienna, Austria  Fariba Karimi  Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Complexity Science Hub, A-1080 Vienna, Austria  Institute of
Interactive Systems and Data Science, Graz University of Technology, 8010
Graz, Austria

###### Abstract

Recent research has challenged the widespread belief that gender inequities in
academia would disappear simply by increasing the number of women. More
complex causes might be at play, embodied in the networked structure of
scientific collaborations. Here, we aim to understand the structural
inequality between male and female scholars in the dissemination of scientific
knowledge. We use a large-scale dataset of academic publications from the
American Physical Society (APS) to build a time-varying network of
collaborations from 1970 to 2020. We model knowledge dissemination as a
contagion process in which scientists become informed based on the propagation
of knowledge through their collaborators. We quantify the fairness of the
system in terms of how women acquire and diffuse knowledge compared to men.
Our results indicate that knowledge acquisition and diffusion are slower for
women than expected. We find that the main determinant of women’s disadvantage
is the gap in the cumulative number of collaborators, highlighting how time
creates structural disadvantages that contribute to marginalize women in
physics. Our work sheds light on how the dynamics of scientific collaborations
shape gender disparities in knowledge dissemination and calls for a deeper
understanding on how to intervene to improve fairness and diversity in the
scientific community.

## Introduction

Gender disparities in academia persist despite growing participation of women.
Research suggests that women are still disadvantaged on multiple dimensions,
such as productivity, salary, online representation, credit, and recognition
[1, 2, 3, 4, 5, 6]. Men and women also differ in how they build collaborations
[7, 8] and receive citations [9], which in turn affect their future prospects
(e.g., grants, faculty positions) in academia [10]. Most evidently, the gap in
the number of men and women in Science, Technology, Engineering, and
Mathematics (STEM) is indisputable. In physics, the percentage of women is as
low as 15%, and the increase in women’s participation in recent years is still
considerably slow [4, 1]. Combined with the historical barriers to entry into
science and the first-mover advantages for men, this gap will not close within
this century [1, 11]. However, past research has questioned the common belief
that increasing the number of female researchers is enough to achieve gender
equality in academia [4]. One reason is that temporal discrepancies in the
arrival and participation of women in STEM have far-reaching implications,
fueling the persistence of stereotypes in STEM (which prevents engagement of
future generations) [12], impacting hiring and promotion decisions [13, 14].

In this context, the question of how disparities between male and female
scholars affect the dissemination of scientific knowledge remains unanswered.
An increasing body of literature hints that knowledge is not “in the air”,
i.e., globally available [15]; rather, it is constrained by geographical and
social proximity [16, 17, 18, 19, 20, 21, 22]. The reason is that acquiring
complex scientific and technical knowledge is a trial-and-error process that
requires effort. Therefore, a close connection to the source of knowledge is
crucial for its deep understanding, transmission, and ultimately its use to
advance science [23]. In academia, the coauthorship network embodies
meaningful social connections that let scientific knowledge circulate. This
network is the expression of the fruitful collaborations that authors had
during their careers, sharing expertise, skills and experience [24, 23, 25,
26]. In this perspective, the structure and evolution of collaboration ties
between scientists [27, 28, 29, 30] become essential ingredients to understand
knowledge flow.

Our hypothesis is that the time-varying collaboration network functions as a
structural barrier by limiting the opportunity of women to acquire and diffuse
knowledge. To this end, we use a large-scale dataset of academic publications
of the American Physical Society (APS) and build a temporal network [31, 32]
of scientific collaborations from 1970 to 2020. Specifically, the network
consists of yearly snapshots where nodes are authors and links connect
researchers who coauthored a paper during that year. We model knowledge flow
as a contagion process on this collaboration network [33, 34]. We quantify the
fairness of the system in terms of how women acquire and diffuse knowledge
compared to men. For the acquisition of knowledge, we check whether the gender
distribution of the authors reached by the spreading matches that of the whole
collaboration network. For the diffusion of knowledge, we estimate whether
women need more time than men to inform the same fraction of individuals in
the network (Fig. 1).

Our analysis of the collaboration network indicates that women in physics are
disadvantaged in acquiring and diffusing knowledge. Exploring the main
determinants of this unfairness, we find that the gap in the number of women
does not have a substantial impact on knowledge acquisition and diffusion;
instead, the network structure intertwined with the dynamics of scientific
collaborations contributes to relegating women to the margins.

## Results

### Knowledge flow in time-varying collaboration networks

Figure 1: Acquisition and diffusion fairness in the circulation of scientific
knowledge. a Schematic of knowledge flow through the collaboration network. An
author (node) starts sharing knowledge at a given year $t$, and the spreading
continues following the author’s connections (links). Once reached by the
knowledge flow, informed nodes (circled in red) start diffusing to their
uninformed neighbors. Authors are split into two groups according to their
gender: Women (yellow) and men (blue). The network varies in time: At each
year, new nodes enter the network and connections may change. b Number of
authors over time. Both male and female scholars have a growing trend. Women
are the minority in the network, although their percentage has increased over
the years. c Acquisition and diffusion of knowledge. To quantify how women
acquire and diffuse knowledge, we evaluate (i) the fraction of female authors
reached when the spreading starts from a random (either male or female) author
in the network (Acquisition), and (ii) the fraction of authors reached when
knowledge spreads from a woman (Diffusion).

We analyze a large dataset of 678,916 scientific papers published in the
journals of the American Physical Society (APS) between 1893 and 2020. Each
article comes with metadata on the date of publication, the papers cited, and
authors’ information, particularly full names and affiliations. However, the
APS dataset contains neither a unique identifier of the authors nor their
gender. Hence, we use a name disambiguation method to extract the identity of
scientists and match them with their publications [35, 36], and we infer
gender based on author names [37] (see Methods for details). The final sample
consists of 34,596 female and 195,490 male authors, plus 200,669 scientists
for whom no reliable gender estimate was possible.

We build a coauthorship network, where nodes represent scientists and links
represent a collaboration among them, i.e., having coauthored a paper
together. We leverage the temporal information of our dataset to build a time-
varying collaboration network (Fig. 1a). Specifically, the network consists of
a series of snapshots, each representing a publication year, where two authors
are connected if they have coauthored a paper during that year. The
coauthorship network grows in time, as new nodes are added at every snapshot.
Specifically, when a scientist publishes for the first time in a given year, a
node is added to the corresponding snapshot. Once in the network, nodes are
not removed, hence isolated nodes in a certain snapshot represent authors that
do not collaborate with others in that year. For the following analyses, we
only consider nodes that are part of the largest connected component of the
network that aggregates all publication years from 1970 to 2020. The final
network consists of 367,836 nodes: 21,801 female and 135,552 male scholars,
the rest representing scientists for whom gender is unknown. We focus on this
time interval because the number of authors, and in particular the number of
women, before the 70s is too small for our analysis of knowledge diffusion.

