Traces of surfactants can severely limit the drag reduction of
superhydrophobic surfaces

François J. Peaudecerfa, Julien R. Landelb, Raymond E. Goldsteina, and Paolo
Luzzatto-Fegizc,1

a Department of Applied Mathematics and Theoretical Physics, Center for
Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge
CB3 0WA, UK

b School of Mathematics, University of Manchester, Manchester M13 9PL, UK

c Department of Mechanical Engineering, University of California, Santa
Barbara, 93106, USA

1<EMAIL_ADDRESS>
## Abstract

Superhydrophobic surfaces (SHSs) have the potential to achieve large drag
reduction for internal and external flow applications. However, experiments
have shown inconsistent results, with many studies reporting significantly
reduced performance. Recently, it has been proposed that surfactants,
ubiquitous in flow applications, could be responsible, by creating adverse
Marangoni stresses. Yet, testing this hypothesis is challenging. Careful
experiments with purified water show large interfacial stresses and,
paradoxically, adding surfactants yields barely measurable drag increases.
This suggests that other physical processes, such as thermal Marangoni
stresses or interface deflection, could explain the lower performance. To test
the surfactant hypothesis, we perform the first numerical simulations of flows
over a SHS inclusive of surfactant kinetics. These simulations reveal that
surfactant-induced stresses are significant at extremely low concentrations,
potentially yielding a no-slip boundary condition on the air–water interface
(the ”plastron”) for surfactant amounts below typical environmental values.
These stresses decrease as the streamwise distance between plastron stagnation
points increases. We perform microchannel experiments with thermally-
controlled SHSs consisting of streamwise parallel gratings, which confirm this
numerical prediction. We introduce a new, unsteady test of surfactant effects.
When we rapidly remove the driving pressure following a loading phase, a
backflow develops at the plastron, which can only be explained by surfactant
gradients formed in the loading phase. This demonstrates the significance of
surfactants in deteriorating drag reduction, and thus the importance of
including surfactant stresses in SHS models. Our time-dependent protocol can
assess the impact of surfactants in SHS testing and guide future mitigating
designs.

##  1 Introduction

Super-hydrophobic surfaces (SHSs) combine hydrophobic surface chemistry and
micro- or nano-scale patterning in order to retain a network of air pockets
when exposed to a liquid (see e.g. reviews 1, 2, 3). Since a large portion of
the interface between the solid wall and the liquid is replaced by an
air–liquid interface, which can be considered almost as a shear-free surface
(known as a “plastron”), SHSs could be used to obtain significant drag
reduction in fluid flow applications (4, 5). Microchannel tests have recorded
drag reductions of over 20% (e.g. 6, 7, 8, 9, 10, 11) and rheometer tests
reported slip lengths of up to 185 $\mu$m (12). Turbulent flow experiments
have reduced drag by up to 75% (13, 14, 15, 16). However, a wide range of
experiments have provided inconsistent results, with several studies reporting
little or no drag reduction (16, 17, 18, 19, 20, 21, 22, 23, 24, 25).

A key step towards solving this puzzle has come with the realization that
surfactants could induce Marangoni stresses that impair drag reduction. This
was first hypothesized to account for experiments (23, 24) which revealed
little measurable slip, in contradiction with available theoretical
predictions based on the absence of surfactants (26, 27, 28, 29, 30, 31).
Following this hypothesis, surfactants naturally present in water would adsorb
onto the air–water interface, as sketched in Fig. 1A, and they would be
advected by the flow and therefore accumulate at downstream stagnation points,
where the interface terminates in a three-phase contact line. The resulting
surfactant gradient would then yield a Marangoni stress resisting the fluid
motion, thereby decreasing slip and increasing drag (Fig. 1B). Traditional
models of SHSs are surfactant-free, and therefore do not account for this
additional drag. This is especially concerning for marine applications, since
it is well documented that seawater contains significant amounts of
surfactants (32). Rivers, estuaries, as well as fog also show significant
levels of both synthetic and natural surfactants (33, 34).

Recent experiments have shown that while a nominally clean flow already
displayed slip that was several times below predictions of surfactant-free
theories, adding large amounts of surfactant had a barely measurable effect
(35). This counter-intuitive result appears to undermine the surfactant
hypothesis, which would come with the expectation of a strong sensitivity to
surfactant concentration. Other phenomena, such as thermal Marangoni effects
(e.g. due to heating from a PIV laser), or interface curvature (36, 37), must
be assessed. Proving or disproving the surfactant hypothesis, while resolving
the above paradox, is essential to design SHSs that can achieve large,
reliable drag reduction.

Figure 1: Presence of surfactants as contaminant can generate Marangoni forces
on a SHS. (A) Surfactants present in water adsorb at the air–water interface
of the plastron of SHSs. (B) In the presence of an external flow, surfactants
distribute in gradients between stagnation points. From these gradients result
Marangoni forces resisting the flow and immobilizing the interface.

##  2 Simulations with detailed kinetics

To investigate the flow at extremely low surfactant concentrations, which are
difficult to achieve and control in experiments, we developed a computational
tool for simulating surfactant-laden flows over a SHS whose plane geometry is
shown in Fig. 2A (see Appendix for details about the simulations). Neglecting
air viscosity, we consider a two-dimensional air–water interface of length $g$
on which surfactants from the bulk can adsorb/desorb and generate Marangoni
forces. This interface (shown in pink in Fig. 2 A–C) is on the top of a
chamber of height $H$ and bounded by no-slip ridges of length $\ell/2$ each.
The top of the chamber, of length $g+l$, represents a basis element of a SHS.
The flow in the chamber is forced by a Poiseuille profile $u(y)$ at $x=0$. In
Fig. 2 B–D we show simulation results from very low surfactant concentration
to high concentration in the bulk, using the properties of the well-
characterized surfactant sodium dodecyl sulfate (SDS) (38). For very low
concentration, $c_{0}=${10}^{-6}\text{\,}\mathrm{m\textsc{m}}$$, the
corresponding flow velocity is shown in Fig. 2B, with gap length
$g=$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, wall length
$\ell=$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, channel height
$H=$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and forced Poiseuille flow at
$x=0$ of peak speed $50\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$. The flow field is essentially identical to that
for pure water, i.e. $c_{0}=0$. Fig. 2D shows the effect of the bulk
surfactant concentration on the characteristic drag $\langle\tau\rangle$,
which has been spatially averaged over one SHS unit of length $g+\ell$. The
data are normalized by the corresponding wall shear stress $\tau_{\mathrm{P}}$
of the forcing Poiseuille flow. The case
$c_{0}=${10}^{-6}\text{\,}\mathrm{m}\mathrm{M}$$ is highlighted by the left
arrow in Fig. 2D. Steady simulations with progressively larger bulk
concentration, holding other parameters fixed, yield the curve shown in Fig.
2D. Fig. 2C shows the flow velocity for
$c_{0}=${10}^{-2}\text{\,}\mathrm{m}\mathrm{M}$$. A large Marangoni stress
appears and the slip velocity is reduced by more than one order of magnitude,
practically reaching a no-slip boundary condition at the plastron. The
corresponding drag is shown by the right arrow in Fig. 2D.

