# Role of surfactant-induced Marangoni stresses in drop-interface coalescence

C. R. Constante-Amores1  A. Batchvarov1  L. Kahouadji1  S. Shin2  J. Chergui3
D. Juric3  O. K. Matar1<EMAIL_ADDRESS>1Department of Chemical
Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ,
United Kingdom 2Department of Mechanical and System Design Engineering, Hongik
University, Seoul 04066, Republic of Korea. 3Université Paris Saclay, Centre
National de la Recherche Scientifique (CNRS), Laboratoire Interdisciplinaire
des Sciences du Numérique (LISN), 91400 Orsay, France.

###### Abstract

We study the effect of surfactants on the dynamics of a drop-interface
coalescence using full three-dimensional direct numerical simulations. We
employ a hybrid interface-tracking/level-set method, which takes into account
Marangoni stresses that arise from surface tension gradients, interfacial and
bulk diffusion, and sorption kinetic effects. We validate our predictions
against the experimental data of Blanchette and Bigioni [Nat. Phys.
2(4):254-257 (2006)] and perform a parametric study that demonstrates the
delicate interplay between the flow fields and those associated with the
surfactant bulk and interfacial concentrations. The results of this work
unravel the crucial role of the Marangoni stresses in the flow physics of
coalescence with particular attention paid to their influence on neck
reopening dynamics in terms of stagnation-point inhibition, and near-neck
vorticity generation. We demonstrate that surfactant-laden cases feature a
rigidifying effect on the interface compared to the surfactant-free case, a
mechanism that underpins the observed surfactant-induced phenomena.

##  1 Introduction

The occurrence of drop-interface coalescence has been observed in a wide range
of natural phenomena and industrial applications, such as rain/cloud formation
(Raes et al., 2000), atomisation (Villermaux, 2007), and also emulsification
or de-emulsification processes (Ziegler & Wolf, 2005). Over half a century
ago, Charles & Mason (1960) observed coalescence in their ground-breaking
experiments, and, ever since, researchers have been in constant pursuit of a
better physical understanding of this phenomenon. However, it was not until
the advent of high-speed imaging that it became possible for Thoroddsen &
Takehara (2000) to observe the self-similar coalescence cascade phenomena of a
drop before its total coalescence. Since then, the significant interest in the
field has led to a recent comprehensive review on the topic by Kavehpour
(2015) who concluded that further work is needed to understand the Marangoni-
effect during the drop-interface coalescence dynamics.

The dynamics commence with the drainage of the fluid between the drop and an
interface separating this fluid from another bulk phase whose extent is
typically much larger than the drop diameter. This drainage leads to the
formation of a thin fluid layer between the drop and the interface. As the
layer thickness decreases, van der Waals forces trigger its rupture with the
generation of a hole which expands driven by capillarity. The hole expansion
has been widely studied by Eggers et al. (1999); Paulsen et al. (2011);
Anthony et al. (2017) among others, concluding that the interfacial dynamics
are solely governed by a balance between viscous and capillary forces, and
subsequently the Ohnesorge number $Oh$ (e.g., ratio of viscous to capillary
forces) is the most appropriate control parameter for this phenomenon.
Different coalescence regimes have been identified depending on the order of
magnitude of $Oh$: (i) the inertial/capillary coalescence regime ($Oh\ll 1$),
which is characterised by a nearly inviscid liquid, and the dynamics are
surface-tension-driven; (ii) the viscous/capillary regime ($Oh>1$) where the
viscous forces play a major role in the interfacial dynamics; finally, (iii)
an intermediate regime which bridges the inertial/capillary and
viscous/capillary regimes (e.g., no dominance by either viscosity or surface
tension).

As pointed out earlier for intermediate values of $Oh$, Thoroddsen & Takehara
(2000) observed the so-called ‘coalescence-cascade of a drop’ in which the
drop coalescence leads to the generation of a smaller daughter droplet which
results in a cascade of self-similar events until this successive coalescence
process is completed. The process of formation of a daughter droplet is known
as a ‘partial coalescence’ phenomenon, and its physical understanding came
from the insightful experimental and numerical results of Blanchette & Bigioni
(2006) who suggested that the occurrence of pinch-off depends solely on the
competition between the vertical (inertia-viscous) and horizontal (capillary)
pulls (i.e., the former aids the total coalescence and the latter the
capillary breakup), rather than the mechanism of Rayleigh-Plateau instability.
Additionally Blanchette & Bigioni (2006) have provided an extensive $Bo$-$Oh$
phase diagram delineating the boundaries between partial and total
coalescence. Here, $Bo$ is the Bond number which compares the importance of
gravitational to surface tension forces.

Importantly, Blanchette et al. (2009) and Sun et al. (2018) have also
considered situations in which there is a surface tension mismatch between the
drop and the interface triggering the generation of tangential Marangoni
stresses in the plane of the common interface formed post coalescence. In
their experimental and numerical investigation of the coalescence of a water
drop with an ethanol reservoir, they reported that Marangoni-induced flow
leads to the ejection of an additional drop from its summit during its
vertical stretching. Similarly, the generation of gradients of surface tension
can also be triggered by the use of surfactants (Manikantan & Squires, 2020).

Current understanding of the coalescence dynamics in such surfactant-laden
systems came from Dong et al. (2019) who experimentally showed for the first
time surfactant concentration profiles for systems characterised by high Bond
numbers. Additionally, they suggested that surfactants have a strong effect on
the interfacial dynamics inducing interfacial rupture (i.e., hole formation)
in an off-axis location. Their interfacial concentration profiles agree
qualitatively with the previous numerical work performed by Martin &
Blanchette (2015). Finally, Shim & Stone (2017) suggested that the presence of
surfactants decreases the air-drainage-time between the drop and interface
(so-called, ‘damped-coalescence-cascade mechanism’).

Craster et al. (2002) have shown that the presence of surfactants does not
affect the dynamics of interfacial singularities, as they are swept away from
the pinch-off point; thus, the scaling laws predicted by Eggers (1993) are
remain unaltered. Nonetheless, recently, Constante-Amores et al. (2020) showed
the deleterious effect of Marangoni stresses on interfacial singularities in
the context of capillary retraction of a liquid thread. They demonstrated that
Marangoni stresses drive reopening of the neck during the capillary retraction
(i.e., escape from the ‘end-pinching’ mechanism) as a result of the
suppression of stagnation point formation by flow-reversal in the vicinity of
the neck, and the higher generation of vorticity from the neck.

Although, Martin & Blanchette (2015) have shown that the presence of
surfactants is responsible for the inhibition of the partial coalescence
event, there is still a lack of explanation of what causes this pinch-off
inhibition. Moreover, the appreciation of Marangoni stresses profiles, and
other significant insights into the flow fields close to the pinch-off are
also missing. The present study aims to clarify and answer these questions and
to overcome all numerical difficulties presented in Martin & Blanchette (2015)
by taking into account the nonlinear relation between the surfactant
concentration and surface tension, which is of central importance in non-
dilute systems. Additionally, we are able to explore parameter ranges
corresponding to large density and viscosity contrasts, corresponding to air-
water systems, without numerical difficulties, and to go beyond the surfactant
elasticity range studied by Martin & Blanchette (2015).

