# Coalescence of Sessile Aqueous Droplets Laden with Surfactant

Soheil Arbabi<EMAIL_ADDRESS>Institute of Physics, Polish Academy of
Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland  Piotr Deuar  Institute of
Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
Rachid Bennacer  Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMPS, 4 Av.
des Sciences, 91190 Gif-sur-Yvette, France  Zhizhao Che  State Key Laboratory
of Engines, Tianjin University, 300350 Tianjin, China  Panagiotis E.
Theodorakis<EMAIL_ADDRESS>Institute of Physics, Polish Academy of
Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland

###### Abstract

With most of the focus to date having been on the coalescence of freely
suspended droplets, much less is known about the coalescence of sessile
droplets, especially in the case of droplets laden with surfactant. Here, we
employ large-scale molecular dynamics simulations to investigate this
phenomenon on substrates with different wettability. In particular, we unravel
the mass transport mechanism of surfactant during coalescence, thus explaining
the key mechanisms present in the process. Close similarities are found
between the coalescence of sessile droplets with equilibrium contact angles
above 90∘ and that of freely suspended droplets, being practically the same
when the contact angle of the sessile droplets is above 140∘. Here, the
initial contact point is an area that creates an initial contact film of
surfactant that proceeds to break into engulfed aggregates. A major change in
the physics appears below the 90∘ contact angle, when the initial contact
point becomes small and line-like, strongly affecting many aspects of the
process and allowing water to take part in the coalescence from the beginning.
We find growth exponents consistent with a 2/3 power law on strongly wettable
substrates but no evidence of linear growth. Overall bridge growth speed
increases with wettability for all surfactant concentrations, but the speeding
up effect becomes weaker as surfactant concentration grows, along with a
general slowdown of the coalescence compared to pure water. Concurrently the
duration of the initial thermally limited regime increases strongly by almost
an order of magnitude for strongly wettable substrates.

††preprint: AIP/123-QED

##  I Introduction

Many natural phenomena involve the coalescence of fluid droplets.[1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] For example, this
is a fundamental process that determines the distribution and coalescence rate
of raindrops in atmospheric aerosols.[21, 22, 23, 24] Apart from such natural
processes, droplet coalescence is relevant for many industrial applications as
well, such as inkjet printing,[25] microfluidics,[26, 27, 28, 29] and water
treatment during crude oil and natural gas separation.[30, 31] Further control
of the process may involve the use of various additives,[32, 33, 34, 35, 36,
37, 38, 39] such as surfactant,[40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 56, 57, 58] which can reduce surface tension at fluid
interfaces, crucial in multi-phase systems. For example, surfactant can
stabilize droplets’ surface or prevent their coalescence, thus improving the
bio-compatibility in certain systems[59] or affecting the fusion, mixing, and
manipulation of droplets in microfluidic devices.[60] While many efforts have
thus far been taken to understand coalescence phenomena at a fundamental
level,[1, 3, 61] there are still numerous aspects of this process that require
further investigations, such as the coalescence of sessile droplets with
surfactant.

In experiments,[62, 63, 64, 65, 66, 67, 68] droplet coalescence has been
studied for various conditions (e.g., in the presence of applied electric
fields) and geometries (e.g., micro-channels, fibers, etc.), mainly by means
of high-speed imaging and electrical sensing. Due to resolution limitations,
the focus of these investigations has mainly been placed on unveiling
macroscopic properties of the process. [50, 52, 49, 69, 70, 71] For example,
the effect of surfactant concentration on droplet coalescence has been
investigated by high-speed imaging.[50] When an asymmetry in surfactant
concentration of the coalescing droplets is present, Marangoni flow was
observed and the curvature on either side of the growing bridge was different.
Numerical continuum approaches have also been employed to complement
understanding of droplet coalescence. [42, 72, 73, 74, 75, 76, 5, 69, 7, 17,
77, 15, 78] However, such methods suffer from inadequate resolution in
capturing the mass transport mechanism of surfactant during coalescence or
resolving the initial contact of the droplets. The singularity at the contact
point still remains a challenge for continuum simulation.[7] On the other
hand, molecular dynamics (MD) simulation can naturally resolve the contact
region at the molecular level, to observe the start of the coalescence
process.

Figure 1:  Different stages of coalescence of surfactant-laden droplets on
non-wettable (a–d, $\theta_{s}\simeq{\color[rgb]{0,0,0}{142}}^{\circ}$) and
wettable (e–h, $\theta_{s}\simeq 49^{\circ}$) substrates, with surfactant
concentration above CAC (35.48 wt%). $b$ is the bridge height and $w$ the
bridge width. $X$ is distance between the centers of mass of the two droplets
in the $x$-direction, while $\theta_{s}$ is the equilibrium contact angle of
the droplet. $\theta_{b}$ is the angle formed at the bridge at the liquid–gas
surface. The stages of coalescence shown here are: (a, e) Initial approach
($t=t_{c}$); (b, f) Moment of pinching ($t=t_{c}$); (c) Developed bridge
($t=t_{c}+196.25~{}\tau$) and (d) final, equilibrium state of the system
($t=t_{c}+1483.75$) in the case of the non-wettable substrate. (g) Developed
bridge ($t=t_{c}+1028.75~{}\tau$) and (h) final, equilibrium state of the
system ($t=t_{c}+3161.25~{}\tau$) in the case of the wettable substrate. The
snapshots of the system were obtained using Ovito software.[79] Figure 2:
(Multimedia view) Coalescence of droplets with surfactant concentration (35.48
wt%) on substrates with different wettability (upper row, non-wettable; middle
row, intermediate; lower row, wettable).