For a bulk concentration of just ${10}^{-2}\text{\,}\mathrm{m\textsc{m}}$
(that is, ${10}^{-5}\text{\,}\mathrm{mol}\text{\,}{\mathrm{L}}^{-1}$,
corresponding to less than 10 grams of surfactants per m3 of water), the drag
flattens to the no-slip asymptote, after which adding more surfactants has
little effect. These results explain why adding surfactants in experiments has
a barely measurable effect in most experiments, since the transition to a
regime where Marangoni stresses dominate occurs at extremely small
concentrations. Such small concentrations are very difficult to achieve in
laboratory experiments, where controlling surfactant contamination (e.g. from
SHS manufacturing and materials, water handling, micro-PIV beads, or any
surface or fluid, including air, in contact with the water flowing through the
microchannel) is exceedingly difficult. Needless to say, such levels of
cleanliness are not found in flow applications. In addition, surfactants at
such low concentrations are essentially impossible to detect using a classical
tensiometer apparatus, since the corresponding decrease in surface tension, in
a static fluid, is negligible (38).

The results presented in Fig. 2D are generic to other types of surfactant, as
shown in Appendix and Fig. 5. We also note that the properties of the SDS
surfactant used in our study do not induce the strongest Marangoni stresses.
In fact, its effect is rather mild compared to the surfactants reported in
(39). In normal flow applications, we should therefore assume that stronger
surfactants than SDS are likely to be present, thus deteriorating even more
the performance of the SHS.

The inevitability of surfactants suggests that the main strategy to minimize
Marangoni stresses is to optimize the SHS geometry. To test this hypothesis,
we simulate flow over SHSs of varying interface length $g$. By increasing the
distance $g$ between the upstream and downstream stagnation points, we reduce
the average surfactant gradient over the plastron. Fig. 2E shows drag versus
interface length for $c_{0}=${10}^{-2}\text{\,}\mathrm{m}\mathrm{M}$$ (plotted
with red squares). We find that the drag over a contaminated surface is very
sensitive to this change of geometry. The drag is also significantly larger
than the idealized results for a perfectly clean flow (plotted with red
diamonds) over a large range: $0.01\leq g\leq$10\text{\,}\mathrm{mm}$$.

Figure 2: Numerical simulations of surfactant-laden flows over a SHS. (A) 2D
geometry of the simulations for a periodic chamber (i.e. one SHS unit) with
gratings of length $g=$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, separated
with ridges of length $\ell=$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, and
with forcing Poiseuille flow at $x=0$. (B) Streamwise velocity $u$ for bulk
surfactant concentration $c_{0}=${10}^{-6}\text{\,}\mathrm{m\textsc{m}}$$,
exhibiting slip on the plastron; (C) at
$c_{0}=${10}^{-2}\text{\,}\mathrm{m\textsc{m}}$$ no slip is reached on the
plastron. (D) Average normalised drag versus surfactant concentration. An
asymptote is reached already at concentrations well below those found in the
environment. (E) Drag versus grating length, for perfectly clean water (red
diamonds) and flow with bulk concentration of
$c_{0}=${10}^{-2}\text{\,}\mathrm{m\textsc{m}}$$ (red squares), showing that
drag reduction in surfactant-laden flows is very sensitive to grating length.
All simulations are performed with a peak velocity of the forcing Poiseuille
flow of $50\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-1}$.

##  3 Experiments show reduced slip

Figure 3: $\mu$-PIV set-up and experimental velocity measurements with
constant background flow. (A) Side view of the inverted PDMS microchannel with
a textured SHS on top. The microchannel is in contact with a thermal
controller and lies above the water objective of the confocal microscope. (B)
Geometry of the streamwise parallel rectangular gratings forming the SHS. (C)
Cross-section in $(y,z)$ plane at $x=g/2$ of the (inverted) microchannel. The
measuring planes of the $\mu$-PIV at heights $z_{i}$ are indicated with
different colors. (D) Lateral distribution of the streamwise velocity at
different heights $z_{i}$ from the plastron, in the case of short lanes
$g=$2\text{\,}\mathrm{m}\mathrm{m}$$. The profile is centered on a grating,
with the edges of the ridges in dashed magenta lines. Experiments show no
signicant slip at the plastron compared with clean case simulations. (E) Same
as in (D) in the case of long lanes $g=$30\text{\,}\mathrm{m}\mathrm{m}$$.
Slip velocity at the plastron is still three times smaller than predictions
from clean case simulations.

To verify the effect of lane length on drag reduction, we performed
experiments utilizing micro-particle image velocimetry ($\mu$-PIV) on a
confocal microscope, and measured the velocity field of gravity-driven
microchannel flows in planes parallel to a SHS (see Fig. 3 A–C and Appendix
for details). Similar to earlier work (13, 10, 24), our SHSs were made of
hydrophobic polydimethylsiloxane (PDMS) using soft photolithography
techniques. The SHSs consisted of streamwise parallel rectangular lanes or
gratings, as shown in Fig. 3B. The lane width
$w=$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and ridge width
$r=$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ were kept fixed in the
experiments. The microchannel height varied slightly, $100\leq
H\leq$120\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. We sealed the PDMS chamber
using a glass coverslip held in place through Van der Waals adhesion, without
surface treatment, to avoid surfactant contamination and to preserve PDMS
hydrophobicity. The glass coverslip and chamber were maintained at a fixed
temperature within $0.1^{\circ}\mathrm{C}$ accuracy through cooling elements.
This minimized any potential thermal Marangoni effect (as discussed in
Appendix) and prevented significant condensation inside the gratings, thus
ensuring stability of the plastron over several hours. We used purified water
in the experiments and cleaned all possible surfaces in contact with water and
micro-beads following a strict cleaning protocol (Appendix). The microchannel
is connected to inlet and outlet reservoirs whose heights are adjusted
separately to allow control of both the background pressure gradient and
hydrostatic pressure in the microchannel. The plastron was maintained flat
through adjustment of the hydrostatic pressure and the interface position was
located with an accuracy of $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

The symbols in Fig. 3D show experimental velocity profiles at several
horizontal slices in the case of short lanes,
$g=$2\text{\,}\mathrm{m}\mathrm{m}$$ (standard deviations in the velocity are
indicated with vertical bars). For comparison, Fig. 3D also shows surfactant-
free numerical predictions (plotted with solid lines), for the same three-
dimensional geometry. Similar to theoretical calculations assuming free slip
at the plastron, the surfactant-free numerical simulations predict a very
large slip, $\max(u/U)\approx 0.85$ at $z_{i}/H=1.6$% from the plastron with
$U$ the mean flow speed, whereas experimental data at $z/H\leq 2\%$ show no
statistically significant slip. In the case of long lanes (Fig. 3E),
$g=$30\text{\,}\mathrm{m}\mathrm{m}$$, experimental data reveal some slip,
which is still significantly lower than numerical and theoretical results
(26), by approximately $70$%. These results agree with our surfactant-laden
simulations reported earlier in Fig. 2E: for a given background flow and
surfactant concentration, increasing the lane length increases the slip
velocity, bringing it closer to predictions from surfactant-free models. This
consistency strongly suggests that the reduced slip velocity observed on our
experimental SHSs with respect to theoretical predictions comes indeed from
the presence of surfactants on the plastron surface.