The rest of this article is organised as follows: Section 2 presents the
governing equations, numerical setup, and the validation of the surfactant-
free case against the experimental observations of Blanchette & Bigioni
(2006). Section 3 provides a discussion of the results which are focused on
the origin of the inhibition of the interfacial singularity, and a parametric
study accounting for the strength of the Marangoni stress, and sorption
dynamics. Finally, concluding remarks are summarised in Section 4.

##  2 Problem formulation and numerical method

With the purpose of studying the dynamics of interfacial coalescence in the
presence of surfactants, we perform direct numerical simulations of the two-
phase Navier-Stokes equations in a three-dimensional Cartesian domain
$\mathbf{x}=\left(x,y,z\right)$ (see Figure 1a). The treatment of the
interface and its surface tension forces is handled using a hybrid front-
tracking/level-set technique, also known as the Level Contour Reconstruction
Method (LCRM) (Shin & Juric (2009), Shin et al. (2017)), with surfactant
transport being resolved both in the bulk and on the interface. More
information on the numerical technique applied to surfactant transport can be
found in the work of Shin et al. (2018). Moreover, the dependence of the
surface tension on the interfacial surfactant concentration is described by a
nonlinear Langmuir equation of state (Muradoglu & Tryggvason, 2014; Shin et
al., 2018).

###  2.1 Scaling

In what follows, all variables will be made dimensionless (represented by
tildes) using

$\quad\tilde{\mathbf{x}}=\frac{\mathbf{x}}{R_{o}},\quad\tilde{t}=\frac{t}{T},\quad\tilde{\textbf{u}}=\frac{\textbf{u}}{U},\quad\tilde{p}=\frac{p}{\rho_{l}U^{2}},\quad\tilde{\sigma}=\frac{\sigma}{\sigma_{s}},\quad\tilde{\Gamma}=\frac{\Gamma}{\Gamma_{\infty}},\quad\tilde{C}=\frac{C}{C_{\infty}},\quad\tilde{C_{s}}=\frac{C_{s}}{C_{\infty}},$
(1)

where, $t$, u, and $p$ stand for time, velocity, and pressure, respectively.
The physical parameters correspond to the liquid density $\rho_{l}$,
viscosity, $\mu_{l}$, surface tension, $\sigma$, surfactant-free surface
tension, $\sigma_{s}$, and gravitational acceleration, $g$;
$T=\sqrt{\rho_{l}R_{o}^{3}/\sigma_{s}}$ is the capillary time scale and
$R_{o}$ is the initial drop radius; hence the velocity scale is
$U=R_{o}/T=\sqrt{\sigma_{s}/(\rho_{l}R_{o})}$. The interfacial surfactant
concentration, $\Gamma$, is scaled on the saturation interfacial
concentration, $\Gamma_{\infty}$, whereas the the bulk and bulk sub-phase (the
region immediately adjacent to the interface) surfactant concentrations given
by $C$ and $C_{s}$, respectively, are scaled on the initial bulk surfactant
concentration, $C_{\infty}$. As a result of the scaling in equation (1), the
dimensionless forms of the governing equations for the flow and the surfactant
transport are respectively expressed as

$\nabla\cdot\tilde{\textbf{u}}=0,$ (2)
$\tilde{\rho}(\frac{\partial\tilde{\textbf{u}}}{\partial\tilde{t}}+\tilde{\textbf{u}}\cdot\nabla\tilde{\textbf{u}})+\nabla\tilde{p}=-Bo\textbf{i}_{z}+Oh~{}\nabla\cdot\left[\tilde{\mu}(\nabla\tilde{\textbf{u}}+\nabla\tilde{\textbf{u}}^{T})\right]+\int\limits_{\tilde{A}\tilde{(t)}}\left(\tilde{\sigma}\tilde{\kappa}\textbf{n}+\nabla_{s}\tilde{\sigma}\right)\delta\left(\tilde{\textbf{x}}-\tilde{\textbf{x}}_{{}_{f}}\right)\mbox{d}\tilde{A},$
(3)
$\frac{\partial\tilde{C}}{\partial\tilde{t}}+\tilde{\textbf{u}}\cdot\nabla\tilde{C}=\frac{1}{Pe_{b}}\nabla^{2}\tilde{C},$
(4)
$\frac{\partial\tilde{\Gamma}}{\partial\tilde{t}}+\nabla_{s}\cdot(\tilde{\Gamma}\tilde{\textbf{u}}_{\rm
t})=\frac{1}{Pe_{s}}\nabla^{2}_{s}\tilde{\Gamma}+Bi\left(k\tilde{C_{s}}(1-\tilde{\Gamma})-\tilde{\Gamma}\right),$
(5) $\tilde{\sigma}=1+\beta_{s}\ln{\left(1-\tilde{\Gamma}\right)},$ (6)

which correspond to the equations of mass and momentum conservation, the
convective-diffusion equations for the surfactant bulk and interfacial
concentrations, and the nonlinear surfactant equation of state, respectively.
Here, the density $\tilde{\rho}$ and viscosity $\tilde{\mu}$ are expressed by
$\tilde{\rho}=\rho_{g}/\rho_{l}+\left(1-\rho_{g}/\rho_{l}\right)\mathcal{H}\left(\tilde{\textbf{x}},\tilde{t}\right)$
and
$\tilde{\mu}=\mu_{g}/\mu_{l}+\left(1-\mu_{g}/\mu_{l}\right)\mathcal{H}\left(\tilde{\textbf{x}},\tilde{t}\right)$
wherein $\mathcal{H}\left(\tilde{\textbf{x}},\tilde{t}\right)$ represents a
smoothed Heaviside function, which is zero in the gas phase and unity in the
liquid phase, where the subscript $g$ designates the gas phase;
$\tilde{\textbf{u}}_{\rm{t}}=\left(\tilde{\textbf{u}}_{\rm{s}}\cdot\textbf{t}\right)\textbf{t}$
represents the velocity vector tangential to the interface in which
$\tilde{\textbf{u}}_{\rm{s}}$ corresponds to the interfacial velocity;
$\kappa$ is twice the mean interface curvature calculated from the Lagrangian
interface grid;
$\nabla_{s}=\left({\mathbf{I}}-\mathbf{n}\mathbf{n}\right)\cdot\nabla$ stands
for the surface gradient operator wherein $\mathbf{I}$ is the identity tensor
and $\mathbf{n}$ is the outward-pointing unit normal to the interface;
$\tilde{\mathbf{x}}_{f}$ is the parameterisation of the interface
$\tilde{A}(\tilde{t})$; finally, $\delta$ represents a Dirac delta function
that is non-zero when $\tilde{\mathbf{x}}=\tilde{\mathbf{x}}_{f}$ only. The
numerical method used to solve the above equations is described in detail by
Shin et al. (2018).