In general, droplet coalescence is a non-equilibrium process that occurs in
three stages. The first stage involves the initial approach of droplets, when
they come close enough to interact (pinching/contact) and form the so-called
bridge (Figure 1a, e and pinching b, f). This is followed by a bridge-growth
stage that leads to the reshaping (third stage) of the droplets (Figure 1c, g)
and eventually the single spherical-cap shape for sessile droplets, which is
the final equilibrium and minimum energy state of the system (Figure 1d,h).
The time evolution of this process on substrates with different wettability is
illustrated in Figure 2 (Multimedia view). The coalescence process is governed
by the interplay of viscous, inertial and surface-tension forces, as the
system tries to decrease the surface tension.[80] In particular, from the
perspective of fluid dynamics, viscous forces are expected to play an
important role in the initial bridge growth, which is driven primarily by
molecular interactions. At a later stage of the bridge growth, inertial
effects are expected to take over. In the case of freely suspended droplets, a
linear scaling of the bridge radius, $b$, in time ($b\propto t$) and
logarithmic corrections $b\propto t\ln t$ have been suggested for the viscous
regime.[81, 6] For the inertial regime, a power law scaling ($b\propto
t^{1/2}$) has been proposed.[5] However, this classification is still under
debate in the literature, and an inertially limited viscous (ILV) regime has
also been reported.[82, 1] Moreover, an initial thermal regime, which is
inaccessible to experiments has been identified by means of all-atom[4] and
coarse-grained (CG)[83, 84] MD simulation. This regime arises due to
collective thermal motion of particles at the droplets’ surface. Here, all-
atom simulation has proposed a scaling law for a length scale characterizing
the extent of the thermal fluctuations, namely
$l_{T}\approx{\left(k_{B}T/\gamma\right)}^{1/4}R^{/1/2}$, where $k_{B}$ is
Boltzmann’s constant, $T$ the temperature, $\gamma$ the surface tension, and
$R$ the droplet radius.[4] When the bridge radius is smaller than this length
scale ($b<{l_{T}}$), the bridge grows mainly due to thermal motion of
particles, while later ($b>{l_{T}}$) hydrodynamic effects are expected to
dominate. Recent MD simulations of a CG model have indicated the presence of
the thermal regime and subsequent inertial scaling law ($b\propto t^{1/2}$)
for the bridge growth for both aqueous droplets with and without
surfactant.[83, 84]

In the case of the coalescence of sessile droplets,[85, 81, 86, 87, 88, 89,
45, 90, 91, 34] for water droplets on non-wettable substrates (contact angle
$\theta_{s}\geq 90^{\circ}$, Figure 1) it has been suggested that the bridge
grows with time as $b\propto t^{1/2}$, as has been observed in the case of
freely suspended droplets. In contrast, for wettable substrates, namely
$\theta_{s}<90^{\circ}$, the bridge is predicted to grow with a new power law
$b\propto t^{2/3}$.[86] Moreover, experiments of droplets with an equilibrium
contact angle in the range $10^{\circ}-56^{\circ}$ suggest that the bridge
height roughly grows in time with power-law exponents between 0.50 and 0.86.
Data was seen to follow the scaling law for the entire range of time in the
case of $10^{\circ}$ contact angle, while the rest of the cases studied
deviated from this law at longer times.[66] The bulk fluid properties also
affect the bridge growth. For example, in the case of polymer droplets[34, 92,
93], their viscosity can result in a lower rate of coalescence in comparison
with water droplets. One finds power-law exponents lower than 1/2 for
nonwettable substrates (equilibrium contact angle greater than $90^{\circ}$)
and power-law exponents lower than 2/3 for wettable substrates.[34] The bridge
width, $w$ (Figure 1), grows as $w\propto t^{0.5}$.[87, 66, 88] However, a
linear scaling $w\propto t$ has also been suggested for droplets of high
viscosity.[81] To our knowledge, the regimes of applicability of the above
findings have not been investigated in experiments nor in modeling that would
include the early-time molecular level physics.