##  4 Pressure-relaxation experiments for surfactant effect

Figure 4: Velocity measurements in pressure-relaxation experiments. (A)
Kymograph showing the time evolution of the lateral profile of the mid-gap
streamwise velocity measured at
$(x=g/2,z=$-5\text{\,}\mathrm{\SIUnitSymbolMicro m}$)$ after a constant
loading speed $U\approx$4.1\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$$.
The two horizontal black dashed lines designate the start and end of the
pressure-head reduction to zero. Bottom inset is a picture showing the
position of the two gratings studied (appearing in light grey). The kymograph
clearly shows a strong backflow over the lanes but not over the ridge. (B)
Profiles from A taken along the red, blue and green dashed vertical lines at
the centerline of each grating and of the ridge, respectively. Red symbols are
hidden by blue symbols as both gratings have the same profile. The standard
deviation of all the data is smaller than the symbol size. (C) Peak backflow
velocity measured when the background pressure vanishes (corresponding to the
second vertical dashed line in B), versus the mean loading phase speed for the
two sets of experiments conducted for
$U\approx$2.3\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$$ and
$U\approx$4.1\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$$. (D) Ensemble
average of the normalised backflow velocity multiplied by time for the two
sets of experiments. Standard deviations are shown in lighter shading.

To prove experimentally the surfactant hypothesis while circumventing the
difficulty of removing significant traces of surfactants, we designed time-
dependent pressure-relaxation experiments to reveal in-situ the presence of
surfactant-induced stresses. All the experiments were conducted with the
apparatus previously presented in Fig. 3 A–C, with controlled temperature and
interface deflection. These experiments begin with a “loading phase”, during
which the flow is driven by a fixed, strong pressure gradient and allowed to
reach steady state (see Appendix for further details about the experimental
protocol). The driving pressure is then rapidly decreased to zero while
keeping the hydrostatic pressure approximately constant. The change in
background pressure-gradient is simply achieved by moving the stage onto which
the inlet reservoir is attached to the exact level for which both inlet and
outlet water levels are at the same vertical position. The motion of the stage
was conducted very fast, within less than four seconds. During the loading
phase, which was maintained for four minutes to ensure a steady state was
reached, we measured a classical Poiseuille profile in the microchannel (Fig.
6), giving a mean loading speed $U$ (see Appendix for further details on the
technique used to measure $U$). Our measurements do not reveal any significant
slip on the SHS side during the loading phase. However, a clear and somewhat
unexpected backflow, with negative values (shown in blue in Fig. 4A), can be
observed after the background pressure-gradient vanishes.

Fig. 4A shows the time evolution of the lateral profile of the streamwise
velocity measured at $x=g/2$ and $z=$-5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ for a grating with $g=$30\text{\,}\mathrm{m}\mathrm{m}$$. The background
pressure-gradient decreases from a large constant value during the loading
phase (mean loading speed in the microchannel
$U\approx$4.1\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$$ for
$t\leq$5\text{\,}\mathrm{s}$$) to zero at $t=$7.2\text{\,}\mathrm{s}$$.

As the flow in the microchannel is in the Stokes regime
($Re=HU_{\mathrm{max}}/\nu\approx 0.6$, with $\nu$ the kinematic viscosity of
water), we expect that, in the absence of surfactants, the velocity should
decrease from positive values to zero instantaneously and everywhere as soon
as the background forcing vanishes. This monotonic behavior can be observed in
Fig. 4A for the velocity measured above the ridge (see also green triangles in
Fig. 4B). In contrast, the velocity above the plastron (plotted in red and
blue in Fig. 4B for the middle of the left and right gratings, respectively)
decreases sharply to large negative values during the stage motion,
corresponding to a flow opposed to the background pressure gradient. This
backflow persists for a long time after the background pressure gradient
vanishes. In this experiment, it decays over approximately one minute. The
same qualitative results and trends are obtained independently of the choice
of grating, the direction of the loading flow and its intensity, as long as it
is sufficiently strong.

During the loading phase, owing to large advection in the flow compared with
diffusion and adsorption/desorption kinetics, surfactants at the interface are
transported towards the downstream stagnation point of the lane. The
concentration of surfactants increases significantly near this point, whereas
it decreases everywhere else along the interface. This regime is likely to be
analogous to the stagnant cap regime described for air bubbles rising in water
(40). When the background pressure gradient vanishes, the surfactant-induced
stresses are not opposed by viscous stresses any longer, and a Marangoni
backflow develops to homogenize the surfactant concentration at the plastron,
as shown in Fig. 4B. A shear flow establishes across the height of the
chamber, owing to the wall shear stress on the opposite side of the plastron
($z=-H$). As the distribution of surfactant becomes more uniform with time,
surfactant-induced stresses decrease and the backflow velocity diminishes, as
shown by the nonlinear temporal trend in Fig. 4B.

We find that the maximum velocity of the backflow $|u_{\mathrm{m}}|$, measured
when the background pressure gradient vanishes
($t\approx$7\text{\,}\mathrm{s}$$), increases with the magnitude of the mean
load speed $U$, as shown in Fig. 4C. This is consistent with a sharpening of
the concentration gradient near the downstream stagnation point at higher
background pressure gradients during the loading phase, which then results in
stronger Marangoni stresses driving the backflow. The observation that the
plastron always displays a transient that reverses direction relating to the
loading flow is a strong indication that Marangoni stresses are driven by
surfactant accumulation, rather than by thermal gradients set up by laser or
by other elements of the experimental apparatus.