The dimensionless groups that appear in equations (2)-(6) are defined as

$Bo=\frac{\rho_{l}gR_{o}^{2}}{\sigma_{s}},~{}~{}~{}Oh=\frac{\mu_{l}}{\sqrt{\rho_{l}\sigma_{s}R}},~{}~{}~{}$
(7)
$Bi=\frac{k_{d}R_{o}}{U},~{}~{}~{}k=\frac{k_{a}C_{\infty}}{k_{d}},~{}~{}~{}Pe_{s}=\frac{UR_{o}}{D_{s}},~{}~{}~{}Pe_{b}=\frac{UR_{o}}{D_{b}},~{}~{}~{}\beta_{s}=\frac{\Re
T\Gamma_{\infty}}{\sigma_{s}},$ (8)

where $Bo$ and $Oh$ are the Bond number (ratio of gravitational to capillary
forces) and Ohnesorge number (ratio of viscous to surface tension forces),
respectively. The surfactant elasticity parameter, $\beta_{s}$, measures the
sensitivity of the surface tension to the surfactant concentration in which
the parameter $\Re$ represents the thermodynamic ideal gas constant value
$8.314$ J K-1 mol-1, and $T$ denotes temperature. The parameters $Pe_{s}$ and
$Pe_{b}$ are the interfacial and bulk Peclet numbers that represent the ratio
of convective to diffusive time-scales in the plane of the interface and the
bulk, respectively. The Biot number, $Bi$, stands for the ratio of
characteristic desorptive to convective time-scales. Finally, $k$ is the ratio
of adsorption to desorption time scales where $k_{a}$ and $k_{d}$ refer to the
surfactant adsorption and desorption coefficients, respectively.

At equilibrium, there is no surfactant exchange between the interface and the
bulk, and the last term on the right-hand-side of equation (5) reduces to the
Langmuir adsorption isotherm

$\chi=\frac{\Gamma_{eq}}{\Gamma_{\infty}}=\frac{k}{(1+k)},$ (9)

where $\chi$ stands for the fraction of the interface covered by adsorbed
surfactant. Furthermore, the Marangoni stress, $\tilde{\tau}$, which appears
in the third term on the right-hand-side of equation (3), is expressed as a
function of $\tilde{\Gamma}$ as follows:

$\tilde{\tau}\equiv\nabla_{s}\tilde{\sigma}\cdot\textbf{t}=-\frac{\beta_{s}}{1-\tilde{\Gamma}}\nabla_{s}\tilde{\Gamma}\cdot\textbf{t},$
(10)

where $\mathbf{t}$ is the unit tangent to the interface. In all cases
considered in the present study, the Marangoni time-scale, $\mu
R_{o}/\Delta\sigma=O(10^{-4})$ s, as compared to the capillary and
sorptive/desorptive time-scales, which are of $O(10^{-3})$ s and
$O(10^{-3})-O(10^{-4})$ s, respectively; thus Marangoni stresses will play a
crucial role in the coalescence phenomenon. Finally, the tildes are dropped
henceforth with the understanding that hereafter all variables discussed are
dimensionless unless stated otherwise.

###  2.2 Numerical setup, validation, and parameters

|  |
---|---|---
(a) | (b) | (c)
Figure 1:  Schematic representation of the flow configuration, and validation
of the numerical procedure: (a) initial shape of the drop resting close to the
interface, highlighting the computational domain of size $12R_{o}\times
12R_{o}\times 6R_{o}$ (not-to-scale) in a three-dimensional Cartesian domain,
$\mathbf{x}=(x,y,z)$, with a resolution of $386^{3}$; (b) direct comparisons
of our numerical predictions for a surfactant-free case (blue line) with
experimental results reported by Blanchette & Bigioni (2006) for the post-
coalescence dynamics of an ethanol drop in air prior to interfacial
singularity formation with $Oh=0.011$ and $Bo=0.09$; also shown in red lines
are the numerical solutions by Deka et al. (2019) for the same case; (c)
evolution of the neck radius as a function of the time to break-up, $t_{o}$,
which agrees with the well-known inertial-viscous scaling theory of (Eggers,
1993) shown as a dashed line with unity slope.

The numerical setup closely follows the work done by Sun et al. (2018), and
Martin & Blanchette (2015). Thus, the size of the dimensionless computational
domain is chosen as $12R_{o}\times 12R_{o}\times 6R_{o}$, which is found to be
sufficiently large to avoid the effect of artificial reflections from the
boundaries. We define a radial component as
$r=\sqrt{\left(x-x_{o}\right)^{2}+\left(y-y_{o}\right)^{2}}$ where $x_{o}$ and
$y_{o}$ are the abscissa and ordinate drop position, respectively. Solutions
are sought subject to Neumann boundary conditions on all variables at the
lateral boundaries, $p=0$ at the top boundary $z=6R_{o}$, and no-slip at the
bottom $z=0$. The initialisation of the interface corresponds to a spherical
drop resting immediately above a horizontal flat interface before its
interfacial rupture (e.g., all the velocities set to zero) where both drop and
liquid pool are made up of the same liquid. Importantly, the drop is connected
to the flat-interface by a neck of radius $0.25R_{o}$ for the initialisation
of the dynamics; a similar approach has been previously used by Blanchette &
Bigioni (2006, 2009) and Martin & Blanchette (2015). The assumption is based
on the time scale associated to the retraction of the neck
$t_{CR}=R_{o}/\sqrt{2\sigma_{s}/\rho\delta}$, which is too short to have an
influence on the phenomenon.

Figures 1b-c highlight qualitative and quantitative validation of our
numerical framework with results from the literature (Eggers, 1993; Blanchette
& Bigioni, 2006; Deka et al., 2019). The numerical simulations have been
benchmarked against the surfactant-free experimental results of Blanchette &
Bigioni (2006) in terms of the temporal interfacial dynamics of the
coalescence of an ethanol drop surrounded by air (displayed in figure 1b). Our
numerical results are provided as snapshots of the interface location at times
corresponding to those given by Blanchette & Bigioni (2006). We have also
included the numerical predictions from Deka et al. (2019). Figure 1b
demonstrates that our numerical framework is capable of predicting accurately
the interfacial dynamics of the coalescence phenomenon for ‘clean’ interfaces.
Additionally, the temporal evolution of the neck towards its pinch-off is in
good agreement with the well-known inertial-viscous scaling theory of Eggers
(1993) (see figure 1c).

The dimensionless quantities for the studied phenomenon are consistent with
experimentally-realisable systems. The Ohnesorge number was set to $Oh=2\times
10^{-2}$ because it allows the observation of the interplay between the full
range of dynamics as there is a competition between inertial, viscous, and
capillary forces. The chosen density and viscosity ratios,
$\rho_{g}/\rho_{l}=1.2\times 10^{-3}$ and $\mu_{g}/\mu_{l}=0.018$,
respectively, are representative of an air-water system. The elasticity number
$\beta_{s}$ depends on the interfacial concentration at saturation,
$\Gamma_{\infty}$, which, in turn, is related to the critical micelle
concentration (CMC) that is of $O(10^{-6})~{}$mol/m2. We have explored the
range of $0.1<\beta_{s}<0.5$ which corresponds to $2.9\times 10^{-6}<$ CMC
$<1.4\times 10^{-5}$ mol/m2. Typical values for the interfacial diffusion
coefficient for surfactants such as Sodium Dodecyl Sulphate (SDS),
N-Dodecyl-N,N-dimethylammonio-3-propane sulfonate (SB12), and similar monomers
in aqueous solution are within the range of $10^{-12}<D_{s}<10^{-8}$ m2/s when
$\Gamma$ is below the CMC (Joos et al., 1982; Siderius et al., 2002); this
range also covers phospholipid-based pulmonary surfactants, such as
N-(7-Nitrobenz-2-Oxa-1,3-Diazol-4-yl)-1,2-Dihexadecanoyl-sn-Glycero-3-
phosphocholine (NBD-PC), which are considered effectively insoluble (Fallest
et al., 2010; Strickland et al., 2015). Therefore, the interfacial Peclet
number $Pe_{s}$ lies in the range $10^{3}<Pe_{s}<10^{6}$. Recently, Batchvarov
et al. (2020) and Constante-Amores et al. (2020) suggested that the
investigated interfacial dynamics reach saturation above $Pe_{s}=100$; thus,
the selected interfacial Peclet is set to $Pe_{s}=100$. In terms of the chosen
bulk Peclet number, Agrawal & Neuman (1988) suggested that the interfacial and
bulk Peclet numbers are of the same order of magnitude; on this basis,
hereafter we set $Pe_{b}=Pe_{s}$. In summary, we have chosen the values of the
surfactant-related parameters to ensure that all of the relevant physical
processes associated with surfactant transport such as Marangoni stresses,
surface/bulk diffusion, and sorption kinetics are represented in the present
study.