In view of the many unknowns in the coalescence of sessile surfactant-laden
droplets, we embarked on investigating this phenomenon by using large-scale
molecular dynamics simulations of a coarse-grained force-field, considering a
comprehensive range of possible scenarios. Hence, our study includes a range
of surfactant concentrations below and above the critical aggregation
concentration (CAC) and substrates with different wettability, both wettable
($\theta_{s}<90^{\circ}$) non-wettable ($\theta_{s}>90^{\circ}$), and those
with equilibrium contact angle of about $90^{\circ}$. In all these cases, we
have explored the mass transport mechanism of surfactant, which provides
insights into the details of the coalescence process, analyzed the dynamics of
the bridge growth, which characterizes the rate of coalescence, as well as
studied the bridge angle and the velocity of approach.[94] It turns out that
while the coalescence of sessile surfactant-laden droplets on non-wettable
substrates shares similarities with the coalescence of freely suspended
droplets, significant differences in the mass transport mechanism and rate of
coalescence appear for wettable substrates. The following section presents our
MD simulation model and methods. Our results are discussed in Section III.
Finally, the conclusions and possible suggestions for follow-up research are
discussed in Section IV.

##  II Simulation model and methods

Our system consists initially of two droplets placed close to each other as
shown in Figure 1a,e, that is a distance below the cutoff range of the
interactions between beads in order to initiate the coalescence of the
droplets. We have considered concentrations above and below the CAC for C10E4
surfactants. The details of the model for the interactions between the
different components and the validation for various surfactants are taken the
same here as used in a number of previous related studies,[95, 96, 97, 98, 99,
100, 101] and were obtained through the Mie-$\gamma$ Statistical Associating
Fluid Theory (SAFT Mie-$\gamma$).[102, 98, 103, 104, 105] The MD simulations
were carried out in the canonical ensemble with the Nosé–Hoover thermostat as
implemented in LAMMPS software.[106, 107] After equilibration of a single
droplet on the specific substrate, it was cloned to produce the second droplet
and the surrounding vapor for the start of the coalescence in silico
experiment (Figures 1a,e).

The force-field has been validated for a range of water–surfactant systems
with a focus on reproducing key properties, such as surface tension and phase
behavior.[96, 95, 97, 108, 109, 99, 100, 101] In particular, interactions
between the various CG beads representing different chemical units of the
system are described via the Mie potential

${U(r_{\rm ij})}=C\epsilon_{\rm ij}\left[\left({\frac{\sigma_{\rm ij}}{r_{\rm
ij}}}\right)^{\lambda_{\rm ij}^{\rm r}}-\left({\frac{\sigma_{\rm ij}}{r_{\rm
ij}}}\right)^{\lambda_{\rm ij}^{\rm a}}\right],\;{\rm for}\;r_{\rm ij}\leq
r_{\rm c},\\\ $ (1)

where

$C=\left(\frac{\lambda_{\rm ij}^{\rm r}}{\lambda_{\rm ij}^{\rm r}-\lambda_{\rm
ij}^{\rm a}}\right){\left(\frac{\lambda_{\rm ij}^{\rm r}}{\lambda_{\rm
ij}^{\rm a}}\right)}^{\frac{\lambda_{\rm ij}^{\rm a}}{\lambda_{\rm ij}^{\rm
r}-\lambda_{\rm ij}^{\rm a}}},$

and i and j represent the bead types. Hence, $\sigma_{\rm ij}$ is an effective
size of these beads, while $\epsilon_{\rm ij}$ sets the interaction strength
between beads of type i and j. One takes $\lambda_{\rm ij}^{a}=6$, which is
connected to representing the dispersive interactions between the different
particles, while $\lambda_{\rm ij}^{r}$ serves as a fitting parameter in the
SAFT model and can vary. Finally, $r_{\rm ij}$ is the distance between any
pair of beads, which interact when their distance is below a cutoff value set
to $r_{c}=4.583$ $\sigma$, where $\sigma$ is the overall unit of length. The
rest of the units are $\epsilon$ for the energy, $m$ for the mass, while
$\tau$ is the natural MD time unit $\tau=\sigma{(m/{\epsilon})}^{0.5}$. In
real units, we consider the simulation to correspond to: $\sigma=0.43635$ nm,
$\epsilon/k_{B}=492$ K, $m=44.0521$ amu and $\tau=1.4062$ ps. The integration
of the equations of motion was carried out with an integration time-step
$\delta t=0.005$ $\tau$, while the temperature was set to
$k_{B}T/\epsilon=0.6057$, which would correspond to $T=25$ ∘C in real units.

Figure 3:  Description of model and notation. (a) Coarse-grained
representation of water and C10E4 surfactant. (b–d) External and cross-section
views are shown to highlight the bulk, interfaces, and contact line of the
droplet ($\theta_{s}=49^{\circ}$, with concentration above CAC). Surrounding
water vapor is omitted for the sake of clarity. (b) Front view; (c) Top view;
(d) Bottom view.

Here, we have chosen for our investigations the C10E4 surfactant, which has
also been previously studied in the context of the coalescence of freely
suspended droplets[83, 84] and superspreading.[95, 96, 97] The above studies
found other CmEn type surfactants to give similar behavior,[83, 84] so here we
only consider C10E4 as representative of the whole family. Then, the CG
representation of the C10E4 (Figure 3) uses hydrophobic alkane CG ‘C’ beads
with each one representing a $\rm-CH_{2}-CH_{2}-CH_{2}-$ group of atoms.
Hydrophilic CG ‘EO’ beads represent oxyethylene groups $\rm-CH_{2}-O-CH_{2}$.
Finally, water CG ‘W’ beads correspond to two water molecules. The values of
the nonbonded interaction parameters between CG beads are reported in Table S1
of the Supplementary Material (SM)  and their masses in Table S2 of the SM.