At a given year $t$, the network will count $N_{f}(t)$ nodes representing
female authors, $N_{m}(t)$ nodes representing male authors, and $N_{u}(t)$
nodes representing authors without gender information. We will denote as
$N(t)=N_{f}(t)+N_{m}(t)$ the number of nodes at time $t$ for which gender is
assigned. Looking at the fraction of nodes with gender information, we observe
that women are a minority in the network (Fig. 1b), although their percentage
has increased over the years (from 2.4% in 1970 to 15.3% in 2020).

We use the coauthorship network as we assume that scientific collaborations
represent a form of social relationships through which (tacit) knowledge can
flow [24, 23, 26]. Furthermore, considering the temporal nature of the
collaboration network, with connections that can change from one year to
another, provides a more realistic model for studying the flow of knowledge,
as ties formed at one time and never repeated are reasonably less effective
than persistent ones in conveying information [19].

We model knowledge sharing as a contagion process on the time-varying network.
We assume that each node can be susceptible (S) or informed (I). Susceptible
nodes represent individuals that have not acquired the information flowing in
the network, while informed nodes represent those who have. At each time,
informed nodes diffuse knowledge, making all of their susceptible neighbors
informed. Informed nodes do not become susceptible again, which means that
they do not stop spreading the knowledge they acquired (Fig. 1a). Although
this model cannot capture the complexity of how knowledge is transmitted from
one individual to another, its simplicity allows us to focus on how the
temporal evolution of the coauthorship network may affect the flow of
knowledge [38, 39].

To study gender disparities in knowledge dissemination, we select a
publication year, $t_{0}$, choose one node in the corresponding network
snapshot and make it informed, while leaving all other nodes susceptible. All
nodes that are added to the network in a given year $t>t_{0}$ will also be
susceptible. We run the contagion process following the temporal sequence of
collaboration ties, and monitor at each year $t$ the number of informed women,
$I_{f}(t)$, and informed men, $I_{m}(t)$, whose sum we denote as
$I(t)=I_{f}(t)+I_{m}(t)$.

In the case of knowledge flows, the system can be unfair towards female
scholars in two directions (Fig. 1c). On the one hand, given their number and
their position in the network, women could be unable to acquire information as
fast as men (Acquisition). On the other hand, a contagion process starting
from a woman could take more time to reach the same number of individuals than
a process originating from a man (Diffusion).

Figure 2: Measuring acquisition and diffusion fairness. Knowledge can spread
either from a female (yellow) or a male (blue) author. After a time $\Delta
t$, among the $N$ authors of the collaboration network, $I$ of them will be
informed (red circled). For the acquisition, we compare the fraction of
informed individuals that are women to the fraction of female authors. For the
diffusion, we compare the amount of time $\Delta t$ needed to reach a given
fraction of the population when the spreading starts from a woman or a man.

To evaluate fairness in knowledge acquisition, we let the spreading start from
a randomly selected node (male or female, but not unknown) and let the process
run until a given fraction of individuals $i_{\mathrm{end}}$ is reached in
year $t_{\mathrm{end}}$. We then count the fraction of informed nodes
representing female authors,
$i_{f}=I_{f}(t_{\mathrm{end}})/I(t_{\mathrm{end}})$, and the fraction of women
in the network, $n_{f}=N_{f}(t_{\mathrm{end}})/N(t_{\mathrm{end}})$ (Fig. 2).
Following the principle of statistical parity [40], we define acquisition
fairness as

$AF=\left\langle\frac{i_{f}}{n_{f}}\right\rangle,$ (1)

where $\langle\cdot\rangle$ indicates an average over multiple repetitions of
the spreading process, i.e., multiple selections of the first spreading node.
When $AF<1$ the fraction of informed individuals who are female is smaller
than the fraction of women, meaning that there are less informed women than
one would expect, i.e., the system is unfair towards the minority. Conversely,
$AF>1$ means that less men are exposed to knowledge flow compared to what we
would expect, while with $AF\approx 1$ the system is fair with regards to
knowledge acquisition. Note that, for a given starting year $t_{0}$, the
acquisition fairness varies as a function of the target fraction of informed
individuals $i_{\mathrm{end}}$.

To investigate fairness in knowledge diffusion, we compare the spreading
processes starting from women with those generated by men. Again, we let both
types of processes propagate until a certain fraction $i_{\mathrm{end}}$ is
reached in year $t_{\mathrm{end}}$ and compute the length of the length of
each process, namely $\Delta t=t_{\mathrm{end}}-t_{0}$. We define diffusion
fairness as

$DF=\frac{\langle\Delta t_{m}\rangle}{\langle\Delta t_{f}\rangle},$ (2)

where the subscripts indicate the gender of the initial spreader. $DF<1$
indicates that knowledge flows initiated by women take on average more time to
reach the same number of individuals than processes started by men, so the
system is unfair to the minority group. $DF>1$ denotes, instead, an unfairness
towards the majority, while a value $DF\approx 1$ hints at fairness with
regard to knowledge diffusion.

### Women are disadvantaged in acquiring and diffusing knowledge

Figure 3:  Fairness in the acquisition and diffusion of knowledge. a Example
of women’s acquisition of knowledge for a contagion process that starts from a
man. The fraction of female scholars in the collaboration network, $n_{f}$
(yellow) is compared to the fraction of informed scientists that are women,
$i_{f}$ (red). Women among the informed individuals are generally less than
expected. b Example of how knowledge diffuses from a female author, compared
to a process that starts from a male author. The process starting from a woman
takes longer than that starting from a man to reach the same fraction of
individuals. c Acquisition fairness as a function of the starting year of the
knowledge flow. The curves indicate different values of the fraction of
individuals reached. The dashed line represents the fair condition in which
the fraction of women among those who acquired knowledge is equal to the
fraction of women in the network. In general, the acquisition fairness is
below the fair condition, meaning that the women reached by the spreading are
less than expected. d Diffusion fairness as a function of the starting year
for different values of the fraction of individuals reached. The dashed line
represents the fair condition in which the average duration of the process is
the same when starting either from a woman or a man. In general, the diffusion
fairness is close but significantly below the fair condition, meaning that
women take slightly longer than men to diffuse their knowledge.

We first show an example of how the system can be unfair to women in terms of
knowledge acquisition and diffusion. Fig. 3a shows how the fraction of women
in the collaboration network, $n_{f}$ (yellow), and the fraction of informed
individuals that are women, $i_{f}$ (red), change in time for a process
starting in 1970 from a male individual. We observe that $i_{f}=0$ for the
first years of the spreading, meaning that all researchers reached by the
knowledge flow in that time range are men. After that initial phase,
information reaches women too, and $i_{f}$ starts to rise. After more than 15
years, $i_{f}$ reaches $n_{f}$, that is, the fraction of informed researchers
that are women is the same as the fraction of women in the collaboration
network, meaning that men do not have an advantage over women anymore. Fig. 3b
shows the evolution of the fraction of informed nodes (both male and female),
for two different processes starting in 1970 from a man (blue) and from a
woman (yellow). The process starting from the female node is slower than that
starting from the male node, with a gap ranging from one to four years for a
given fraction of informed individuals. While this difference widens in time,
we observe that it already emerges at the early stages of the spreading (inset
of the panel, focusing on the first 10 years of the process). This suggests
that women are disadvantaged in diffusing their knowledge, as they require
more time than men to reach the same number of researchers in the network.