In terms of mesh resolution studies, we have ensured that our numerical
simulations are mesh-independent, and subsequently for a resolution of
$(386)^{3}$, the results do not change with decreasing cell size. We have also
ensured that the liquid volume and surfactant mass conservation are satisfied
with an error under $10^{-3}~{}\%$ (see Appendix for more information).
Extensive mesh studies for surface tension-driven phenomena using the same
computational method have been published previously (Batchvarov et al. (2020);
Constante-Amores et al. (2020)). A discussion of the results is presented
next.

##  3 Results

$t=0.17$ | $t=0.69$ | $t=1.20$ | $t=1.69$
---|---|---|---
|  |  |
(a) | (b) | (c) | (d)
|  |  |
(e) | (f) | (g) | (h)
|  |  |
(i) | (j) | (k) | (l)
|  |  |
(m) | (n) | (o) | (p)
Figure 2:  Effect of $\beta_{s}$ on the drop-interface coalescence dynamics
for insoluble surfactants. Spatio-temporal evolution of the three-dimensional
interface shape for surfactant-free, (a)-(d), and surfactant-laden coalescence
for $\beta_{s}=0.1$, (e)-(h), $\beta_{s}=0.3$, (i)-(l), and $\beta_{s}=0.5$,
(m)-(p). Here, the dimensionless parameters are $Oh=0.02$ and $Bo=10^{-3}$,
and for the surfactant-laden cases, $Pe_{s}=100$ and
$\Gamma_{o}=\Gamma_{\infty}/2$. The colour indicates the magnitude of
$\Gamma$, and legend is shown in (e).

Following the good agreement between the surfactant-free coalescence
simulation and the experimental results of Blanchette & Bigioni (2006), key
surfactant effects will be investigated in this section. We first display our
results related to the effect of insoluble surfactant, e.g. NBD-PC (Fallest et
al., 2010; Strickland et al., 2015), showing ultimately the insights regarding
the surfactant-driven-escape from a potential pinch-off singularity such as
the one depicted in figure 1b. We then present the effect of the sorptive
kinetics on the phenomenon throughout the use of soluble surfactants, e.g. SDS
(Siderius et al., 2002). It is also worth mentioning that all surfactant
simulations have been carried out until the neck has either pinched off or
undergone reopening. Additionally, we provide with conclusive evidence that
the neck-reopening is driven by Marangoni stresses.

###  3.1 Insoluble surfactants

We start the discussion of the results by presenting the effect of the
surface-active agents through the analysis of the elasticity parameters
$\beta_{s}$ with $Oh=0.02$, $Bo=10^{-3}$, $Pe_{s}=100$, and
$\Gamma_{o}=\Gamma_{\infty}/2$. Figure 2 shows the spatio-temporal interfacial
dynamics for the surfactant-free and surfactant-laden cases as a function of
the elasticity parameter. At the early stages of the dynamics, the neck
expands as a result of the capillary retraction of the liquid bridge which
separates the drop from the underlying liquid pool. The capillary retraction
gives rise to the formation of capillary waves that travel upwards towards the
drop summit. As pointed out by Blanchette & Bigioni (2006), the oscillations
caused by the travelling capillary waves yield vertical stretching forming a
nearly-cylindrical drop, as shown in figure 2c, before capillarity acts to
drive the dynamics towards a more energy-favourable state by pulling on the
sides of the drop. This capillary action leads to pinch-off of the liquid
bridge via a singularity which culminates in the formation of a secondary
droplet; this, in turn, follows a ‘cascade of coalescence events’ until the
coalescence process is completed as also shown by Thoroddsen & Takehara
(2000), Blanchette & Bigioni (2006), and Blanchette & Bigioni (2009).

For all surfactant-laden cases, the generation of a secondary droplet is
avoided even for the lower end of the elasticity parameter range. For
$\beta_{s}=0.1$, the dynamics follow closely those of the surfactant-free
case, where significant vertical stretching of the original droplet is
observed (see figures 2e-h). At the point where surface tension is expected to
dominate the narrowing of the neck, the presence of non-uniform surfactant
concentration generates Marangoni stresses that change the outcome of the
dynamics. By increasing $\beta_{s}$, surfactant redistribution along the
interface is enhanced as displayed in figures 2i-l and figures 2m-p, for
$\beta_{s}=0.3$ and $\beta_{s}=0.5$, respectively. The surfactant
concentration gradients, and associated transport, is seen to suppress the
capillary waves and to limit the vertical stretch of the drop.

|  |
---|---|---
(a) | (b) | (c)
Figure 3: Effect of $\beta_{s}$ on the temporal dynamics of the vertical
extent of the drop, (a), its minimum neck radius, (b), and the system kinetic
energy $E_{k}$, (c), for $Oh=0.02$, $Bo=10^{-3}$, $Pe_{s}=100$, and
$\Gamma_{o}=\Gamma_{\infty}/2$.

Figure 3 shows the immobilising effect brought about by the presence of
surfactants through the analysis of the temporal dynamics of the maximum
vertical stretch of the droplet, $z_{max}$, the neck radius, $r_{min}$, and
the kinetic energy, $E_{k}=\int_{V}(\rho\textbf{u}^{2}/2)dV$. Here, the
$E_{k}$ values have been normalised by the surface energy $E_{s}=S\sigma_{s}$,
where $S$ is the initial superficial area of the system. The evidence for
damping of the upward drop oscillation can be seen in figure 3a. Here, the
increase in $\beta_{s}$ is seen to depress the maximum crest location of the
drop. These observations confirm the expectations of Martin & Blanchette
(2015) of suppression of the axial oscillation with an increase in the
surfactant strength (though these authors were only able to run simulations
for $\beta_{s}\leq 0.2$). Interestingly, the temporal evolution of $z_{max}$
exhibits a non-monotonic dependence on $\beta_{s}$, with the most suppressed
crest being associated with the intermediate value of $\beta_{s}=0.3$. The
physical explanation of this outcome will be provided in the discussion of
figure 4 below. Furthermore, investigation of the temporal variation of the
minimum neck radius, $r_{min}$ (see figure 3b), confirms neck reopening for
all surfactant-laden cases, with a 50% rise in the time associated with the
onset of re-opening, $t_{r}$, corresponding to an increase in $\beta_{s}$ from
0.1 to 0.5. The non-monotonic dependence on $\beta_{s}$ is also exhibited by
$t_{r}$: even though the longest delay in neck closure is observed for the
highest $\beta_{s}$ studied, the $t_{r}$ value for $\beta_{s}=0.3$ is
associated with the largest $r_{min}$. Further quantification of these
physical phenomena is provided below.