To link the alkane and oxyethylene beads together, each consecutive pair of
beads along the surfactant chain interacts via a harmonic potential of the
form

$V(r_{\rm ij})=0.5k(r_{\rm ij}-\sigma_{\rm ij})^{2}$ (2)

where $k=295.33$ $\epsilon/\sigma^{2}$. In addition, each consecutive triad of
EO beads along the chain experiences a harmonic angle potential, i.e.

$V_{\theta}(\theta_{\rm ijk})=0.5k_{\theta}(\theta_{\rm ijk}-\theta_{0})^{2}$
(3)

where $\theta_{\rm ijk}$ is the angle between consecutive beads i, j=i+1 and
k=j+1. Constants are $k_{\theta}=4.32$ $\epsilon/$rad2, and $\theta_{0}=2.75$
rad for the equilibrium angle.

The total number of beads per initial droplet in each simulation was $10^{5}$
and two different surfactant concentrations were considered, namely 6.25 wt%,
which is below CAC, and 35.48 wt%, which is well above, with CAC roughly being
7.5 wt%.[83, 84] The latter concentration is taken as an average, since the
number of water molecules in the liquid phase (droplet) fluctuates. The
wettability of the smooth, unstructured substrate was also tuned, as described
below, covering a range of wettable ($\theta_{s}<90^{\circ}$) and non-wettable
($\theta_{s}>90^{\circ}$) types of substrate (Figure 1). The exact values of
the equilibrium contact angle of single droplets are reported in Table S3 of
the SM along with the corresponding droplet dimensions (Table S4 of the SM).
Finally, to determine the beads belonging to the liquid phase (droplet), a
cluster analysis has been performed.[110, 111]

To vary the wettability of the substrate, we need to define the interactions
between the droplet beads and the substrate. This can be done by using the
combining rules defined in SAFT$\gamma$-Mie[103]:

$\sigma_{\rm ij}=\frac{\sigma_{\rm ii}+\sigma_{\rm jj}}{2},$ (4)

$\varepsilon_{\rm ij}=\frac{\sqrt{\sigma_{\rm ii}^{3}\sigma_{\rm
jj}^{3}}}{\sigma_{\rm{\color[rgb]{0,0,0}{ij}}}^{3}}\sqrt{\varepsilon_{\rm
ii}\varepsilon_{\rm jj}}.$ (5)

and start from the defined interactions for the liquid phases as given in
Table S1 of the SM. By tuning the cross interaction $\varepsilon_{ws}$ (“s”
indicates the substrate), the empirical relationship between
$\varepsilon_{ws}$ and the contact angle of pure water droplets (Figure S1 of
the SM) can be found, and is shown in Figure S2 of the SM. Then, by using the
above combining rules, the parameter $\varepsilon_{ss}$ corresponding to a
given $\varepsilon_{ws}$ can be obtained. We shall note here that the exact
value of the $\sigma_{ss}$ parameter is not important for our studies and is
set to unity for simplicity. Similarly, $\lambda_{sj}^{r}=9$ is set for all
interactions involving the substrate. Based on the knowledge of the latter
parameters and the interactions $\sigma_{ww}$ and $\varepsilon_{ww}$ (Table S1
of the SM), the interaction of the substrate with the surfactant beads can be
obtained as well, again using Equations 4-5. After obtaining all the
interaction parameters, equilibrium simulations are run and the surfactant-
laden droplet’s contact angle is determined by using the method of Ref. [96]
(see also text and Figure S1 of the SM) with data reported in Table S3 of the
SM. It might be argued however that estimating the angles can in general be
sensitive to the details of the definition of a sharp interface, as well as to
the fitting procedure.[91, 112] Also, models that could take into account the
disjoining pressure effects, very relevant in the context of droplets on
solids substrates, might perform better than fitting spherical caps to
nanodroplets.[91] Moreover, according to a previous study,[96] the size of the
droplet is large enough to guarantee that the equilibrium contact angle does
not depend on the size of the droplet, which makes the interaction values
valid for both larger and smaller droplets. Table S3 of the SM provides the
exact values for all of the cross interactions between the beads and the
substrate used in the simulations. Our choice here covers the relevant
parameter space for surfactant concentration (above and below CAC) and
substrate types (wettable, non-wettable, and about $90^{\circ}$).