To provide additional support to the surfactant hypothesis, as well as to show
that the occurrence of the backflow is largely independent of the type of
surfactant (and therefore of the associated kinetics), we develop a model for
the backflow, and compare the resulting scaling for the plastron velocity to
experiments. As surfactant diffusion is negligible at the air–water interface
compared with advection, we model the backflow using a one-dimensional time-
dependent advection equation for the interfacial surfactant concentration
(this model is explained in detail in Appendix). We assume
adsorption/desorption fluxes are also negligible. The surfactant transport
equation is coupled with a viscous-Marangoni stress balance. Using a
similarity solution, we find that the magnitude of the backflow decreases in
time such that $|u|\propto 1/t$, which is in agreement with the experimental
results plotted in Fig. 4D. Fig. 4D shows the time evolution of the ensemble
average (symbols) and standard deviations (light shadings) of the normalized
backflow velocity $|u|/U$ multiplied by time. We show data for two sets of
experiments conducted with
$U\approx$2.3\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$$ and
$U\approx$4.1\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$$ in the loading
phase, plotted respectively in orange and black. The model is not valid at
early time, $t\leq$10\text{\,}\mathrm{s}$$, due to the non-instantaneous
decrease of the background pressure-gradient occurring between the two
vertical dashed lines, owing to the finite time required for the stage motion
(approximately 3 to 4 seconds). At late times $t\geq$60\text{\,}\mathrm{s}$$,
the plateau observed for the large forcing experiments seems to end, as the
data decrease again. This is consistent with the fact that the model, which
assumes a semi-infinite lane, is not valid at late times when the effect of
the opposite stagnation point (i.e. the upstream stagnation point during the
loading phase) is felt. Moreover, we can notice in Fig. 4D that the
characteristic time scale of the backflow, measured as the mean of $(|u/U|t)$
in the plateau region, increases with the forcing speed $U$. This indicates a
nonlinear effect probably due to changes in the surfactant distribution at
$t=0$ for different $U$.

##  5 Outlook

An important conclusion is that quantitative models of superhydrophobic slip
must also account for surfactant effects, or they can drastically overestimate
drag reduction. This is consistent, for example, with established approaches
for predicting the rise speed of small gas bubbles, which are known to be
sensitive to surfactant-induced Marangoni stresses (40). It would be valuable
to build reduced-order models to enforce efficiently surfactant effects in
Navier-Stokes solvers, obviating the need for a full solution of the complex
coupled surfactant transport problem.

From a practical standpoint, our numerical and experimental findings point to
the need to focus efforts on textures with large distances between upstream
and downstream stagnation points. This explains why several experimental works
have reported that the most consistent performance was achieved with long
gratings (16, 41), whereas disorganized textures (which are attractive from a
manufacturing standpoint) have been the least effective (25). We can also note
that some of the largest slip lengths reported in the literature were achieved
with annular gratings in a circular rheometer (12). Annular gratings are
effectively infinitely long lanes, without stagnation points, thus preventing
the formation of surfactant concentration gradient and adverse Marangoni
stresses. However, applications of SHSs with annular geometries are limited.

From a methodological standpoint, our experimental tests involving pressure
relaxation also provide a new and simple way of measuring the magnitude of
surfactant-induced effects, for a SHS immersed in a given liquid. The presence
of a backflow immediately reveals a plastron whose dynamics are strongly
susceptible to surfactants. This can be a useful test to guide the design of
SHSs that are unimpaired by surfactant Marangoni stresses. Overall, we believe
this work constitutes a significant advance in our fundamental understanding
of superhydrophobic drag reduction, moving the field closer to SHSs that
perform reliably in realistic applications.

##  6 Materials and Methods

###  6.1 Numerical simulations

The numerical simulations involve solving eight partial differential equations
for the transport of mass, momentum and surfactant in the fluid interior and
along the air–water interface (40) (see Appendix for more details). The ridges
and bottom walls of the chamber have a no-slip boundary condition. At the
air–water interface we enforce continuity of the viscous stress with the
Marangoni stress, and continuity of the bulk velocity with the interfacial
velocity. Note that air drag is negligible here, and is not considered.
Surfactants present in the interior layer near the plastron are adsorbed or
desorbed following specific kinetics. We have chosen a well-characterized
surfactant, namely sodium dodecyl sulfate (SDS), as our case study surfactant.
The SDS properties are well described by Frumkin kinetics (38). These
equations are solved using a finite element method from COMSOL Multiphysics®
software, with a Poiseuille profile as a forcing inlet condition and a weak
constraint added to a free-slip boundary condition in order to capture the
Marangoni effects at the air–water interface. A refined mesh at the interface
and near the endpoints, together with higher order elements of discretisation,
guarantee accuracy of the numerical solutions (see Appendix for more details).

###  6.2 Microchannel preparation

Microchannels of PDMS (Sylgard® 184) with a top surface consisting of long
rectangular gratings were fabricated using soft photolithography (42) (see
Appendix). They were bonded to glass coverslips through natural PDMS adhesion.
All glass slides and tubing were washed thoroughly with purified water before
use in the experiments. The cleaning protocol is described in Appendix.

###  6.3 Laser Scanning Confocal Microscopy

To measure the fluid velocity in the microchannel, we washed and suspended
micro-beads (LifeTechnologies FluoSphere® carboxylate
$0.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter yellow/green 505/515) in
purified water (using Milli Q® water purification system, EMD Millipore) and
injected the resulting solution at $0.08\%$ solids in the chamber. Using a
spinning disk confocal microscope for green fluorescence (Yokogawa CSU-X1
mounted on Zeiss Axio Observer Z1, with laser line $488\text{\,}\mathrm{nm}$
with power at the objective level less than
$155\text{\,}\mathrm{\SIUnitSymbolMicro W}$), we captured image sequences at
distinct $z$-planes of the chamber on high sensitivity EMCCD (Photometrics
Evolve 512 Delta), at a rate of approximately 10 frames per second. We used a
water objective (Olympus LUMPLFLN 60XW) to guarantee a proper determination of
the position of the imaging plane and allow deep imaging without loss of
fluorescence brightness. A bright field picture of the ridges and gratings
allowed to determine the position of the SHS with respect to the field of
imaging. To prevent heating of the microchannel during imaging, it was placed
in thermal contact with a heat conducting plate, itself in contact with
Peltier elements driven by a thermoelectric controller (Laird Technologies
MTTC1410). To absorb the heat from the objective, we mounted a second Peltier
element around it, coupled to a similar thermoelectric controller to maintain
its temperature at $23\pm 0.1^{\circ}\mathrm{C}$. Fans were also used to cool
the hot plates of the Peltier elements.

###  6.4 Flow control

To ensure the driving pressure has no fluctuations, the flow of water with
fluorescent particles is driven by gravity. The microchannel is connected with
flexible tubing (Tygon® ND-100-80) to two reservoirs (polypropylene tube,
Eppendorf), whose absolute and relative heights can be varied independently.
By varying the absolute position of the two reservoirs while keeping their
relative position such that no flow is observable in the chamber, we minimized
the curvature of the air–water interface in the chamber by adjusting the
hydrostatic pressure. The maximum interface deflection over the width of the
interface was less than $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$. By
adjusting the relative position of the reservoirs, varying background pressure
gradients could be imposed in the channel to generate flow. The variation of
relative position was performed using a motorized stage (Thorlabs NRT150/M)
driven by a precision controller (Thorlabs APT BSC201) guaranteeing a position
precision of $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and rapid changes in
pressure gradients (maximum acceleration:
$50\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-2}$, maximum speed:
$50\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$).