Finally, inspection of the kinetic energy plots shows that the presence of
surfactants induces a monotonically-decreasing overall value of $E_{k}$ with
$\beta_{s}$ over the range in time encompassing the creation of the
cylindrically-shaped drop (see figure 3c). This reduction of the kinetic
energy is a result of the rigidification of the interface brought on by the
tangential Marangoni stresses in agreement with Asaki et al. (1995). It is
evident, however, that for the surfactant-free case, $E_{K}$ decreases
rapidly, as the drop breaks up via neck pinch-off, eventually dipping below
those associated with $\beta_{s}=0.1$ and $\beta_{s}=0.3$.

|  |  |
---|---|---|---
(a) | (b) | (c) | (d)
|  |  |
(e) | (f) | (g) | (h)

|
---|---
(i) |

Surfactant-free Surfactant-laden | Surfactant-free Surfactant-laden
---|---
|
(j) | (k)

Figure 4:  Panels (a-i): Effect of the elasticity parameter $\beta_{s}$ on the
flow and surfactant concentration fields associated with the drop-interface
coalescence phenomenon. Two-dimensional representation of the interface
location, $\Gamma$, $\tau$, and the radial component of the interfacial
velocity $u_{tr}$ are shown in (a)-(d) and (e)-(h) for $t=1.20$ and $t=1.68$,
respectively. Note that the abscissa in (a) and (e) corresponds to the radial
coordinate $r$, and in (b)-(d) and (f)-(h) to the arc length $s$. The panel
(i) represents a magnified view of (h). The arrows in (g) indicate the
directions of motion driven by the Marangoni stresses $\tau$; in (h) and (i),
point P1, and S1 and S2 designate the peak in $u_{tr}$ and the stagnation
points in the surfactant-free $u_{tr}$ profile, respectively. The diamond
shapes in (i) show the location of the necks. The parameter values and the
times for the radius are the same as in figure 3. Panels (j) and (k): Effect
of surfactants on the azimuthal vorticity $\omega_{\theta}$ for the
surfactant-free (left panels), and the surfactant-laden cases (right panels),
for $\beta_{s}$=0.5, at $t=1.20$, $t=1.68$, respectively. All other parameters
remain unchanged from figure 3. The colour indicates the value of the
azimuthal vorticity $\omega_{\theta}$.

The next part of the analyses focuses on the time evolution of a two-
dimensional projection of the interfacial shape, $\tilde{\Gamma}$, $\tau$, and
the radial component of the interfacial velocity, $u_{tr}$, presented in
panels a-i of figure 4. We also show the interplay between the surface and the
azimuthal component of the vorticity field (i.e., vorticity is defined as
$\omega=\nabla\times\textbf{u}$), displayed in figures 4j and 4k. In the
surfactant-free case, it is seen from figure 4d that $u_{tr}<0$ and $u_{tr}>0$
upstream and downstream of the developing neck, which drives flow away from
this region. The narrowing of the neck induces capillary-driven flow that
leads to further neck-thinning and the development of large peaks in $u_{tr}$,
as shown in figure 4e and h, which are typical of singularity formation. Close
inspection of the $u_{tr}$ profile in figure 4i for the surfactant-free case
reveals that it is characterised by the presence of a large velocity peak
($P1$) and two stagnation points (labelled $S1$ and $S2$) with the neck
sandwiched between them. Over time, the inertio-capillary-induced flow
ultimately culminates in interfacial breakup to form a daughter droplet. From
the vorticity plots in figures 4j and 4k, it is seen that for the surfactant-
free case, the vorticity generation is confined to the vicinity of the neck as
the two stagnation points aid the fluid recirculation around the neck (so-
called ‘vortex-ring’, displayed in the left-panel of figure 4j). As time
evolves, the interfacial curvature of the neck increases, and a large
vorticity generation can be observed on the side of the bulk accompanying the
eventual neck pinch-off, as depicted in the left-panel of figure 4k. More
information regarding the mechanisms which induce the generation of vorticity
at the liquid-gas interface is provided below.

For the surfactant-laden cases, the accumulation of $\Gamma$ near the nascent
neck can be seen in figure 4b thus giving rise to a local decrease of
$\sigma$. The presence of $\Gamma$ gradients results in the generation of a
large positive peak in the $\tau$ profile in the vicinity of the neck region,
which is largest for the intermediate value of $\beta_{s}=0.3$, as shown in
figure 4c (i.e., consistent with the non-monotonic response of dynamics
observed in figure 3). Upstream and downstream of the neck, $\tau>0$ and
$\tau<0$, respectively, which drives flow towards the drop summit and tail,
reflected by $u_{tr}<0$ and $u_{tr}>0$, respectively. Although the overall
shape of the $u_{tr}$ curve for the surfactant-free case is robust to the
addition of insoluble surfactant, it is evident that the magnitude of $u_{tr}$
decreases with increasing $\beta_{s}$, particularly in the neck region;
moreover, the oscillation in $u_{tr}$ in the surfactant-free case is damped
out for $\beta_{s}>0$.

Additionally, by close inspection of the $u_{tr}$ plots for the surfactant-
laden cases in figure 4i, it becomes clear that only one stagnation point is
present near the neck for $\beta_{s}=0.3$ and $0.5$; thus, $u_{tr}>0$ towards
its tail. The Marangoni-induced flow has therefore led to the suppression of
one of the stagnation points. Furthermore, by comparing the vorticity field
pattern of the surfactant-free and the surfactant-laden cases, a change is
observed as a result of the presence of surfactants, and the inhibition of a
stagnation point. The generation of vorticity is also confined to the vicinity
of the free-surface; however, the ‘vortex-ring’ no longer exists, as shown in
the right panel of figure 4j.

Further in time, when the escape of capillary singularity commences, we
observe that vorticity is separated from the vicinity of the interface, and
advected towards the bulk of liquid reservoir, which is consistent with the
findings of Ananthakrishnan & Yeung (1994), supporting the reopening of the
neck, as displayed in the right panel of figure 4k. The reason behind the
vorticity separation is the inhibition of one of the stagnation points. This
behaviour is similar to the phenomenon explained by Hoepffner & Paré (2013) in
terms of capillary retraction of surfactant-free viscous ligaments where they
suggested that the advection of $\omega_{\theta}$ plays a crucial role in
their escape from breakup. These observations also agree with the recent
studies reported by Constante-Amores et al. (2020) and Constante-Amores et al.
(2021) in terms of the escape of capillary singularity during the capillary
retraction of a liquid thread, and the inhibition of jet-drops formation from
bursting bubbles, respectively.