The mass transport of surfactant molecules is investigated by tracking the
motion of each individual molecule between the various parts of the droplet,
which are illustrated in Figures 3c-d. These different parts are the
liquid–gas (LG) surface for the left and right droplets (the center of the
coordinate system is taken to be in the middle between the two droplets where
the bridge forms), the bulk of the left and right droplets, the solid–liquid
(SL) interface for the left and right droplets, the contact line for the left
and right droplets, and the LG and SL interfaces, bulk, and contact line of
the bridge as shown in Figure 3. Hence, this makes a total of twelve different
regions where each surfactant molecule can belong. Further discussion and
details on the calculation of the probabilities related to the motion of
surfactant between these regions describing the mass transport mechanism of
surfactants can be found in the SM. Finally, we have calculated the density
profiles of the water and surfactant during coalescence, and the approach
distance and velocity of the droplets[34], which will be further discussed in
the following section.

Figure 4:  The pinching moment ($t\approx t_{c}$) in the case of (a) a non-
wettable substrate ($\theta_{s}>90^{\circ}$), (b) an intermediate case
($\theta_{s}\approx 90^{\circ}$), and (c) a wettable substrate
($\theta_{s}<90^{\circ}$). Water participation in the coalescence process is
more pronounced in the case $\theta_{s}<90^{\circ}$, while the cases
$\theta_{s}\geq 90^{\circ}$ behave very similarly to freely suspended
droplets.[83, 84] Snapshots were obtained using Ovito software.[79]

##  III Results and discussion

###  III.1 Surfactant mass transport mechanism

Our previous studies[83, 84] of the coalescence of freely suspended droplets
have shown that surfactant plays an ever increasing role at the pinching point
as its concentration in the droplets increases, while water has a smaller
effect in initiating the coalescence process. Similar behavior is also
observed in the case of $\theta_{s}>90^{\circ}$ shown in Figure 4 for sessile
droplets, since the bridge formation starts far from the substrate (Figure
4a). For this reason, the bridge region initially is not affected by the
presence of the substrate and the bridge angle is very steep
($\theta_{b}\approx 80^{\circ}$, for example, see Figure S6 of the SM) when
pinching occurs. For the same reason, we find that the growth of the bridge
occurs symmetrically in both the $y$ and $z$ directions ($b$, $w$, Figure 1).
In contrast, in the case of wettable substrates, water molecules participate
in the pinching process from its onset (Figure 4c).

Figure 5:  Snapshots of coalescing droplets with C10E4 surfactant and
surfactant mass transport on substrates of different wettability. The
surfactant concentration in the liquid phase is above CAC, namely 35.48 wt%.
The size of the arrow heads reflects the probabilities associated with
surfactant transport to the different droplet areas (see Table S5 of the SM
for further details). (a–c) corresponds to the case
$\theta_{s}\simeq{\color[rgb]{0,0,0}{142}}^{\circ}$ with snapshots obtained at
times (a) $t_{\rm c}+17.5~{}\tau$, (b) $t_{\rm c}+185.0~{}\tau$, and (c)
$t_{\rm c}+1733.7~{}\tau$. (d–f) $\theta_{s}\simeq 94^{\circ}$, with snapshots
shown at times (d) $t_{\rm c}+28.75~{}\tau$, (e) $t_{\rm c}+315.0~{}\tau$, and
(f) $t_{\rm c}+1415.0~{}\tau$. (g–i) $\theta_{s}\simeq 49^{\circ}$ with
snapshots at times (g) $t_{\rm c}+200.0~{}\tau$, (h) $t_{\rm
c}+1358.7~{}\tau$, and (i) $t_{\rm c}+2200.0~{}\tau$. Snapshots in (a), (d),
and (g) are soon after the end of the thermal regime. Snapshots (b) and (e)
illustrate a clearly developed bridge with new aggregates formed in its bulk
or additional monomers remaining at the bridge region. (h) shows a clearly
developed bridge highlighting the absence of aggregates in the case
$\theta_{s}<90^{\circ}$. (c), (f), and (i) correspond to cases of a fully
developed bridge. Magnified views of the bridge region and its cross-section
(showing only surfactant hydrophobic beads in the bridge region, red) are
attached above and to the right of the snapshots, respectively. Snapshots were
obtained using Ovito software.[79] Figure 6:  (Multimedia view) Cross-section
of the bridge during coalescence on different substrates (first two columns,
$\theta_{s}>90^{\circ}$, middle two columns, $\theta_{s}\approx 90^{\circ}$,
and last two columns, $\theta_{s}<90^{\circ}$). Upper row shows the case of
droplets with concentration lower than CAC (6.25 wt%), while the lower row
with concentration above CAC (35.248 wt%). All beads (odd columns) or
surfactant beads only (even columns) are shown at each cross-section at the
bridge.