We now perform a systematic analysis of the system unfairness using the
measures we have introduced, beginning with an evaluation of the acquisition
fairness. We consider a publication year $t_{0}$, and let the knowledge spread
until a target fraction of informed individuals $i_{\mathrm{end}}$ is reached.
We perform 1000 simulations, each time randomly selecting the node from which
the spreading starts. The acquisition fairness is then evaluated as the
average over those repetitions. We consider different starting years for the
spreading, from 1970 to 2010 at steps of five years, ignoring starting years
after 2010, so that the process can last for at least ten years. We estimate
the acquisition fairness for different values of $i_{\mathrm{end}}$, ranging
from $1\%$ to $50\%$ of the nodes for which we have gender information. In
this way, we assess the fairness of the system among those who acquire
knowledge at the early stages of the diffusion process and those reached later
on.

Results are shown in Fig. 3c. Each curve represents the value of the
acquisition fairness as a function of the starting year $t_{0}$, for different
values of the fraction of individuals to be reached $i_{\mathrm{end}}$. The
dashed line at $AF=1$ shows the fair condition, i.e., the final fraction of
women reached is equal to the fraction of women present in the network at that
time. In general, the acquisition fairness is significantly below the fair
condition, which means that the women reached by the knowledge flow are less
than expected. This is especially pronounced for those who acquire knowledge
at the early stages of the contagion: For instance, when a process starts in
1980, among the first 1% of informed individuals women are on average 25% less
than expected. Such result can have far-reaching implications, as acquiring
knowledge early in time can generate competitive advantages that eventually
exacerbate gender disparities. Besides early acquisition, the larger the
fraction of individuals reached, the closer we get to the fair condition
$AF=1$. In addition, we note that the acquisition fairness has a nontrivial
dependence on the starting year. While we observe a U-shaped trend between
1970 and 1990, we note a decreasing trend afterwards, which questions whether
the simple increase in the number of women is sufficient to reduce gender
differences in physics [4].

Focusing on the diffusion fairness of the system, we run two different sets of
simulations. In the first, we fix a starting year $t_{0}$, choose a random
female author, and let the process run until a given fraction of scientists
$i_{\mathrm{end}}$ is reached. We run 4000 simulations, each time picking a
different node to initiate the diffusion. In the second, we follow the same
procedure, selecting a starting node that instead represents a male author.
Again, we repeat this 4000 times, choosing different nodes at the beginning of
the contagion process. We then evaluate the diffusion fairness $DF$, comparing
the average duration of the spreading starting from a woman, $\Delta t_{f}$,
to that starting from a man $\Delta t_{m}$. As for the acquisition fairness,
we consider different starting years ranging from 1970 to 2010 in five-year
intervals, assessing $DF$ across various $i_{\mathrm{end}}$, from $1\%$ to
$50\%$.

In Fig. 3d we show the results. Again, each curve portrays the value of the
diffusion fairness as a function of the starting year $t_{0}$, for different
fractions of individuals to be reached $i_{\mathrm{end}}$. The dashed line at
$DF=1$ shows the fair condition, in which the average duration of the
diffusion process is the same when starting from a woman or a man. Though the
curves lie close to $DF=1$, they significantly differ from it for most
starting years, meaning that men can have an advantage in knowledge
dissemination compared to women. Moreover, we note that the curves tend to
overlap, suggesting that the diffusion fairness does not vary appreciably as a
function of $i_{\mathrm{end}}$. In other words, regardless of the number of
individuals that one aims at reaching, women are slower than men in
disseminating their knowledge through the collaboration network.

Fig. 3 gives us an overall picture of women’s disadvantages in acquiring and
diffusing knowledge. Specifically, among those who acquire knowledge early in
time, women are less represented than men. Moreover, women take slightly more
time than men to diffuse their knowledge. In the Supplementary Information we
complement this analysis by investigating the contagion process on a time-
aggregated version [31] of the collaboration network (Fig. S1).

### The determinants of unfairness

Figure 4: Determinants of the unfairness of the system. a Probability
distributions of the number of coauthors for men (blue) and women (yellow). On
average, female scholars have less collaborators than male scholars. b
Evolution of homophily in the APS collaboration network. The tendency of both
male (blue) and female (yellow) scholars to collaborate with scientists having
their same gender has increased over the decades. See Supplementary
Information for details on how to evaluate homophily. c-d The determinants of
acquisition and diffusion unfairness. Bars represent the value of acquisition
and diffusion unfairness for the null models (shades of blue) and the data
(red), for three starting years of the spreading process. The “Balance” model
shows no unfairness, i.e., $AF=1$ and $DF=1$ (black dashed lines). The “Late
Arrival” model does not show unfairness, suggesting the gap in the number of
women in the collaboration network does not impact the way women acquire and
diffuse knowledge. The “Coauthors” is in good agreement with the data, hinting
that the gap in the number of coauthors has a significant negative effect on
acquisition and diffusion fairness.

The unfairness in the acquisition and diffusion of knowledge could be
determined by different factors. First, there is an imbalance in the number of
men and women in the system, which is not constant over time; indeed, women
arrive later in the network, and this might affect their opportunities to
receive and send information. Additionally, women and men differ in their
collaboration patterns (Fig. 4a), as women have usually fewer coauthors over
their careers [41]. This gap could contribute to the acquisition and diffusion
unfairness. We investigate the influence of those factors by designing three
null models where we assign gender to the nodes so as to preserve (or remove)
certain characteristics of the system (see Methods for details). The first
null model (to which we refer as “Balance”) generates an “ideal” scenario
where women are 50% of the researchers at any point in time, and where women
and men have the same distributions of coauthors, i.e., the same degree
distributions in the time-aggregated network [31]. With the second null model
(“Late Arrival”) we preserve how the fraction of women in the network evolves
while removing gender differences in the number of collaborators. Finally, the
third model (“Coauthors”) preserves both features.

For each model, we consider three publication years, i.e., 1970, 1985, and
2000, and let the knowledge flow until a target fraction of informed
individuals $i_{\mathrm{end}}$ is reached. We consider $i_{\mathrm{end}}=0.05$
for acquisition fairness, while $i_{\mathrm{end}}=0.25$ for diffusion
fairness. The two values chosen remark the different potential advantages in
knowledge dissemination: Researchers should aim at acquiring knowledge earlier
in time and diffusing it to as many peers as possible. We evaluate the
acquisition fairness as the average over 1000 simulations (10 network
randomizations, 100 runs of the spreading per randomization), in which the
starting node is selected at random each time. For the diffusion fairness, we
compare the average spreading time of 5000 simulations (50 network
randomizations, 100 runs of the spreading per randomization) starting from a
random woman to the average spreading time of 5000 simulations (same 50
network randomizations, 100 runs of the spreading per randomization) starting
from a random man.

Fig. 4c-d show the results obtained for acquisition and diffusion,
respectively. In the “Balance” model there is neither acquisition nor
diffusion unfairness, i.e., $AF=1$ and $DF=1$, no matter the starting year. As
the model generates a scenario where there are no appreciable differences
between men and women, this result does not come as a surprise and it acts as
a validation of our measures of acquisition and diffusion fairness.