###  6.5 Image analysis and micro-PIV

The images obtained by confocal microscopy were analyzed with MATLAB® using a
customized version of the particle image velocimetry (PIV) toolbox mPIV (43).
The correlation functions between adjacent frames were rescaled to correct for
non uniform time steps between frames, and the resulting correlations
functions were added over a pack of 10 to 30 frames before searching for peak
correlations. This allowed to correct for Brownian motion uncertainties. For
the time-relaxation experiments, the velocity profile in the chamber during
the loading phase at high flow rate was obtained by measuring the mean length
of the fluorescent particle streaks on the image and dividing by the exposure
time.

##  7 Acknowledgments

We thank D. Page-Croft and the engineers of the G.K. Batchelor Laboratory for
their technical support. We gratefully acknowledge financial support from the
Raymond and Beverly Sackler Foundation, the Engineering and Physical Sciences
Research Council, the European Research Council Grant 247333, Mines ParisTech,
the Schlumberger Chair Fund, as well as from Churchill and Magdalene Colleges,
Cambridge.

## References

  * (1) D. Quéré.  Wetting and Roughness.  Ann. Rev. Materials Research, 38(1):71–99, August 2008.
  * (2) J. P. Rothstein.  Slip on Superhydrophobic Surfaces.  Ann. Rev. Fluid Mech., 42(1):89–109, January 2010.
  * (3) M. A. Samaha, H. V. Tafreshi, and M. Gad-el Hak.  Superhydrophobic surfaces: From the lotus leaf to the submarine.  Comptes Rendus Mécanique, 340(1–2):18 – 34, 2012.  Biomimetic flow control.
  * (4) T. L. Liu and C. J. C. Kim.  Turning a surface superrepellent even to completely wetting liquids.  Science, 346(6213):1096–1100, November 2014.
  * (5) Xuelin Tian, Tuukka Verho, and Robin H. A. Ras.  Moving superhydrophobic surfaces toward real-world applications.  Science, 352(6282):142–143, 2016.
  * (6) J. Ou, B. Perot, and J. P. Rothstein.  Laminar drag reduction in microchannels using ultrahydrophobic surfaces.  Phys. Fluids, 16(12):4635, 2004.
  * (7) J. Ou and J. P. Rothstein.  Direct velocity measurements of the flow past drag-reducing ultrahydrophobic surfaces.  Phys. Fluids, 17(10):103606, 2005.
  * (8) C. H. Choi and C. J. Kim.  Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface.  Phys. Rev. Lett., 96:066001, 2006.
  * (9) P. Joseph, C. Cottin-Bizonne, J. M. Benoit, C. Ybert, C. Journet, P. Tabeling, and L. Bocquet.  Slippage of water past superhydrophobic carbon nanotube forests in microchannels.  Phys. Rev. Lett., 97(15), 2006.
  * (10) P. Tsai, A. M. Peters, C. Pirat, M. Wessling, R. G. H. Lammertink, and D. Lohse.  Quantifying effective slip length over micropatterned hydrophobic surfaces.  Physics of Fluids, 21:112002, 2009.
  * (11) Elif Karatay, A. Sander Haase, Claas Willem Visser, Chao Sun, Detlef Lohse, Peichun Amy Tsai, and Rob G. H. Lammertink.  Control of slippage with tunable bubble mattresses.  Proceedings of the National Academy of Sciences of the United States of America, 110(21):8422–8426, 2013.
  * (12) C. Lee and C. J. Choi, C.-H. Kim.  Structured surfaces for a giant liquid slip.  Phys. Rev. Lett., 101:064501, 2008.
  * (13) R. J. Daniello, N. E. Waterhouse, and J. P. Rothstein.  Drag reduction in turbulent flows over superhydrophobic surfaces.  Phys. Fluids, 21(8):085103, 2009.
  * (14) B. Woolford, Prince J., Maynes D., and Webb B. W.  Particle image velocimetry characterization of turbulent channel flow with rib patterned superhydrophobic walls.  Physics of Fluids, 21(8):085106, 2009.
  * (15) Y. C. Jung and B. Bhushan.  Biomimetic structures for fluid drag reduction in laminar and turbulent flows.  Journal of Physics: Condensed Matter, 22(3):035104, 2010.
  * (16) H. Park, G. Sun, and C.-J. Kim.  Superhydrophobic turbulent drag reduction as a function of surface grating parameters.  J. Fluid Mech., 747:722–734, April 2014.
  * (17) C. Lee, C.-H. Choi, and C.-J. Kim.  Superhydrophobic drag reduction in laminar flows: a critical review.  Experiments in Fluids, 57(12):176, 2016.
  * (18) K. Watanabe, Yanuar, and H Udagawa.  Drag reduction of Newtonian fluid in a circular pipe with a highly water-repellent wall.  J. Fluid Mech., 381:225–238, 1999.
  * (19) S. Gogte, P. Vorobieff, R. Truesdell, A. Mammoli, F. van Swol, P. Shah, and C. J. Brinker.  Effective slip on textured superhydrophobic surfaces.  Phys. Fluids, 17(5):051701, April 2005.
  * (20) C. Henoch, T. N. Krupenkin, and P. Kolodner.  Turbulent drag reduction using superhydrophobic surfaces.  In 3rd AIAA Flow Control Conference, 5-8 June 2006, San Francisco, California, page 3192, 2006.
  * (21) J. Zhao, X. Du, and X. Shi.  Experimental research on friction-reduction with superhydrophobic surfaces.  Journal of Marine Science and Application, 6:58–61, 2007.
  * (22) C. Peguero and K. Breuer.  On drag reduction in turbulent channel flow over superhydrophobic surfaces.  In B. Eckhardt, editor, Advances in Turbulence XII, pages 233–236, 2009.
  * (23) T. J. Kim and C. Hidrovo.  Pressure and partial wetting effects on superhydrophobic friction reduction in microchannel flow.  Physics of Fluids, 24(11):112003, 2012.
  * (24) G. Bolognesi, C. Cottin-Bizonne, and C. Pirat.  Evidence of slippage breakdown for a superhydrophobic microchannel.  Phys. Fluids, 26(8):082004, 2014.
  * (25) B. Gruncell.  Superhydrophobic surfaces and their potential application to hydrodynamic drag reduction.  PhD thesis, University of Southampton, 2014.
  * (26) J. R. Philip.  Flows satisfying mixed no-slip and no-shear conditions.  Z. Angew. Math. Physik, 23(3):353–372, May 1972.
  * (27) J. R. Philip.  Integral properties of flows satisfying mixed no-slip and no-shear conditions.  Z. Angew. Math. Physik, 23(6):960–968, November 1972.
  * (28) C. Cottin-Bizonne, C. Barentin, E. Charlaix, L. Bocquet, and J.-L. Barrat.  Dynamics of simple liquids at heterogeneous surfaces: Molecular-dynamics simulations and hydrodynamic description.  The European Physical Journal E, 15:427–438, 2004.
  * (29) E. Lauga and H. A. Stone.  Effective slip in pressure-driven Stokes flow.  J. Fluid Mech., 489:55–77, 2003.
  * (30) Christophe Ybert, Catherine Barentin, Cottin-Bizonne C., Pierre Joseph, and L. Bocquet.  Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries.  Physics of Fluids, 19(12):123601, 2007.
  * (31) A. M. J. Davis and E. Lauga.  Geometric transition in friction for flow over a bubble mattress.  Physics of Fluids, 21:011701, 2009.
  * (32) R. A. Kropfli, L. A. Ostrovski, T. P. Stanton, E. A. Skirta, A. N. Keane, and V. Irisov.  Relationships between strong internal waves in the coastal zone and their radar and radiometric signatures.  J. Geophys. Res., 104:3133–3148, 1999.
  * (33) Michael A Lewis.  Chronic and sublethal toxicites of surfactants to aquatic animals: a review and risk assessment.  Water Research, 25(1):101–113, 1991.
  * (34) Maria Cristina Facchini, Stefano Decesari, Mihaela Mircea, Sandro Fuzzi, and Giuseppe Loglio.  Surface tension of atmospheric wet aerosol and cloud/fog droplets in relation to their organic carbon content and chemical composition.  Atmos. Env., 34(28):4853–4857, 2000.
  * (35) D. Schäffel, K. Koynov, D. Vollmer, H. J. Butt, and Schöeneker C.  Local flow field and slip length of superhydrophobic surfaces.  Phys. Rev. Lett., 116:134501, 2016.
  * (36) M. Sbragaglia and A. Prosperetti.  A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces.  Physics of Fluids, 19:043603, 2007.
  * (37) C. J. Teo and B. C. Khoo.  Flow past superhydrophobic surfaces containing longitudinal grooves:effects of interface curvature.  Microfluid Nanofluid, 9:499–511, 2010.
  * (38) A. J. Prosser and E. I. Franses.  Adsorption and surface tension of ionic surfactants at the air–water interface: review and evaluation of equilibrium models.  Colloids and Surfaces A: Physicochemical and Engineering Aspects, 178(1):1–40, 2001.
  * (39) C. H. Chang and E. I. Franses.  Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms.  Colloids and surfaces A, 100:1–45, 1995.
  * (40) R. Palaparthi, D. T. Papageorgiou, and C. Maldarelli.  Theory and experiments on the stagnant cap regime in the motion of spherical surfactant-laden bubbles.  J. Fluid Mech., 559:1–44, July 2006.
  * (41) R. Truesdell, A. Mammoli, P. Vorobieff, F. van Swol, and C. J. Brinker.  Drag reduction on a patterned superhydrophobic surface.  Phys. Rev. Lett., 97(4), 2006.
  * (42) J. Cooper McDonald and George M Whitesides.  Poly ( dimethylsiloxane ) as a Material for Fabricating Microfluidic Devices.  Accounts of Chemical Research, 35(July):491–499, 2002.
  * (43) Nobuhito Mori and Kuang-An Chang.  Introduction to mpiv, 2003.  http://www.oceanwave.jp/softwares/mpiv.
  * (44) M. J. Rosen and J. T. Kunjappu.  Surfactants and interfacial phenomena.  Wiley, 2012.
  * (45) A. Hourlier-Fargette, A. Antkowiak, and S. Neukirch.  Role of uncrosslinked chains in the sliding dynamics of droplets on elastomers.  ArXiv e-prints, November 2016.