To quantify mass transport, we count the surfactant molecule movements between
the 12 regions identified in Figure 3. The raw numbers are reported in Table
S5 of the SM. These set the intensity of the arrows in Figure 5, indicating
the dominant direction of surfactant transport. Figures 5a–c present the
dominant surfactant transport processes for the case $\theta_{s}>90^{\circ}$.
As in the case of freely suspended droplets, an interface film of surfactant
initially forms consisting of surfactant from the LG surfaces of the droplets
that come into contact. Figure 6 (Multimedia view) illustrates this in the
case of non-wettable substrates, as well as the absence of this film when
coalescence takes place on wettable substrates. The perimeter of the bridge is
expected to linearly grow with $b$, while the area of the film increases as
$b^{2}$. Moreover, the data in Table S5 of the SM shows that the dominant
movement of surfactant is towards the LG surface of the droplets, as in the
case of freely suspended droplets. Since this movement towards the LG surface
occurs and the area of contact between the droplet grows, the surfactant
concentration in the film decreases, the film ruptures and some surfactant
remains in the form of aggregates. Due to the lack of space at the LG
interface of the merged droplets, surfactant from the newly formed aggregates
in the bridge bulk cannot be accommodated at the LG interface. Also, we
observe surfactant transport away from the bridge from the SL interface toward
the LG surface through the contact line. Other transport processes away from
the bridge are insignificant during coalescence in the case of non-wettable
droplets. When the contact angle $\theta_{s}\approx 90^{\circ}$, the pinching
is similar to the case of droplets with contact angle $\theta_{s}>90^{\circ}$
(Figure 4) as well as the case of freely suspended droplets.[83, 84] Finally,
$\theta_{b}$ has a value of around $90^{\circ}$ at the pinching moment (for
example, see Figure S6 of the SM).

The case $\theta_{s}<90^{\circ}$ shows a somewhat different behavior. The
transport toward the LG surface is higher than in the cases where
$\theta_{s}\geq 90^{\circ}$, and unlike the previous cases, we do not see the
formation of new aggregates as coalescence takes place [Figure 6 (Multimedia
view)]. This can be attributed to several causes. Firstly, the amount of
surfactant at the bridge is smaller in the case of $\theta_{s}<90^{\circ}$,
due to the higher curvature of the droplets. In this case, there is only a
small portion of the droplet surfaces that come into contact, at the contact
line of the droplets. In contrast, for non-wettable substrates, a large
portion of the surfaces of the two droplets come into contact forming a film.
This major contact area difference may also explain the higher degree of
participation of water molecules in the pinching process in the case of
wettable substrates. Secondly, there is much less space in the bridge to form
aggregates from any excess of surfactant that does not start at the bridge’s
LG surface. Finally, we note that surfactant transport from the contact line
towards the SL and LG interface, is overall more pronounced than in the case
of non-wettable substrates. This might be due to the immediate start of the
decrease of the contact-line length as the droplets merge, resulting in a
greater excess of surfactant in the contact-line region, and a greater
migration to the LG and SL interfaces.

Figure 7:  Bridge height, $b$, as a function of time, $t$, starting from the
pinching moment, $t_{c}$. Inset shows the bridge width, $w$. Results for
droplets with surfactant concentration 35.48 wt% on substrates with different
wettability, as indicated. The values of the power-law exponents for $b$ and
$w$ are $\beta_{b}$ and $\beta_{w}$, respectively, and are reported in the
plot. Additional data for droplets without surfactant and with surfactant
concentrations below CAC are reported in Figures S3 and S4 of the SM.

###  III.2 Bridge dynamics

In the case $\theta_{s}>90^{\circ}$, the pinching of the two droplets takes
place well above the substrate (Figure 1b), and only later does the bridge
region make contact with the substrate (Figure 1c). In the case
$\theta_{s}\leq 90^{\circ}$, the bridge starts to form on the substrate from
the onset of the coalescence process, which affects the bridge dynamics. The
rate of coalescence can generally be described by the pace of the bridge
growth in the direction normal as well as parallel to the substrate. The size
of the bridge in these directions is $b$ and $w$, respectively, both defined
as shown in Figure 1. Figure 7 plots $b$ and $w$ above the CAC as a function
of time (see Figures S3 and S4 in the SM for pure water and concentration
below CAC.) In the cases above CAC and for nonwettable substrates
($\theta_{s}>90^{\circ}$), exponents for $b$ are about 0.5, which are in line
with the case of freely suspended droplets.[83, 84] In contrast, in the case
of wettable substrates, the exponent is higher reaching values of about 0.72,
which suggests a much faster dynamics in comparison with the nonwettable
substrates. Similarly, the parameter $w$ shows a similar but weaker tendency
for growth with exponents of about 0.5 for both freely suspended droplets[83,
84] and droplets on nonwettable substrates, up to about 0.65 for the wettable
case. These higher values are agreeable with the 2/3 values seen for polymer
droplets on wettable substrates.[34] In the case of water droplets, as a
sanity check, exponents of $b$ as well as $w$ for both wettable and
nonwettable substrates are in the range 0.60–0.66, which is in line with
results reported in the literature,[86] where a power law of 2/3 has been
suggested (see Figure S3a of the SM).

When surfactant concentration is below the CAC (Figure S3b of the SM),
exponents for $b$ are in the range 0.57–0.60 for non-wettable substrates
(similar to pure water), but a significant increase of the exponent is noted
for substrates reaching about 0.87, when the equilibrium contact angle of the
droplet is about $52^{\circ}$. The exponents for $w$ are in the range
0.55–0.68, with the highest exponents observed in the case of equilibrium
contact angles close to $70^{\circ}$.