Remarkably, also the “Late Arrival” model does not show unfairness. This
suggests that neither the gap in the number of women nor its temporal
evolution has an impact on the way female researchers acquire and diffuse
knowledge. This outcome is not entirely unexpected. In the case of
acquisition, the flow of knowledge reaches more men than women ($i_{f}<i_{m})$
but almost in the same proportion as the number of men and women in the
network ($i_{f}/n_{f}\approx i_{m}/n_{m}$); therefore, the system is fair. For
diffusion, instead, the number of women does not play a significant role, as
we compare the average time that men and women need to inform a certain
fraction of individuals, no matter their gender.

The “Coauthors” model proves that the gap in the number of collaborators has a
significant negative effect on the fairness of the system. The difference in
coauthor distributions seems to explain most of the unfairness of the system,
as the model is in good agreement with empirical data. The analysis of this
model hallmarks how the structure of the collaboration network weakens the
ability of women to acquire and diffuse knowledge from and to their peers.

The mismatch we observe between this last model and the data suggests that
other network features might affect the system fairness. First, the
distribution of the number of collaborators for both genders is not constant
over time (see Fig. S2 in Supplementary Information). Given that the null
model only controls for the distributions in the network that aggregates all
publication years, it flattens out these temporal differences. Second, beyond
the number of coauthors, the exact position of the nodes in the network can
add to the system unfairness (see Fig. S3 in Supplementary Information).
Finally, the “Coauthors” model shuffles the gender of the nodes without
controlling for homophily. However, gender homophily in scientific
collaborations substantially increased [7] for both men and women over the
years (Fig. 4b; see Supplementary Information for details on how we measure
homophily). Therefore, it is likely that homophily generates a gap between the
null model and the data.

Together, the analysis of the null models sheds light on the factors
underlying gender differences in the acquisition and diffusion of knowledge
over time. Specifically, we find that unfairness is not driven by the gap in
the number of women, rather it arises from the structure of the collaboration
network.

## Discussion

In this work, we investigated gender inequalities in the dissemination of
scientific knowledge through the APS collaboration network. Although, in
principle, knowledge flow is not contingent on gender, we discovered a
systematic trend suggesting that women may face challenges in acquiring and
diffusing knowledge. These two dimensions of unfairness are not symmetric:
Whereas the difference in knowledge diffusion is less marked and seems to
improve over time, women’s disadvantage in knowledge acquisition is
considerable and persistent. This disparity is critical, as it might prevent
women from accessing novel ideas, new collaborations, or even finding job
opportunities. Exploring the potential drivers of unfairness, we found that
the gap in the number of female scholars does not explain women’s
disadvantage; instead, the difference in the cumulative number of
collaborators jeopardizes women’s opportunities to acquire and diffuse
knowledge.

These findings challenge the hypothesis that increasing the number of women in
research is enough to provide equal opportunities to acquire and diffuse
scientific knowledge. This lack of opportunities is not simply related to when
women enter the system but to how they take part in it, that is, the evolution
of their network of collaborators. Differences between women and men in the
size of their networks may stem from multiple factors. First, as a
historically disadvantaged minority in a male-dominated field and with more
parental responsibilities, women might be excluded from cutting-edge
collaboration opportunities [42, 43]. Second, women are more likely to drop
out of academia in all career ages compared to men [4], preventing them from
growing their network (see Fig. S4 in Supplementary Information for the
analysis of career lengths in our data). Moreover, women tend to have less
academic mobility than men [44]: Since mobility plays a crucial role in
expanding scholars’ network, this might contribute to widen the gap in the
number of collaborators. Remarkably, these differences are intertwined with
time, which results in slowing down the acquisition and diffusion of
knowledge, thus creating a temporal glass ceiling [45] for women in physics.

Our work sheds light on how the dynamics of scientific collaborations shape
gender disparities in knowledge dissemination and calls for a deeper
understanding on how to intervene to guarantee fairness and diversity in the
scientific community. Specifically, it highlights that policies intended to
increasing the number of female scholars in academia may not be sufficient to
reduce gender disparities in the acquisition and diffusion of scientific and
technical knowledge; instead, it points to interventions towards prevention of
women career dropout and promotion of academic mobility and diverse
collaborations.

## Methods

### Name disambiguation and gender inference in APS dataset

We construct the collaboration network from 678,916 publication records of the
American Physical Society, ranging from $1893$ to $2020$. This dataset
contains, for each publication, a list of coauthor names and their
affiliations, alongside other metadata. We apply a recently developed
implementation [36] of an existing name disambiguation solution [35] to link
publications to authors. For a pair of publications with matching author
names, this solution uses affiliations, citations, and coauthors to decide
whether the two names and the respective publications belong to the same
author. It applies a conservative strategy by merging only authors for which
there are no conflicting first names. This reduces the number of authors with
extreme publication counts. In the collaboration network, this limits the
number of nodes with very high degree, which could deeply affect the contagion
dynamics.

Similar to previous studies [11, 36, 4], we infer gender labels from author
names. Using a recently developed open-source solution [37], we estimate
34,596 scientists to be women, 195,490 to be men, while the gender of 200,669
disambiguated authors remains unknown. For authors with multiple names, we
remove names for which no gender could be inferred and assign gender based on
a majority vote, i.e., if the majority of names are labeled as male (female),
the scientist is considered male (female). Authors with a draw in the number
of female and male labeled names or those for whom only “unknown” labels could
be inferred, most often due to providing only initials or unisex names, are
labeled as “unknown”. Note that these authors remain present in the network,
namely they participate in the knowledge spreading process. However, they are
ignored when 1) seeding the contagion process and 2) evaluating the fairness
of the system.

### Null models of time-varying collaboration networks

We explore the factors underlying the emergence of acquisition and diffusion
unfairness in the flow of knowledge by designing three null models of the APS
collaboration network. For each model, the structure of the temporal network,
namely the nodes and the connections between them in every snapshot, is
unchanged while the node attributes, meaning the authors’ gender, are
reshuffled so as to preserve certain characteristics of the system.

  * •

Balance. This model generates an ideal scenario where 1) 50% of the authors
are women at any time; 2) men and women do not differ in any network feature.
Starting from the first snapshot of the temporal network, we randomly assign
the gender to the nodes with a 50/50 ratio. In the following snapshots, old
nodes keep their gender, while the gender of new nodes is again attributed
randomly with a 50/50 ratio. The main purpose of this model is to validate our
measures of unfairness, as the scenario it creates should be perfectly fair.

  * •

Late arrival. This model preserves the proportion of women in each year, so as
it remains the same as in the data, and removes gender differences in the
network features. We reshuffle the gender of the nodes in the first snapshot
of the network. In the following snapshots, we keep the gender of old nodes
and reshuffle that of new ones. The model allows us to investigate how the
late arrival of women in the collaboration network affects their opportunity
to acquire and diffuse scientific knowledge, while controlling for the
structure of the network.