##  Appendix A Supporting Model

###  A.1 Equations for surfactant-laden flows

We consider a two-dimensional microchannel flow over a finite-length
superhydrophobic surface (SHS), with the geometry presented in Fig. 2A. Since
we maintained a flat plastron in the experiments, and the present focus is on
Marangoni stresses, here we assume that the air–water interface is flat.

The continuity and Navier-Stokes equations for the flow of mass and momentum
are coupled with the transport equation for a surfactant. The two-dimensional
velocity field is ${\boldsymbol{u}}=({u},{v})$ for the $(x,y)$ directions, and
the surfactant concentration field is ${c}$. The fluid has density $\rho$ as
well as dynamic and kinematic viscosities $\mu$ and $\nu$. The surfactant has
bulk diffusivity $D$. The transport equations are

$\displaystyle\nabla\cdot\boldsymbol{u}$ $\displaystyle=$ $\displaystyle 0,$
(1) $\displaystyle\frac{\partial\boldsymbol{u}}{\partial
t}+\nabla\cdot(\boldsymbol{u}\boldsymbol{u})$ $\displaystyle=$
$\displaystyle-\frac{\nabla{p}}{\rho}+\nu\nabla^{2}\boldsymbol{u},$ (2)
$\displaystyle\frac{\partial c}{\partial t}+\nabla\cdot(\boldsymbol{u}c)$
$\displaystyle=$ $\displaystyle D\nabla^{2}c.$ (3)

The inlet conditions consist of a Poiseuille flow with mean velocity $U$ and
maximum velocity $u_{\mathrm{max}}$, and a specified bulk concentration
$c=c_{0}$, whereas at the outlet we set

$\displaystyle\frac{\partial u}{\partial x}$ $\displaystyle=$ $\displaystyle
0,$ (4) $\displaystyle\frac{\partial c}{\partial x}$ $\displaystyle=$
$\displaystyle 0.$ (5)

On the solid surfaces at $y=0$ and $y=H$

$\displaystyle\boldsymbol{u}$ $\displaystyle=$ $\displaystyle\boldsymbol{0},$
(6) $\displaystyle\frac{\partial c}{\partial y}$ $\displaystyle=$
$\displaystyle 0.$ (7)

At the plastron, an adsorption/desorption model is used to couple the
surfactant transport between the bulk and the interface

$\displaystyle D\left.\frac{\partial c}{\partial y}\right|_{I}$
$\displaystyle=$ $\displaystyle-S(c_{I},\Gamma),$ (8)
$\displaystyle\frac{\partial\Gamma}{\partial
t}+\frac{\partial({u_{I}}\,\Gamma)}{\partial x}$ $\displaystyle=$
$\displaystyle D_{\mathrm{s}}\frac{\partial^{2}\Gamma}{\partial
x^{2}}+S(c_{I},\Gamma),$ (9)

where $\Gamma$ is the interfacial concentration, $D_{\mathrm{s}}$ is the
surface diffusivity, the subscript $I$ denotes quantities at the interface,
and $S(c_{I},\Gamma)$ encapsulates the adsorption model. In practice, the
choice of adsorption kinetics turns out to be of weak importance, since
surfactant effects are already very strong at extremely low surfactant
concentrations (as shown in Fig. 2). For such low values of concentration,
different kinetics models are essentially equivalent (39). For definiteness,
here we use Frumkin kinetics

$S(c_{I},\Gamma)=\kappa_{\mathrm{a}}c_{I}(\Gamma_{\mathrm{m}}-\Gamma)-\kappa_{\mathrm{d}}\,\Gamma\,\textrm{e}^{A\Gamma/\Gamma_{\mathrm{m}}},$
(10)

where $\kappa_{\mathrm{a}},\kappa_{\mathrm{d}}$ are the adsorption and
desorption coefficients, ${\Gamma}_{\mathrm{m}}$ is the maximum packing
interfacial concentration and $A$ is the interaction coefficient (38).