Finally, the initial thermal regime is well visible for sessile droplets as in
the case of freely suspended droplets[83, 84] or sessile polymer droplets.[34]
However, we notice both a decreasing extent of these fluctuations (smaller
$b$, $w$) for sessile droplets when $\theta_{s}<90^{\circ}$, due to the
additional attraction and the contact with the substrate which suppresses
these fluctuations, but also concurrently a strong lengthening of this regime
in time, by almost an order of magnitude.

Figure 8:  (a) Rate of change of angle $\theta_{b}$,
$m=d\theta_{b}/d(t-t_{c})$, as a function of the equilibrium contact angle,
$\theta_{s}$, that expresses the wettability of the substrates with
$\theta_{s}>90^{\circ}$ reflecting non-wettable cases. $m$ is calculated from
linear fits to the data of Figures S5 and S6 of the SM, where a linear change
of $\theta_{b}$ with time is observed to a good approximation. The
$\theta_{s}=180^{\circ}$ data is for freely suspended droplets (no substrate).
As an example here, panel b shows data for droplets with surfactant
concentration 6.26 wt%.

###  III.3 Angle formed at the bridge

We monitored the angle $\theta_{b}$ (Figure 1) as a function of time for all
cases by employing the method of Ref. [96] (see Figure S1 and text in the SM).
In particular, a layer parallel to the substrate at a distance $b$ was
considered, as indicated by the dashed line of Figure 1g. The dimensions of
the spherical-cap liquid-phase above this layer were recorded in the $x$ and
$y$ directions, as well as the distance of the apex from the layer, which are
used as input to calculate $\theta_{b}$.[96] The approach has been applied in
both the right and the left parts of the merged droplets and the average was
taken, resulting in the final values reported for $\theta_{b}$. Time traces of
$\theta_{b}$ are plotted in Figures S5 and S6 of the SM for all cases
considered here, while an example is shown in Figure 8. We can observe that
$\theta_{b}$ exhibits to a large degree a linear behavior with time, which
allows us to gather the slopes of the various curves and thus monitor the rate
of change (dynamics) of this angle, $m$, during coalescence.

These rates of change data are plotted in Figure 8. The rate slows down as the
substrate becomes more wettable, whether surfactant is present or not, but for
$\theta_{s}\gtrsim 140^{\circ}$, the freely suspended behavior is already
reached. This is also directly seen in Figure S5 of the SM. Greater surfactant
concentration also slows down the process, as could be suspected from the
earlier suspended droplet studies [83, 84], with pure water droplets being the
fastest. Hence, the use of surfactant facilitates the smoothing of the wedge
formation at the bridge apex.

Figure 9:  The distance between the centers of mass of the two droplets in the
$x$-direction, $X$, (see Figure 1)  as a function of time, $t$, starting from
the moment of first contact of the droplets, $t_{c}$, for surfactant-laden
droplets of concentrations above CAC (35.48 wt%). The inset shows the
instantaneous velocity, $u=\dot{X}$. Data for water droplets and surfactant-
laden droplets with concentration below CAC are shown in Figure S7 of the SM.

###  III.4 Velocity of approach

We calculated the coalescing system length, $X$, as defined in Figure 1, and
done in our previous work in the context of sessile polymer droplets.[34] Its
value is approximately one droplet diameter. Its time derivative, $u=\dot{X}$,
provides a measure of how fast the droplets approach each other. Figure 9
presents $X$ for droplets above the CAC on various substrates with different
wettability, while data for water droplets and droplets with surfactant with a
concentration below CAC are given in Figure S7 of the SM. Overall, the
velocity of approach is smaller in the case of wettable substrates,
irrespective of surfactant concentration, paralleling what we saw with bridge
sizes $b$ and $w$ and the angle formed at the bridge $\theta_{b}$. This is
also true for water droplets. The plots of Figure S8 of the SM show that an
increase of surfactant concentration also significantly slows down the
approach of the two droplets. This is mostly noticeable in the case of high
concentrations, while surfactant-laden droplets with concentration below CAC
show a similar behavior to pure water. Moreover, surfactant smooths the
approach of the two droplets, as can be seen by the change of shape of the
velocity of approach curves, particularly visible in Figure S8 of the SM.
Finally, there is a clear shift in the moment of maximum velocity to later
times for more wettable substrates at higher surfactant concentration, though
this shift is much smaller in the latter case.

Figure 10:  Surfactant density at the moment of pinching (upper panels) and at
a time when the bridge has been clearly developed (bottom panels) for above
droplets with surfactant concentration amounting to 35.48 wt%. Here, cases for
substrates with different wettability are shown, namely, (a)
$\theta_{s}\simeq{\color[rgb]{0,0,0}{142}}^{\circ}$, snapshots taken at
$t_{c}$ and $t_{c}+100~{}\tau$, (b) $\theta_{s}\simeq 90^{\circ}$, snapshots
taken at $t_{c}$ and $t_{c}+412.5~{}\tau$, and (c) $\theta_{s}\simeq
50^{\circ}$, snapshots taken at $t_{c}$ and $t_{c}+1352.5~{}\tau$.