Next, we turn our attention to the role of the Marangoni stresses in the
generation of vorticity at the gas-liquid interface; this is consistent with
the work of Batchelor (2007) who concluded that vorticity in a homogeneous
fluid is generated at the boundaries only. Furthermore, we can assume that the
interface behaves as a viscous free-surface because of the vanishing viscosity
and density ratios (similar assumptions have been made previously by Dooley et
al. (1997) and Xia et al. (2017)). Kamat et al. (2020) demonstrated that for
surface-tension-driven phenomena, vorticity generation depends solely on the
interfacial boundary conditions when $Oh<<1$. As a result, the tangential
stress at the interface is balanced by the surface tension gradients,
resulting in

$\textbf{t}\cdot\textbf{D}\cdot\textbf{n}=\textbf{t}\cdot\nabla_{s}\sigma$
(11)

in which, D represents the rate of deformation tensor (the symmetric part of
the velocity gradient tensor). By further mathematical manipulation (Lundgren
& Koumoutsakos, 1999), the generation of vorticity at the free-surface depends
entirely on the velocity field, interfacial geometry, and surface tension
gradients:

$\omega=\omega_{n}+\omega_{t}+\omega_{\tau}=-2\frac{\partial\textbf{u}\cdot\textbf{n}}{\partial
s}+2\textbf{u}\cdot\textbf{t}\kappa+\textbf{t}\cdot\nabla_{s}\sigma.\ $ (12)

Similar results for $\omega_{n}$ and $\omega_{t}$ have been reported by
Lundgren & Koumoutsakos (1999) and Brøns et al. (2014). The first two terms on
the right-hand-side of equation (12) correspond to the the normal and
tangential velocity-driven vorticity generation, respectively, whereas the
last term is representative of the Marangoni stress vorticity contribution.

Figure 5 shows the vorticity distribution along the interface according to
equation (12). By close inspection of the profiles, we observe that the
vorticity generation at the interface is highly dominated by the interfacial
curvature term, $\omega_{t}$. Moreover, in the vicinity of the neck we
discover the existence of a positive peak in the Marangoni stress-driven
vorticity production (i.e., $\omega_{\tau}$). The peak of $\omega_{\tau}$ has
a different sign in comparison to $\omega_{n}$. The leading cause for this
behaviour stems from the suppression of the stagnation points on both sides of
the neck, as shown in figure 4i. Ultimately, this analysis demonstrates the
positive effect of surface tension-driven vorticity generation on the neck
reopening process. Additionally, this finding is in agreement with Kamat et
al. (2020), who concluded that the generation of vorticity arising from the
presence of surfactants is generated over a time scale of similar magnitude to
the capillary time scale.

---
Figure 5:  Vorticity production, $\omega$, along the gas-liquid interface
expressed in terms of the local normal and tangential components from the
velocity field and the Marangoni stresses, represented by $\omega_{n}$,
$\omega_{t}$ and $\omega_{\tau}$, respectively. The surfactant-laden case is
characterised by $\beta_{s}=0.5$ at $t=1.68$. All other parameters remain
unchanged from figure 3. The diamond shape shows the location of the neck.

Finally, we aim to provide more conclusive evidence that the interfacial
singularity inhibition is Marangoni-driven rather than a result of the
reduction of the surface tension (i.e., capillary pressure reduction). For
this reason, we have performed an additional simulation in which, Marangoni
stresses have been suppressed. Figure 6 reports the temporal evolution of the
maximum axial position $z_{max}$, the neck radius $r_{min}$, and the kinetic
energy $E_{k}$ for the surfactant-free and Marangoni-suppressed cases. Similar
flow behaviour between the surfactant-free and Marangoni-suppressed cases are
observed. The most remarkable finding is that for the Marangoni-suppressed-
case, it is observed that the mean reduction of the surface tension does not
prevent the horizontal collapse of the droplet (see figure 6b). The inspection
of the kinetic energy plot shows that the Marangoni-suppressed and surfactant-
free cases have almost identical behaviour (see figure 6c). Therefore, when
Marangoni stresses are enabled fully, a change of the fate of the coalescence
is observed via the reopening of the neck.

|  |
---|---|---
(a) | (b) | (c)
Figure 6: Demonstration that tangential Marangoni stresses are responsible for
the inhibition of the interfacial singularity. Temporal evolution of the
maximum vertical displacement of the interface, neck radius and kinetic
energy; (a)-(c), respectively, for the surfactant-free, full-Marangoni
$|\tau|>0$, and no-Marangoni cases $|\tau|=0$, for $Oh=0.02$, $Bo=10^{-3}$,
$\beta_{s}=0.5$, $Pe_{s}=100$ and $\Gamma_{o}=0.5\Gamma_{\infty}$.

###  3.2 Soluble surfactants

In this subsection we present a discussion of the results associated with the
effects of surfactant solubility and sorption kinetics, parameterised by $Bi$
and $k$, respectively. Unless stated otherwise, the parameters remain fixed to
their ‘base’ values: $Oh=0.02$, $Bo=10^{-3}$, $\beta_{s}=0.5$, and
$Pe_{s}=100$; the interfacial surfactant concentration is initialised using
its equilibrium surfactant concentration, thus $\Gamma_{o}=\chi=k/(1+k)$. Once
again, simulations are carried out until either neck pinch-off or reopening
has been observed.

####  3.2.1 Effect of the Biot number, $Bi$

|  |
---|---|---
(a) | (b) | (c)
Figure 7:  Effect of $Bi$ on the temporal dynamics of the vertical strength of the drop, (a), minimum neck radius, (b), and kinetic energy (c), when $Oh=0.02$, $Bo=10^{-3}$, $Pe_{s}=100$, $\beta_{s}=0.5$, $k=1$ and $\Gamma_{o}=\chi$.  |  |  |
---|---|---|---
(a) | (b) | (c) | (d)
|  |  |
(e) | (f) | (g) | (h)
Figure 8:  Effect of the solubility parameter $Bi$ on the flow and surfactant
concentration fields associated with the drop-interface coalescence
phenomenon. Two-dimensional representation of the interface location,
$\Gamma$, $\tau$, and the radial component of the interfacial velocity
$u_{tr}$ are shown in (a)-(d) and (e)-(h) for $t=1.20$ and $t=1.68$,
respectively. Note that the abscissa in (a) and (e) corresponds to the radial
coordinate $r$, and in (b)-(d) and (f)-(h) to the arc length $s$. Here, all
other parameters remain unchanged from figure 7.

Figure 7 shows the effect of varying $Bi$ in the range $0.1-10$ on the drop
maximal vertical extent, $z_{max}$, the neck radius $r_{min}$, and the kinetic
energy $E_{k}$ with $k=1$; also shown are the curves associated with the
insoluble surfactant and surfactant-free cases which respectively correspond
to the $Bi\rightarrow 0$ and $Bi\rightarrow\infty$ (and/or
$\beta_{s}\rightarrow 0$) limits. At the lower end of this range (i.e.,
$Bi=0.1$), the sorptive timescales are much larger than those associated with
interfacial effects; therefore, the dynamics are dominated by capillarity and
Marangoni stresses, and are therefore expected to be similar to those observed
for the insoluble surfactant case. This is confirmed upon inspection of figure
7 as the curves associated with the $Bi=0.1$ case practically overlap with
those generated for the insoluble surfactant case. For large $Bi$, the
monomers desorb rapidly from the interface, which represents the case of a
highly-soluble surfactant characterised by dynamics that are similar to those
that accompany the surfactant-free case. This is also confirmed by comparing
the curves associated with the surfactant-free and $Bi=10$ cases, the latter
corresponding to the largest $Bi$ value studied.

As depicted in figure 7a, increasing the level of solubility leads to a
decrease in the surfactant mass at the interface available to induce Marangoni
stresses and, consequently, delays the retardation in the initialisation of
the vertical stretch of the drop. Interestingly, the lowest $z_{max}$ is
associated with the intermediate $Bi=1$ case, thus $z_{max}$ exhibits a non-
monotonic dependence on the surfactant solubility. Turning attention towards
the effect of solubility on the neck-size, $r_{min}$ displayed in figure 7b,
it is clearly seen that Marangoni-induced flow results in the inhibition of
the capillary-driven singularity over the entire range of $Bi$ values studied.
This effect becomes increasingly more pronounced with decreasing $Bi$ and the
$r_{min}$ vs $t$ profiles for the surfactant-laden cases are bounded between
the insoluble and the surfactant-free cases. Finally, the $E_{k}$ profiles
depicted in 7c show that decreasing $Bi$ leads to an overall reduction in the
kinetic energy, which stems from the rigidifying effect of the Marangoni
stresses; this is weakened by increasing the solubility and the enhanced
surfactant desorption from the interface which leads to a reduction in $\tau$.