On the air–water interface, the velocity field is coupled to the interfacial
surfactant distribution through a balance between viscous and Marangoni
stresses (44)

$\displaystyle v$ $\displaystyle=$ $\displaystyle{0},$ (11)
$\displaystyle\mu\left.\frac{\partial{u}}{\partial y}\right|_{I}$
$\displaystyle=$
$\displaystyle-n{R}{T}\left(\frac{\Gamma_{\mathrm{m}}}{\Gamma_{\mathrm{m}}-\Gamma}+A\frac{\Gamma}{\Gamma_{\mathrm{m}}}\right)\frac{\partial\Gamma}{\partial
x},$ (12)

where $n$ is the surfactant style constant (39), ${R}$ is the universal gas
constant and ${T}$ is the absolute temperature. The boundary condition for
$\Gamma$, where the interface meets each solid boundary at a stagnation point
(i.e. upstream stagnation point: $(x=\ell/2,y=H)$, and downstream stagnation
point: $(x=g+\ell/2,y=H)$, see Fig. 2A), is given by

$\frac{\partial\Gamma}{\partial x}=0.$ (13)

##  Appendix B Supporting Materials and Methods

###  B.1 Surfactant-laden simulations

The model presented above was implemented in COMSOL Multiphysics® in a two-
dimensional finite element numerical simulation. The geometry corresponding to
Fig. 2A was created using the values for the gap length $g$, the ridge length
$\ell$, the chamber height $H$ and the maximum forcing speed of the Poiseuille
flow $u_{\mathrm{max}}$ presented in Table 1. All the other physical
parameters of the simulations are also presented in Table 1, and correspond to
the well-characterized surfactant sodium dodecyl sulfate (SDS). The SDS
properties are well described by Frumkin kinetics (38).

When designing the mesh of the domain, we were particularly careful about
strong possible variations of some variables near the stagnation points at the
beginning and end of the gap. The maximum size of elements near these points
is equal to $0.01\text{\,}\mathrm{\SIUnitSymbolMicro m}$. For all simulations
with $g<$1\text{\,}\mathrm{mm}$$, the maximum element size on the interface is
$0.05\text{\,}\mathrm{\SIUnitSymbolMicro m}$. For simulations with
$g=$1\text{\,}\mathrm{mm}$$ and $g=$2\text{\,}\mathrm{mm}$$, the maximum
element size on the interface is $0.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
Finally, for $g=$5\text{\,}\mathrm{mm}$$ and $g=$10\text{\,}\mathrm{mm}$$, a
coarser mesh is used in the central part of the interface,
$1\text{\,}\mathrm{mm}$ away from the endpoints, with a maximum element size
of $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$. In the bulk, the maximum element
size is $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for all simulations.

To implement the model, we combine the Laminar Flow module with a Dilute
Species Transport module of COMSOL for the transport equations in the bulk
(1–3). The equation for the transport of surfactant on the interface (9) is
implemented through a General Form Boundary PDE, with a source term
corresponding to the kinetic flux $S$. This flux also serves to implement the
boundary condtion (8) at the interface for the Dilute Species Transport
module. The Marangoni forces resulting from the non-uniform distribution of
surfactants at the interface modify the Laminar Flow, as stated in (12),
through a weak contribution at the interface coupled to a free-slip boundary
condition.

The flow in the simulated chamber is forced by an inlet velocity boundary
condition corresponding to a Poiseuille velocity profile
$u(y)=4u_{\mathrm{max}}y(H-y)/H^{2}$. The initial guess velocity profile for
the stationary solver is set to this reference Poiseuille profile in the
entire chamber.

In order to increase the accuracy of the computation, we discretize the fluid
flow with quadratic elements for the velocity field and linear elements for
the pressure field, quadratic elements for the concentration field in the bulk
and the concentration field on the interface.

We use the MUMPS solver of COMSOL to solve for the steady-state of the system,
with a relative tolerance of $10^{-5}$.

In order to check how the results obtained would change for surfactants of
different strengths, we also ran simulations for two extreme sets of
parameters values for the surfactant choice, using the Frumkin kinetics
framework. The first set corresponds to a model strong surfactant with high
affinity to the interface and low diffusivity
($\kappa_{\mathrm{d}}=$1\text{\,}{\mathrm{s}}^{-1}$$,
$\kappa_{\mathrm{a}}=${10}^{6}\text{\,}{\mathrm{m}}^{3}\text{\,}{\mathrm{mol}}^{-1}\text{\,}{\mathrm{s}}^{-1}$$,
$\Gamma_{\mathrm{m}}=${10}^{-5}\text{\,}\mathrm{mol}\text{\,}{\mathrm{m}}^{-2}$$,
$A=-3$,
$D=D_{\mathrm{s}}=${10}^{-11}\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$),
the second corresponds to a model weak surfactant with weak affinity to the
interface and high diffusivity which promotes the smoothing of any interfacial
gradients ($\kappa_{\mathrm{d}}=$100\text{\,}{\mathrm{s}}^{-1}$$,
$\kappa_{\mathrm{a}}=${10}^{-1}\text{\,}{\mathrm{m}}^{3}\text{\,}{\mathrm{mol}}^{-1}\text{\,}{\mathrm{s}}^{-1}$$,
$\Gamma_{\mathrm{m}}=${10}^{-6}\text{\,}\mathrm{mol}\text{\,}{\mathrm{m}}^{-2}$$,
$A=3$,
$D=D_{\mathrm{s}}=${10}^{-9}\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$).
These parameters were selected from the extreme values of data reported in
Tables 1 and 3 in (39). We performed simulations with the rest of the
parameters as in the simulation for Fig. 2D, except for the bulk concentration
$c_{0}$ whose range was extended to cover the transitions for the strong and
weak surfactant respectively. For both cases, we observe progressive
immobilization of the interface and increase of viscous stress with increasing
bulk concentration of surfactant (Fig. 5). For the strong surfactant, the
transition towards a no-slip boundary condition occurs at minute
concentrations, below $c_{0}\approx${10}^{-12}\text{\,}\mathrm{m\textsc{m}}$$,
and well below the transition value for SDS
($c_{0}\approx${10}^{-4}\text{\,}\mathrm{m\textsc{m}}$$). For the weak
surfactant, the transition occurs at
$c_{0}\approx$1\text{\,}\mathrm{m\textsc{m}}$$, which is much higher than for
SDS. In general, one cannot of course guarantee that no traces of strong
surfactants are present in a given flow.