  * •

Coauthors. With this model, we preserve the proportion of women in each year
and the difference between genders in the distribution of the number of
coauthors. First, we order nodes by their total number of connections and
divide them into groups, each representing 2% of the nodes. As for the Late
arrival model, we start from the first snapshot and reshuffle gender of nodes
within the same group. In this way, we explore how gender difference in the
distribution of coauthors impacts the fairness of the system on top of the
late arrival of women in the network.

## Data Availability

The APS datasets are available upon request to the American Physical Society
(https://journals.aps.org/datasets).

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## Acknowledgments

C.Z. acknowledges the support of 101086712-LearnData-HORIZON-
WIDERA-2022-TALENTS-01 financed by EUROPEAN RESEARCH EXECUTIVE AGENCY (REA)
(https://cordis.europa.eu/project/id/101086712). C.Z. and L.G. thank César
Hidalgo and Balázs Lengyel for the insightful comments and discussion. L.G.
and F.B. acknowledge support of the Air Force Office of Scientific Research
under award number FA8655-22-1-7025. J.B. was supported by the Austrian
Science Promotion Agency FFG under project No. 873927 ESSENCSE. and is a
recipient of a DOC Fellowship of the Austrian Academy of Sciences at the
Complexity Science Hub.


# Tuning the receding contact angle on hydrogels by addition of particles

François Boulogne  Department of Mechanical and Aerospace Engineering,
Princeton University, Princeton, NJ 08544, USA  Laboratoire Matière et
Systèmes Complexes (MSC), UMR 7057 CNRS, Université Paris Diderot, Bâtiment
Condorcet, 10 rue Alice Domon et Léonie Duquet, Paris, France  François
Ingremeau  Department of Mechanical and Aerospace Engineering, Princeton
University, Princeton, NJ 08544, USA  LIPhy, CNRS, and Université Grenoble
Alpes, 140 Rue de la Physique, 38402 Saint-Martin-d’Hères, France  Laurent
Limat  Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS,
Université Paris Diderot, Bâtiment Condorcet, 10 rue Alice Domon et Léonie
Duquet, Paris, France  Howard A. Stone  Department of Mechanical and Aerospace
Engineering, Princeton University, Princeton, NJ 08544, USA

###### Abstract

Control of the swelling, chemical functionalization, and adhesivity of
hydrogels are finding new applications in a wide range of material systems. We
investigate experimentally the effect of adsorbed particles on hydrogels on
the depinning of contact lines. In our experiments, a water drop containing
polystyrene microspheres is deposited on a swelling hydrogel, which leads to
the drop absorption and particle deposition. Two regimes are observed: a
decreasing drop height with a pinned contact line followed by a receding
contact line. We show that increasing the particles concentration increases
the duration of the first regime and significantly decreases the total
absorption time. The adsorbed particles increase the pinning force at the
contact line. Finally, we develop a method to measure the receding contact
angle with the consideration of the hydrogel swelling.

##  1 Introduction

Hydrogels are promising materials for numerous applications such as
superabsorbant materials, soft contact lenses [1, 2], drug delivery systems
[3], human implants or tissue engineering [4]. The control of interfacial
properties represents a major breakthrough in technological innovations with a
particular focus on the control of roughness, adhesion [5], friction [6],
wetting [2] or optical properties. However, it is challenging to control these
interfacial characteristics due to the complexity of the physical and chemical
properties of hydrogels. Among the possibilities to control these properties,
some techniques consist in grafting polymers at the gel surface so as to
control adhesion properties [7, 8] or in patterning the surface to tune the
adhesion [9].

In the literature on colloidal deposition, it is well-known that particles can
pin a contact line [10, 11, 12, 13]. Thus, synthetic textured surfaces have
been designed to achieve control of the roughness and to study contact angle
hysteresis [14, 15, 16, 17]. The effect of defects on a receding contact line
has been studied theoretically [18, 19, 20, 21, 22, 23] and observed
experimentally [24, 25, 26, 27, 28, 29]; see [30, 31] for reviews.

Surprisingly, despite the fact that hydrogels contain a large quantity of
water, a water drop deposited on its surface has a non-zero contact angle.
This effect is due to the free polymer chains, which are present at the gel
interface and pin the contact line [32]. Therefore, to tune the wetting
properties of hydrogels, we propose to investigate the functionalization of
gel interfaces by the deposition of particles.

Previously, we have studied hydrogels, which swell following the absorption of
a water drop containing a dilute suspension of micron-size particles [33].
Unlike the so-called coffee stain effect observed during the evaporation of
such drops on non-porous substrate, we have shown that the absorption leads to
a nearly uniform deposition of particles. In our earlier study, we highlighted
that the final deposition is the result of a combination of two successive
regimes: a pinned contact line followed by a receding contact line as the
absorption proceeds. We argued and observed that solvent absorption into the
hydrogel occurs with a nearly homogeneous flux that decays as $t^{-1/2}$ for
most of the absorbing surface. The larger flux observed near the contact line
brings more particles to the drop edge during the pinned contact line regime.
However, the second regime of the time dependent dynamics populates the center
of the wetting area more than the edge, which finally erases the larger
concentration at the drop edge.

In this paper, we show that a higher particle concentration leads to a
stronger pinning of the contact line, i.e. a smaller depinning contact angle.
As a result, it delays the transition between the two regimes and it affects
the final deposition morphology. Moreover, we quantify this effect, by
measuring the liquid-gel contact angle with a novel method based on the
detection of the hydrogel interface on which particles are adsorbed. We
rationalize our findings with the literature related to the pinning of a
contact line on surfaces containing defects.

##  2 Experimental details

Our experiments consist in a colloidal drop containing micron-size particles,
dispensed on a swelling hydrogel (figure 1). We made a fully closed glass cell
constituted of two concentric compartments. The gel is placed in the center
and is surrounded by a reservoir of water. Therefore, the environment is
maintained at a relative humidity of 100% to prevent evaporation of the drop
and the hydrogel.

Figure 1: Experimental setup. A drop is dispensed on an absorbing hydrogel in
an environment saturated in vapor to prevent evaporation. The light green area
shows the typical swelling front [33].

###  2.1 Hydrogels

Absorbing substrates are made of polydimethyl acrylamide (PDMA) hydrogel [34].
We denote $m_{mono}$ the mass of monomer (N,N-dimethylacrylamide) and $m_{w}$
the mass of water. In addition, $[\rm{MBA}]$ and $[\rm{mono}]$ are,
respectively, molar concentrations of the crosslinker (N,N’-methylene-bis-
acrylamide) and the monomer. The composition of the gel is set by
$m_{mono}/(m_{mono}+m_{w})=0.15$ and $[\rm{MBA}]/[\rm{mono}]=2$%. The quantity
of initiator (potassium persulfate and N,N,N’,N’-tetramethylethylenediamine)
is set to a molar ratio of 0.01 of the monomer quantity. All chemicals are
purchased from Sigma-Aldrich, USA. The solution is poured in a mold made of
two glass plates ($75\times 50$ cm2, Dow Corning) separated with a rubber
spacer of $3$ mm thickness (McMaster-Carr). Hydrogels are stored in a vapor
saturated environment and they are used three days after preparation.