Evidence of surfactant-induced immobilisation with decreasing $Bi$ is further
provided in figure 8a,e, and 8d,h, which depict the interface shape and
$u_{tr}$, respectively; the rest of the parameters remain unaltered from
figure 7. It is also clearly seen from figure 8b,f that higher surfactant
desorption is observed as $Bi$ increases driven by the mass transfer between
the interface and bulk. Although the largest $\Gamma$ is associated with the
smallest $Bi$ values, the largest gradients, and, therefore, Marangoni
stresses, $\tau$, are found for the intermediate Biot number, $Bi=1$, as shown
in figure 8c,g; this is consistent with with the non-monotonic dependence of
the vertical stretch of the drop on $Bi$, described in figure 7a. Notably, a
comparison of panels (b) and (f), and (c) and (g) of figure 8 reveals that
over time, the $\Gamma$ gradients become sharper, particularly for small and
intermediate $Bi$ leading to an increase in $\tau$, as was also observed in
the insoluble surfactant case. These Marangoni stresses counteract the
direction of the inertio-capillary-induced flow, dampen the oscillations in
$u_{tr}$ (see figure 8d,h), and act to prevent neck pinch-off where the
efficacy is once more dependent on the magnitude of $Bi$. Finally, the same
analysis regarding the role of surfactants in the inhibition of stagnation
points, explained in subsection 3.1, can be extrapolated to the solubility
parameter, $Bi$.

####  3.2.2 Effect of the adsorption parameter, $k$

Figure 9 shows the effect of the adsorption parameter, $k$, on the temporal
dynamics of the drop-interface coalescence phenomenon through the analysis of
the vertical stretch of the droplet, $z_{max}$, the neck radius, $r_{min}$,
and the kinetic energy, $E_{k}$, for $Bi=1$, and $k=(0.01,1,5)$. We note that
as $k\rightarrow 0$, $\chi\rightarrow 0$, and this corresponds to vanishingly
small equilibrium interfacial concentrations, which were used to initialise
the simulations. In addition, from eq. (2.5), $k\rightarrow 0$ implies that
$\Gamma$ will remain small, and thus, in this limit, we expect the dynamics to
be consistent with those associated with the surfactant-free case. For $k\gg
1$, on the other hand, the flow behaviour is similar to that observed in the
insoluble surfactant case.

We start the discussion of the effect of the $k$ parameter by analysing its
effect on the vertical stretch of the droplet (shown in figure 9a). By
inspection of the profiles, a monotonic response of the vertical stretch is
observed with decreasing $k$ values (e.g., $k=0.1$) the dynamics are similar
to that of the surfactant-free case characterised by neck formation and pinch-
off, as shown in figure 10a,e; this arises due to the increase in mass
transfer from the interface to the bulk decreasing the interfacial
concentration (see figure 10b,f) and reducing the magnitude of the Marangoni
stress, which is maximal for $k=1$, as shown in figure 10c,g. The trends
highlighted in figure 9a are mirrored in figure 9b that displays the temporal
dynamics of the $r_{min}$ which also exhibits a monotonic dependence on $k$.
Increasing $k$ alters the $u_{tr}$ profile in figure 10d,h in a similar manner
to that observed upon increasing $\beta_{s}$ and/or decreasing $Bi$ as was
shown previously in figures 4d,h and 8d,h, respectively. As a result, it is
seen clearly that the presence of surfactants alters the fate of the
coalescence phenomenon as Marangoni-driven flow induces neck reopening.
Finally, the $E_{k}$ plots shown in figure 9c support, once more, the high
interfacial rigidification brought about by the presence of surfactants.

|  |
---|---|---
(a) | (b) | (c)
Figure 9:  Effect of the adsorption parameter $k$ on the temporal dynamics of the vertical strength of the drop, (a), minimum neck radius, (b), and kinetic energy (c), when $Oh=0.02$, $Bo=10^{-3}$, $\beta_{s}=0.5$, $Pe_{s}=Pe_{b}=100$, $Bi=1$ and $\Gamma_{o}=\chi$. |  |  |
---|---|---|---
(a) | (b) | (c) | (d)
|  |  |
(e) | (f) | (g) | (h)
Figure 10:  Effect of the adsorption parameter $k$ on the flow and surfactant
concentration fields associated with the drop-interface coalescence
phenomenon. Two-dimensional representation of the interface location,
$\Gamma$, $\tau$, and the radial component of the interfacial velocity
$u_{tr}$ are shown in (a)-(d) and (e)-(h) for $t=1.20$ and $t=1.68$,
respectively. Note that the abscissa in (a) and (e) corresponds to the radial
coordinate $r$, and in (b)-(d) and (f)-(h) to the arc length $s$. Here, all
other parameters remain unchanged from figure 9.

##  4 Conclusions

A study of the effect of Marangoni-induced flow as a result of the presence of
surfactants on the drop-interface coalescence was presented using a hybrid
front-tracking/level-set method (Shin & Juric (2009), Shin et al. (2017), Shin
et al. (2018)). The surfactant transport equations were fully-coupled to the
Navier-Stokes equations in which the surface tension depends on the
interfacial surfactant concentration through a nonlinear Langmuir equation of
state. The numerical framework has been validated against the experimental
work presented by Blanchette & Bigioni (2006) for the surfactant-free
coalescence dynamics, and the inertio-viscous scaling-lows regarding the
temporal evolution of the neck towards its capillary singularity presented by
Eggers (1993). We have selected a surfactant-free base case characterised by
the dimensionless quantities of $Oh=0.02$ and $Bo=10^{-3}$. The former
parameter ensures a rich dynamics in the inertia-viscous-capillary flow
regime, whereas the latter parameter ensures that gravity forces do not affect
the dynamics of the system, which could mask effects related to the presence
of surfactants.

For insoluble surfactants, we have demonstrated that Marangoni stresses drive
motion from regions of high-surfactant concentration (low surface tension) to
low concentration (high tension) regions, resulting in retardation of the
interfacial dynamics. This immobilising effect of the surfactants as a result
of the Marangoni stresses is also observed via the strong reduction in the
maximum stretching (by dampening the strength of the capillary waves which
converge in the drop summit) of the drop and kinetic energy. We have also
shown, that the condition for the capillary singularity is the existence of
two stagnation points close to the drop neck, which leads to the generation of
vorticity in this area (‘vortex-ring’). In the presence of surfactants,
Marangoni-induced flow suppresses one of the stagnation points resulting in
the advection of vorticity towards the liquid bulk and the reopening of the
neck. This effect is strongest for insoluble surfactants and, for soluble
surfactants, is maximal for an intermediate range of solubility and sorption
kinetic parameter values.