###  III.5 Density Profiles

The density of surfactant molecules on a cross-section parallel to the $x-z$
plane and passing through the center of mass of the droplets is presented in
Figure 10. Also, Figure S9 of the SM shows the corresponding density
distribution of water molecules for various substrates. We observe the
formation of surfactant aggregates during the coalescence for substrates with
equilibrium contact angles larger than $90^{\circ}$, while it is absent for
wettable substrates (see for example Fig. 5a–b and d–e versus g–h). Large
concentrations of surfactant are distributed at the LG and SL interfaces.
Interior to this lies an inner layer devoid of aggregates, which is
particularly wide for the wettable droplets. Notably the wettable case has a
much smaller number of aggregates, despite the fact that all cases shown have
the same concentration. This is due to the larger area of the LG and SL
interfaces on wettable substrates, which might suggest that the CAC has a
higher value. The smaller thickness of the wettable droplets, also, hinders
the accommodation of a larger number of surfactant aggregates.

##  IV Conclusions

In this study, we have investigated the coalescence of sessile surfactant-
laden droplets on substrates with different wettability, including those with
equilibrium contact-angle higher as well as lower than $90^{\circ}$. We have
explored the influence of surfactant concentration both below and above the
CAC, and we juxtaposed our results with the case of pure water droplets. In
particular, we have elucidated the mass transport mechanism in all these cases
and explored the dynamics of the coalescence process by following the height
and width of the bridge, the rate of change of the bridge angle, as well as
the velocity of approach of the droplets.

Overall, sessile droplets with $\theta_{s}\geq 90^{\circ}$ share similarities
with freely suspended droplets, and for $\theta_{s}\simeq 140^{\circ}$ already
behave practically identically as if they were freely suspended. In this case,
the influence of the substrate on the coalescence process is rather small. For
example, the pinching region is mainly driven by the interaction of the
surfactant molecules at the droplets’ LG surface, as in the case of the freely
suspended droplets. In contrast, in the case of wettable substrates
($\theta_{s}<90^{\circ}$), we see that water molecules are part of the
pinching process, a significant departure from the physics of the freely
suspended case. The mass transport of surfactant molecules during coalescence
also shows some differences between wettable and nonwettable substrates, which
mostly relates to their intensity. A more notable difference is the absence of
newly formed aggregates as the bridge grows during coalescence in the case of
wettable substrates. This is due to the lower amount of surfactant at the
pinching region as the initial contact film has a far smaller area. In fact,
the nature of the surface of first contact changes dramatically from a
circular region to a thin contact line when the threshold of contact angle
$\theta_{s}\approx 90^{\circ}$ is crossed. The smaller amount of space
available in the bridge region between the LG and SL interfaces during later
evolution also contributes. The latter is related to a still open question of
identifying changes in CAC when the equilibrium contact angle of the droplets
changes.

We also found that an increased wettability of the substrate leads to higher
rate exponents for the growth of the bridge radius, $b$, and its width, $w$,
but generally exponents for $w$ are lower. We confirm the approach to a
$\approx\tfrac{2}{3}$ rate exponent at low contact angles suggested by several
previous studies,[86, 34] but see no evidence for the previously proposed
linear growth.[81] Overall, surfactant will decelerate the coalescence of the
sessile droplets, when above CAC, but we observed higher exponents in the case
of concentration below the CAC for wettable substrates, which is not fully
understood at the moment. Similarly, the bridge angle, $\theta_{b}$, changes
at a faster pace in the case of water droplets and low concentration and when
the wettability of the substrate is lower (larger $\theta_{s}$). Finally, by
analyzing the velocity of approach of the two droplets, which is generally
high at the initial stages of coalescence when the bridge forms, we found that
more wettable substrates and a higher surfactant concentration will lead to
smoother changes in the velocity, less acceleration. We anticipate that our
study provides fundamental insights into the coalescence of sessile
surfactant-laden droplets, an important phenomenon that has previously mostly
remained unexplored at the molecular-scale level.

##  V Supplementary Material

The SM contains: details on the estimation of the contact angle of a sessile
droplet. Calibration of the dependence of the contact angle of a water droplet
as a function of the droplet–substrate attraction parameter
$\varepsilon_{ws}$. Table of the water–substrate and surfactant–substrate
interaction parameters. Further data and methodology details on the mass
transport mechanism along with the probabilities of surfactant moving between
different areas of the droplets. Data on the bridge angle, the velocity of
approach, and the density profiles of the droplets, for a wider set of
parameters than shown in the main text.

###### Acknowledgements.

This research has been supported by the National Science Centre, Poland, under
grant No. 2019/34/E/ST3/00232. We gratefully acknowledge Polish high-
performance computing infrastructure PLGrid (HPC Centers: ACK Cyfronet AGH)
for providing computer facilities and support within computational grant no.
PLG/2022/015747.

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