###  2.2 Colloidal drop

Colloidal suspensions consist of hydrophilic fluorescent particles of $2a=1$
$\mu$m diameter (Lifetechnologies, Orange with absorption/emission wavelengths
at 540/560 nm) purchased as an aqueous suspension containing 0.02 %
thimerosal. The suspension is diluted with deionized water at a concentrations
$C_{p}$ from $2\times 10^{6}$ to $1\times 10^{9}$ particles/ml, which
corresponds to a volume fraction range of $[9\times 10^{-6},4\times 10^{-3}]$.
In all experiments reported here, the drop volume is $0.8$ $\mu$l. For
complementary investigations (see the appendix), we used hydrophilic
fluorescent particles of $2a=0.2$ $\mu$m diameter (Lifetechnologies, Orange
(540/560 nm), 2 mM azide). Blended suspensions of non-fluorescent and
fluorescent particles were prepared by mixing polystyrene-carboxylated
particles of $2a=1$ $\mu$m diameter (Polysciences, Polybead carboxylate) with
fluorescent particles of $2a=1$ $\mu$m diameter. The concentration of
fluorescent particles is set to $5\times 10^{6}$ particles/ml and the non-
fluorescent particle concentrations is adjusted to obtain a given final
concentration.

###  2.3 Optical microscopy

Bright field and fluorescent images are captured with a Hamamatsu camera
(Digital camera ORCA-Flash4.0 C11440, $2048\times 2048$ pixels) mounted on an
inverted microscope (Leica DMI4000 B) with a $5\times$ objective. The
acquisition is automated with the software Micromanager [35].

To measure the gel interface position, we use a confocal microscope Leica TCS
SP5 with a dry objective PL Fluotar, 10$\times$. The microscope is equipped
with a translation stage and a $z$-piezo stage.

##  3 Observations

###  3.1 Absorption regimes

For a drop deposited on an absorbing hydrogel, two regimes can be
distinguished, independently of the presence or absence of particles [36].
During regime I, the contact line is pinned, the drop has a constant radius
$R$ and the drop height decreases in time. At the end of this regime, at
$t=t_{I}$, the contact line recedes (regime II) toward the center of the
wetting area until the drop vanishes at time $t=t_{f}$.

Figure 2: (a) Schematics illustrating the two regimes for low and high
particle concentrations. (b) Durations of regime I, $t_{I}$, and the total
absorption time, $t_{f}$, as a function of the particle concentration $C_{p}$.
The error bars represent the standard deviation.

The schematic illustrations in figure 2(a) depict the qualitative effect of
the particle concentration on the two regimes of drop dynamics. As the
particle concentration increases, particles reinforce the pinning of the
contact line already observed on bare hydrogels. Thus, the duration $t_{I}$ of
regime I increases, which allows more efficient absorption of the drop solvent
over a large surface area (figure 2(b)). When the contact line recedes, the
area for liquid absorption decreases. However, for larger particle
concentrations, the remaining volume of water in the drop is smaller at
$t=t_{I}$ and a shorter absorption time $t_{f}$ is observed, as reported also
in figure 2(b).

For different particle concentrations $C_{p}$, at low particle concentrations,
typically $C_{p}<1\times 10^{7}$ particles/ml, $t_{I}$ and $t_{f}$ are
comparable to the values obtained for a pure water drop of the same volume. At
such low concentrations, the particles weakly affect the transition of regimes
I and II. For a larger concentration, the duration of regime I increases while
the total absorption time decreases. For $C_{p}>2\times 10^{8}$ particles/ml,
the drop height at $t=t_{I}$ is comparable to the particle size (figure 2(a)
and SI Movie). Therefore, we observe $t_{I}\approx t_{f}$.

###  3.2 Particle deposition

As the gel absorbs water, particles are carried toward the gel interface. In
our experiments, we observed that the particles are irreversibly absorbed on
the gel, i.e. particles are never resuspended once they are adsorbed even when
the contact line recedes. Thus, the final deposition pattern depends on the
absorption flux $J(r,t)$ and on the transition between regimes I and II, which
drives the wetting area where particles can be deposited [33].

The final deposition patterns for four different concentrations ranging from
$C_{p}=2\times 10^{6}$ to $10^{9}$ particles/ml are shown in figure 3(a). To
quantify the effect of the particle concentration on the particle deposition
profile, we used mixtures of non-fluorescent and fluorescent particles. In the
fluorescent mode of the microscope, we detect the individual positions of
fluorescent particles. The blended suspension allows us to apply the same
algorithm regarding the particle concentration. In figure 3(b), we show the
radial concentration profile obtained from binning the deposit area in
concentric rings of the same areas. For a particle concentration of
$C_{p}=10^{7}$ particles/ml, we recover the profile obtained previously [33],
which shows a small decay along the radius. We explained this profile from the
combination of the particle deposition that occurs during regimes I and II. As
the particle concentration increases, the trend of absorbed particle number
density versus radii is inverted. A nearly flat profile is observed for
$C_{p}=10^{8}$ particles/ml and a larger deposition is obtained near the
contact line for $C_{p}=10^{9}$ particles/ml. Indeed, the radial profile is
similar to the observation made at the end of the regime I for a low particle
concentration [33]. Thus, for $t_{I}\approx t_{f}$, a peak at the drop edge is
observed because of a larger absorption flux near the contact line. However,
for lower concentrations, additional particles are deposited when the contact
line recedes for $t>t_{I}$, which allows more particles to deposit in the
center.

Figure 3: (a) Final deposit for four initial concentrations (from the top
right, in the clockwise order): $C_{p}=2\times 10^{6}$, $10^{7}$, $10^{8}$ and
$10^{9}$ particles/ml. Images are obtained by fluorescent microscopy. (b)
Number of particles $N$ non-dimensionalized by the total number $N_{tot}$ as a
function of the radial position for three different concentrations.

In the next sections, we rationalize the effect of the particle concentration
on the duration of the regime I, namely $t_{I}$. In particular, we investigate
how the particle concentration affects the contact angle of the liquid on the
hydrogel when the contact line depins. It is particularly challenging to
measure the liquid contact angle $\theta_{liq}(t)$ on the hydrogel because the
substrate continuously deforms as the absorption proceeds. Also, direct
optical methods are difficult to employ here for two reasons. First, the
contrast of refractive index between the hydrogel and the drop is low, such
that the position of the gel-liquid interface cannot be measured by methods
based on refraction. Second, as we study the effect of particles, these
particles may interfere with direct visualization. Thus, we developed another
method relying on the presence of particles to measure the gel-liquid
interface position and, with further modeling, we obtain the liquid contact
angle $\theta_{liq}(t)$ for any particle concentration.

##  4 Results and interpretation

###  4.1 Model for the contact angle dynamics

Figure 4: Notations for the calculations including the effect of the swelling
of the gel on the liquid contact angle. The volume $V_{gel}$ represents the
volume of the swollen gel in the spherical cap drawn in black.