Future directions are related to the performance of numerical simulations
featuring three-dimensional behaviours occurring for large Bond numbers.
Recently, the experimental work performed by Dong et al. (2019) suggested that
the presence of surfactants induces the rupture of the interface (i.e., hole
formation) in an off-axis location at high Bond numbers. Thus, a fully three-
dimensional retracting capillary wave will certainly affect the behaviour of
the system rising a more complex coalescence dynamics, and constitute a
fruitful area of future research.

### Acknowledgements

This work is supported by the Engineering & Physical Sciences Research
Council, United Kingdom, through a studentship for CRCA in the Centre for
Doctoral Training on Theory and Simulation of Materials at Imperial College
London funded by the EPSRC (EP/L015579/1), and through the EPSRC MEMPHIS
(EP/K003976/1) and PREMIERE (EP/T000414/1) Programme Grants. CRCA also
acknowledges the funding and technical support from BP through the BP
International Centre for Advanced Materials (BP-ICAM), which made this
research possible. OKM also acknowledges funding from PETRONAS and the Royal
Academy of Engineering for a Research Chair in Multiphase Fluid Dynamics. We
also acknowledge HPC facilities provided by the Research Computing Service
(RCS) of Imperial College London for the computing time. DJ and JC acknowledge
support through computing time at the Institut du Developpement et des
Ressources en Informatique Scientifique (IDRIS) of the Centre National de la
Recherche Scientifique (CNRS), coordinated by GENCI (Grand Equipement National
de Calcul Intensif) Grant 2020 A0082B06721. The numerical simulations were
performed with code BLUE (Shin et al. (2017)) and the visualisations have been
generated using ParaView.

Declaration of interests: The authors report no conflict of interest.

### APPENDIX: mesh study

On account of showing mesh independent results, we have monitored the temporal
variation of the liquid-volume of the system for a resolution of $(386)^{3}$,
which has been used throughout the entire study. Figure 11 shows the plot
profiles for the surfactant-free and the surfactant-laden cases. It is evident
that the numerical method is capable of capturing the rich interfacial
dynamics with a conservation of volume under $10^{-3}\%$. With respect to the
accuracy of the surfactant equations, we refer to Shin et al. (2018), who
carefully benchmarked the formulation and numerical implementation of the
surface gradients of surfactant concentration and surface tension. For the
studied phenomenon, we observed the conservation of surfactant mass under
$10^{-2}\%$ for all the surfactant-laden cases. Additionally, extensive mesh
studies for capillary phenomena, using the same numerical method, had been
previously reported (Batchvarov et al., 2020; Constante-Amores et al., 2020,
2021).

Figure 11: Relative variation of the liquid volume for the surfactant-free and
full-Marangoni case $|\tau|>0$, for $Oh=0.02$, $Bo=10^{-3}$, $\beta_{s}=0.5$,
$Pe_{s}=100$ and $\Gamma_{o}=0.5\Gamma_{\infty}$.

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# NullaNet Tiny: Ultra-low-latency DNN Inference Through Fixed-function
Combinational Logic

Mahdi Nazemi*, Arash Fayyazi*, Amirhossein Esmaili,
Atharva Khare, Soheil Nazar Shahsavani, and Massoud Pedram *M. Nazemi and A.
Fayyazi contributed equally to this work.  School of Electrical and Computer
Engineering
University of Southern California, Los Angeles, California 90089-0001
Email: {mnazemi, fayyazi, esmailid, aakhare, nazarsha<EMAIL_ADDRESS>
## Extended Abstract

While there is a large body of research on efficient processing of deep neural
networks (DNNs) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], ultra-low-latency
realization of these models for applications with stringent, sub-microsecond
latency requirements continues to be an unresolved, challenging problem.
Field-programmable gate array (FPGA)-based DNN accelerators are gaining
traction as a serious contender to replace graphics processing unit/central
processing unit-based platforms considering their performance, flexibility,
and energy efficiency. NullaNet (2018) [32], LUTNet (2019) [33], and LogicNets
(2020) [34] are among accelerators specifically designed to benefit from
FPGAs’ capabilities.

This paper presents NullaNet Tiny, an across-the-stack design and optimization
framework for constructing resource and energy-efficient, ultra-low-latency
FPGA-based neural network accelerators. The key idea is to replace expensive
operations required to compute various filter/neuron functions in a DNN with
Boolean logic expressions that are mapped to the native look-up tables (LUTs)
of the FPGA device (examples of such operations are multiply-and-accumulate
and batch normalization).

Fig. 1 depicts different steps of NullaNet Tiny’s design and optimization
flow.

Figure 1: A high-level overview of NullaNet Tiny’s design and optimization
flow.

The training module is responsible for both quantization-aware training (QAT)
and fanin-constrained pruning (FCP) while the logic minimization module
performs two-level and multi-level logic minimization in addition to retiming.
These modules’ ultimate goal is to convert each filter/neuron into a highly
optimized LUT-based implementation by enumerating all its possible input
combinations, recording encountered output values, and optimizing truth tables
reflecting those input-output combinations.

QAT refers to the quantization of activations to binary, bipolar, or multi-bit
values during neural network training. One of the critical differences between
NullaNet Tiny and prior work is that it employs different activation functions
for different layers to yield higher accuracy. For example, if the inputs to a
DNN assume both negative and positive values, it uses an activation function
such as the sign function to capture the range of inputs better. On the other
hand, if a set of values can only assume non-negative numbers, it relies on
the parameterized clipping activation (PACT) [9] function to quantize
activations.

FCP applies fanin constraints to individual filters/neurons such that the
number of inputs to each filter/neuron is small enough to make a realization
based on input enumeration (as described in NullaNet [32]) feasible. In this
work, FCP is either based on the alternating direction method of multipliers
[35] or gradual pruning [11].

Finally, functions of different filters/neurons are represented using truth
tables which are then fed to the logic minimization module. This paper employs
the ESPRESSO-II logic minimizer [36] for two-level logic minimization and
Xilinx Vivado for multi-level logic minimization and retiming.

Our experimental evaluations summarized in Table I demonstrate the superior
performance of NullaNet Tiny in terms of accuracy, inference latency, and
resource utilization compared to prior art FPGA-based DNN accelerators on the
jet substructure classification (JSC) [37] task (the three architectures are
based on those described in LogicNets [34]).

TABLE I: Comparison of accuracy & hardware realization metrics of NullaNet
Tiny and LogicNets [34] on the JSC task.

Arch. | Accuracy | LUTs | FFs | $\bm{f_{max}}$
---|---|---|---|---
(% Inc.) | (Dec.) | (Dec.) | (Inc.)
JSC-S | 69.65% | 39 | 75 | 2,079 MHz
(+1.85) | (5.50$\bm{\times}$) | (3.30$\bm{\times}$) | (1.30$\bm{\times}$)
JSC-M | 72.22% | 1,553 | 151 | 841 MHz
(+1.73) | (9.30$\bm{\times}$) | (2.90$\bm{\times}$) | (1.40$\bm{\times}$)
JSC-L | 73.35% | 11,752 | 565 | 436 MHz
(+1.55) | (3.20$\bm{\times}$) | (1.40$\bm{\times}$) | (1.02$\bm{\times}$)

At about the same level of classification accuracy, compared to Xilinx’s
LogicNets [34], our design achieves 2.36$\times$ lower latency and
24.42$\times$ lower LUT utilization (please note that not all reported clock
frequencies are realizable on the target Xilinx VU9P FPGA). Similarly,
compared to Google’s optimized design [38], our design has a 9.25$\times$
lower latency.

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