We now consider the change in the contact angles with the notations presented
in figure 4. We denote $\theta_{cap}$ the apparent contact angle of the liquid
drop, which is defined as

$\theta_{cap}(t)=\theta_{liq}(t)+\theta_{gel}(t),$ (1)

where $\theta_{liq}$ is the angle between the gel and the liquid-vapor
interface and $\theta_{gel}$ is the angle of the gel at the contact line.

We now consider the time evolution of a spherical cap volume defined as
$V_{cap}=V_{liq}+V_{gel}$ where $V_{liq}$ is the liquid volume during the
regime I and $V_{gel}$ is the swollen volume of gel beneath the drop (figure
4). The drop has a radius $R$ smaller than the capillary length
$\ell_{c}=\sqrt{\gamma/(\rho_{\ell}g)}\approx 2.7$ mm, where $\gamma$ is the
liquid-vapor surface tension, $\rho_{\ell}$ is the liquid density and $g$ is
the gravity. Thus, the drop shape is approximated as a spherical cap, and for
small angles, we have

$V_{cap}=\frac{\pi R^{3}\theta_{cap}}{4}=V_{liq}+V_{gel}.$ (2)

During regime I, we have shown previously that the flux $J(r,t)$ of water from
the drop to the gel is spatially homogeneous in the center of the drop [33].
The flux is larger near the contact line but this deviation is localized to
$1/10$ of the drop radius [33]. Therefore, we neglect this deviation and we
consider the time evolution of the flux as

$J(t)\approx\frac{\sqrt{{\cal D}}\delta}{\sqrt{t}},$ (3)

where ${\cal D}=250$ $\mu$m2/s is an effective diffusion coefficient and
$\delta\approx 0.18$ is the swelling ratio. The swelling ratio is defined as
$\delta=\frac{V_{sw}^{w}}{V_{sw}^{tot}}-\frac{V_{i}^{w}}{V_{i}^{tot}}$, where
$V_{i}^{w}$ and $V_{sw}^{w}$ are, respectively, the volumes of water in the
initial and swollen states, and $V_{i}^{tot}$ and $V_{sw}^{tot}$ are,
respectively, the volumes of gel at the initial and swollen states. Thus, the
time evolution of the liquid volume $V_{liq}$ during this regime is

$V_{liq}(t)\approx V_{0}-2\delta S\sqrt{{\cal D}t},$ (4)

where $V_{0}$ is the initial volume and $S=\pi R^{2}$ is the wetting area. By
combining equations (2) and (4) into equation (1), we obtain

$\theta_{liq}(t)\approx\frac{4}{\pi R^{3}}\left(V_{0}-2\pi
R^{2}\delta\sqrt{{\cal D}t}+V_{gel}(t)\right)-\theta_{gel}(t).$ (5)

Also, we remark that neglecting the swelling of the substrate is equivalent to
set $V_{gel}(t)=0$ and $\theta_{gel}(t)=0$ in equation (5), i.e.

$\theta_{liq}(t)\approx\frac{4}{\pi R^{3}}\left(V_{0}-2\pi
R^{2}\delta\sqrt{{\cal D}t}\right).$ (6)

By inspection of equations (5) and (6), we can see that the swelling
correction $V_{gel}(t)-\frac{\pi R^{3}}{4}\theta_{gel}(t)$ included in
equation (5) is the difference between the actual shape of the gel and a
spherical cap with the gels angle at the contact line. In the next section, we
present the measurements to obtain $\theta_{gel}(t)$ and $V_{gel}(t)$ to
compute $\theta_{liq}(t)$ from equation (5).

###  4.2 Measurement of $\theta_{liq}(t)$

To estimate the quantities $V_{gel}(t)$ and $\theta_{gel}(t)$, we first
measure the time evolution of the gel-liquid interface position
$h_{gel}(r,t)$; see figure 4. To describe the depinning transition, we assume
that the absorption dynamics is independent of the particle concentration.
This is a reasonable assumption since the highest concentration of particles
leads to a deposit that covers around $\approx 20$% of the surface. This
deposit has a permeability very high compare to the one of the hydrogel.
Therefore, the particles cannot apply a significant resistance to the flow and
can be neglected when considering the absorption dynamic. Thus, the
measurement of $\theta_{liq}(t)$ is also independent of $C_{p}$. Since $t_{I}$
increases with $C_{p}$, the measurement of $\theta_{liq}(t)$ for the largest
concentration will provide $\theta_{liq}(t_{I})$.

To measure $h_{gel}(r,t)$ (see figure 4), we use a suspension at a
concentration $C_{p}=1\times 10^{9}$ particles/ml. As some particles are
rapidly adsorbed at the gel interface, we use their fluorescence to detect the
position of the interface.

With a confocal microscope, we take images in a vertical plane perpendicular
to the gel surface by scanning along the $z$ direction with steps of $2.7$
$\mu$m. Images are taken along the drop radius with a resolution of 512 pixels
and with a width of 50 pixels. At this magnification, 512 pixels represent 1.5
mm (i.e. a resolution of 3$\mu$m/px), which is larger than the drop radius
$R\approx 1.1$ mm. The $z$-scan is repeated every 30 s.

Figure 5: Time evolution of the gel interface position $h_{gel}(r,t)$. The
reference is the vertical position of the contact line $h_{cl}(R,t)$, which is
set to zero. Times range from $t=2$ min to $t=20$ min, with a time increment
of 3 min. The inset shows the time evolution of the contact line vertical
position $h_{cl}(t)$. The vertical dashed line represents $t=t_{I}\approx
t_{f}\approx 20$ min.

For each picture, the sum of pixel intensities is calculated in a sliding
windows of ($30\times 50$) pixels moving along the radius with a step of $10$
pixels ($\approx 30$ $\mu$m). Thus, we obtain an intensity $i(r,z,t)$ for
which the radial position corresponds to the center of the sliding windows.
For a given radial position $r_{i}$ and time $t_{i}$, the intensity
$i(r_{i},z,t_{i})$ presents a maximum located at the gel-liquid interface due
to fluorescent particle adsorption. The position of the maximum is determined
from a fit with a Lorentzian curve to achieve a subpixel precision. As a
result, we obtain the interface position of the gel $h_{gel}(r,t)$ reported in
figure 5.

Figure 6: Time evolution of the liquid contact angle $\theta_{liq}$ on the gel
for data shown in figure 5. The solid line represents equation (6) where the
swelling of the gel is neglected and the green points are given by equation
(5) accounting for the gel swelling. Blue points represent the time evolution
of the gel angle $\theta_{gel}$ at the contact line and the orange points, the
sum $\theta_{gel}+\theta_{liq}$.

From figure 5, we use a trapezoidal method to calculate the volume
$V_{gel}(t)=2\,\pi\int_{0}^{R}h_{gel}(r,t)\,r\,{\rm d}r$ and we also calculate
the gel angle $\theta_{gel}(t)$ from the tangent of $h_{gel}(r,t)$ at the
contact line. These values are reported in figure 6 and shows only a small
variation during our observations. The first minute of swelling is not
available due to the time resolution of our experiment.

Then, with equation (5), we estimate $\theta_{liq}(t)$, which is given in
figure 6. We also plot in figure 6, the estimate of $\theta_{liq}(t)$ from
equation (6), which does that not account for the swelling effect. For a given
volume of liquid, i.e. a given time (see equation (4)), the contact angle is
smaller due to the inclined position of the gel interface. Nevertheless, both
curves have a similar trend and the correction coming from the swelling of the
hydrogel remains small.

###  4.3 Effect of the particle concentration on the contact line depinning

In the literature on contact line hysteresis, it is common to define the
hysteresis $H$ as the difference between the advancing and the receding force
that has been applied on a contact line to trigger its motion. A clever
measurement of this hysteresis has been provided by di Meglio and Quéré for
contact line motion on fibers [24]. For a surface covered with dilute defects,
is has been shown theoretically [18, 19] and experimentally [37] that the
hysteresis is proportional to the density of defects. For larger defect
surface concentration $C_{s}$, numerical and experimental studies [25, 38]
indicate that the hysteresis grows as $C_{s}^{0.7}$.

Figure 7:  The points represent $\cos(\theta_{liq}(t_{I}))$ as a function of
the initial particle concentration $C_{p}$ for 1 $\mu$m diameter particles.
The error bars represent the uncertainty reported from data presented in
Figure 2(b). The solid line is a guide for the eye. The inset shows
$\cos(\theta_{liq}(t_{I}))$ as a function of the defect surface concentration
$C_{s}$ calculated from equation (7). The solid line represents the equation
$\cos(\theta_{liq}(t_{I}))\propto C_{s}^{0.7}$.

As we established the evolution of the liquid contact angle $\theta_{liq}(t)$
on the hydrogel (figure 6), we can interpret the durations of regime I that
depends on the particle concentration (figure 2(b)), as a critical receding
contact angle. Thus, using $t_{I}(C_{p})$ from figure 2(b) and
$\theta_{liq}(t_{I})$ from figure 6, we derive $\cos(\theta_{liq}(t_{I}))$ at
the receding time as a function of the particle concentration $C_{p}$ as
represented in figure 7.

For particle concentrations typically larger than $5\times 10^{7}$
particles/ml, the direct particle identification and counting is particularly
difficult. To estimate the particle surface concentration at time $t_{I}$ for
all experiments, we consider the absorbed volume of the drop $2\delta
S\sqrt{{\cal D}t_{I}}$ [33]. Thus, the particle surface concentration adsorbed
at the surface at $t=t_{I}$ is

$C_{s}=2\delta\sqrt{{\cal D}t_{I}}C_{p}.$ (7)

Our data shows that the hysteresis increases with the particle concentration
and then saturates for $C_{p}>2\times 10^{8}$ particles/ml or for an average
coverage of $C_{s}\approx 3\times 10^{10}$ particles/m2. At this
concentration, the particles apply a sufficient force to maintain pinned the
contact line during the entire absorption process. Furthermore, as suggested
by the experimental literature on solid surfaces, a power-law
$\cos(\theta_{liq}(t_{I}))\propto C_{s}^{0.7}$ also successfully describes our
results.

Our observations are in agreement with the results obtained by Kajiya et al.
[36] for pure liquid water drops. For very dilute suspensions, the critical
liquid contact angle below which the contact line recedes is small.
Nevertheless, this angle is not zero and for our gels is $\theta_{liq}\approx
14^{\circ}$. The results presented in figure 7 shows that the presence of
micron size particles at the contact line allows us to reduce continuously
this critical angle to zero.

##  5 Conclusion

In this paper, we studied the effect of the particle concentration on the
dynamics of a drop absorbed by a swelling hydrogel. The dynamics of a pure
water drop is composed of a first regime where the contact line is pinned on
the hydrogel and a second regime where the contact line recedes. As the
particle concentration increases, the duration of regime I increases. Since
the absorption area is larger during this first regime, the total absorption
time decreases with the particle concentration. For micron size particles, we
observed that above a surface concentration of $2\times 10^{10}$ particles/m2,
the contact line remains pinned throughout the entire absorption process. To
quantify the critical receding contact angle $\theta_{liq}(t_{I})$, we
developed a method based on the detection of the gel-water interface position.
We have shown that $\cos(\theta_{liq}(t_{I}))\propto C_{s}^{0.7}$, in
agreement with the literature related to the contact line hysteresis on solid
substrates containing defects [25, 38] .

The results reported here illustrate that the addition of particles on the
surface of a hydrogel modifies the receding contact angle. In the situation of
a colloidal drop absorbed by the hydrogel, we have shown that the particle
surface concentration profile is modified by changing the initial particle
concentration. We related this effect to the variation of the receding contact
angle due to the particles adsorbed during the swelling process. In
particular, we found for our system that a concentration of $C_{s}\approx
3\times 10^{10}$ particles/ml leads to an homogeneous coating, while a lower
concentration generates a larger deposition in the center of the deposit.

These results extend our previous findings on the homogeneous coatings of
particles on hydrogels [33], which can be useful for the realization of
functionalized interfaces of hydrogels. In terms of applications, these
questions are particularly crucial for biomedical applications such as soft
contact lenses [1, 2], functional biogels [40, 41] or tissue engineering [5].
More generally, we believe that adsorption of particles can inspire future
studies to tune the wetting properties of hydrogels. From the fundamental
point of view, the field of solid capillarity raises important questions on
the wetting of soft materials, the role of soft solid interfaces and the
importance of physical and chemical properties [39].

Future work will be devoted to the importance of chemical and physical
properties of the particles on the final wetting properties. Indeed, we expect
that the size of the defects as well as the hydrophilicity of the chemical
groups grafted on the particles are significant. Since the experiments
described in this paper allows measurement of the pinning force caused by the
defects, it is possible to compare this force for different particle sizes or
different particle surface chemistry. These measurements might be useful to
predict how a deposit of particles changes the spreading of water on a
hydrogel surface.

##  6 Appendix

In addition to the results presented in the main text, which are obtained for
$1$ $\mu$m diameter particles, we reproduced the experiments for $0.2$ $\mu$m
diameter particles. Figure 8 shows equivalent data to figures 2(b) and 7. Both
plots show trends similar to the observations made for $1$ $\mu$m diameter
particles. However, the particle concentration must be increased by a factor
$\approx 10$ to obtain $t_{I}\approx t_{f}$.

Figure 8:  Data for $0.2$ $\mu$m diameter particles. (a) Durations of regime
I, $t_{I}$, and of the total absorption time $t_{f}$ as a function of the
particle concentration $C_{p}$. (b) The points represents
$\cos(\theta_{liq}(t_{I}))$ as a function of the initial particle
concentration $C_{p}$. The solid line is a guide for the eye. The inset shows
$\cos(\theta_{liq}(t_{I}))$ as a function of the defect surface concentration
$C_{s}$ calculated from equation (7). The solid line represents the equation
$\cos(\theta_{liq}(t_{I}))\propto C_{s}^{0.7}$.

##  7 Acknowledgements

F.B. acknowledges that the research leading to these results received funding
from the People Programme (Marie Curie Actions) of the European Union’s
Seventh Framework Programme (FP7/2007-2013) under REA grant agreement 623541.

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