Figure 5: Simulations of surfactant-laden flows in the model SHS chamber shown
in Fig. 2A. We compare the average normalised drag versus the surfactant
concentration for varying surfactant properties using Frumkin kinetics. The
SDS properties are $\kappa_{\mathrm{d}}=$500\text{\,}{\mathrm{s}}^{-1}$$,
$\kappa_{\mathrm{a}}=$89.5\text{\,}{\mathrm{m}}^{3}\text{\,}{\mathrm{mol}}^{-1}\text{\,}{\mathrm{s}}^{-1}$$,
$\Gamma_{\mathrm{m}}=$3.92\text{\times}{10}^{-6}\text{\,}\mathrm{mol}\text{\,}{\mathrm{m}}^{-2}$$,
$A=-2.4$,
$D=D_{\mathrm{s}}=$7\text{\times}{10}^{-10}\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$.
The model strong surfactant (defined using the extreme properties reported in
(39)) has high affinity to the interface and low diffusivity:
$\kappa_{\mathrm{d}}=$1\text{\,}{\mathrm{s}}^{-1}$$,
$\kappa_{\mathrm{a}}=${10}^{6}\text{\,}{\mathrm{m}}^{3}\text{\,}{\mathrm{mol}}^{-1}\text{\,}{\mathrm{s}}^{-1}$$,
$\Gamma_{\mathrm{m}}=${10}^{-5}\text{\,}\mathrm{mol}\text{\,}{\mathrm{m}}^{-2}$$,
$A=-3$,
$D=D_{\mathrm{s}}=${10}^{-11}\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$.
Similarly, the model weak surfactant is defined with weak affinity to the
interface and high diffusivity:
$\kappa_{\mathrm{d}}=$100\text{\,}{\mathrm{s}}^{-1}$$,
$\kappa_{\mathrm{a}}=${10}^{-1}\text{\,}{\mathrm{m}}^{3}\text{\,}{\mathrm{mol}}^{-1}\text{\,}{\mathrm{s}}^{-1}$$,
$\Gamma_{\mathrm{m}}=${10}^{-6}\text{\,}\mathrm{mol}\text{\,}{\mathrm{m}}^{-2}$$,
$A=3$,
$D=D_{\mathrm{s}}=${10}^{-9}\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$.
The behaviour of the transition is similar for all three surfactants, but it
occurs at different concentration thresholds. The threshold is particularly
low for the strong surfactant.

###  B.2 Surfactant-free simulations

To obtain a reference flow profile for long rectangular gratings located on
one side of a three-dimensional microchannel, in the idealized case of pure
water ($c_{0}=0$), we solved the Navier-Stokes equation using COMSOL
Multiphysics®. Our domain corresponds to the portion of a microchannel below
half of a grating element, with plane of symmetry $(x,y=w/2,z)$ (see Fig. 2 B
and C). We use symmetry boundary conditions on each side of the domain to
solve for the flow on a large number of parallel gratings as in the
experiment. The rectangular grating has a length $g=$1\text{\,}\mathrm{mm}$$,
a width $w=$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and is bounded by
lateral ridges of width $r=$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. No-
slip boundary conditions are imposed on the ridges, while free-slip is imposed
on the plastron. The grating is preceded and followed by a no-slip area of
length $\ell/2=$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The channel height
is $H=$120\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The flow was forced by a
three-dimensional Poiseuille profile $u(z)=4u_{\mathrm{max}}z(H-z)$ with
$u_{\mathrm{max}}=$120\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$$. This same velocity profile was also chosen as
initial guess for the steady-state solution. Water viscosity was
$\mu=$9.3\text{\times}{10}^{-4}\text{\,}\mathrm{N}\text{\,}\mathrm{s}\text{\,}{\mathrm{m}}^{-1}$$,
corresponding to a temperature of water of
$23\text{\,}\mathrm{\SIUnitSymbolDegree}$. We use a Physics-controlled mesh
with Finer element size, with a linear discretization of elements. The results
of this simulation were used for both plots of Fig. 3 D and E, as the velocity
profile at the middle of the grating was found independent of the grating
length in the simulation for $g\gg H$ and $g\gg w$.

Table 1: Parameters for surfactant-laden simulations. Parameter | Symbol | Value | Unit | Use
---|---|---|---|---
Grating length | $g$ | $1\text{\times}{10}^{2}$ | $\mathrm{\SIUnitSymbolMicro m}$ |  Fig. 2 B–D
|  | $20\;\mathrm{to}\;100\times 10^{2}$ | $\mathrm{\SIUnitSymbolMicro m}$ |  Fig. 2E
Bulk concentration | $c_{0}$ | $1\text{\times}{10}^{-2}$ | $\mathrm{mol}\text{\,}{\mathrm{m}}^{-3}$ |  Fig. 2D
|  | $10^{-6}\;\mathrm{to}\;1$ | $\mathrm{mol}\text{\,}{\mathrm{m}}^{-3}$ |  Fig. 2 A, B, E
|  | $10^{-12}\;\mathrm{to}\;10$ | $\mathrm{mol}\text{\,}{\mathrm{m}}^{-3}$ |  Fig. 5
Chamber height | $H$ | $1\text{\times}{10}^{2}$ | $\mathrm{\SIUnitSymbolMicro m}$ |
Ridge length | $\ell$ | $5\text{\times}{10}^{1}$ | $\mathrm{\SIUnitSymbolMicro m}$ |
Maximum forcing speed | $u_{\mathrm{max}}$ | $5\text{\times}{10}^{1}$ | $\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-1}$ |
Mean forcing speed | $U$ | $3.3\text{\times}{10}^{1}$ | $\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-1}$ |
Water viscosity | $\mu$ | $8.9\text{\times}{10}^{-4}$ | $\mathrm{N}\text{\,}\mathrm{s}\text{\,}{\mathrm{m}}^{-2}$ |
Water surface tension | $\sigma_{0}$ | $72\text{\times}{10}^{-3}$ | $\mathrm{N}\text{\,}{\mathrm{m}}^{-1}$ |
Bulk diffusivity | $D$ | $0.7\text{\times}{10}^{-9}$ | ${\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$ |
Surface diffusivity | $D_{\mathrm{s}}$ | $0.7\text{\times}{10}^{-9}$ | ${\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$ |
Desorption coefficient | $\kappa_{\mathrm{d}}$ | $500$ | ${\mathrm{s}}^{-1}$ |
Adsorption coefficient | $\kappa_{\mathrm{a}}$ | $89.5$ | ${\mathrm{m}}^{3}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{mol}}^{-1}$ |
Max. packing concentration | $\Gamma_{\mathrm{m}}$ | $39.2\text{\times}{10}^{-7}$ | $\mathrm{mol}\text{\,}{\mathrm{m}}^{-2}$ |
Interaction coefficient | $A$ | $-2.4$ | $-$ |
Surfactant style constant | $n$ | $2$ | $-$ |

##  Appendix C Supporting experimental protocols

###  C.1 Cleaning protocols

Two different cleaning protocols were followed in the preparation of the
experiments. For both cleaning protocols, as well as all the experiments
conducted, only purified water (using Milli Q® water purification system, EMD
Millipore) at $23\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$ with
resistivity $18.2\text{\,}\mathrm{M\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$-1
and less than 5 parts per billion of total organic content was used.

A strict 10-day cleaning protocol was designed in an attempt to avoid any
contamination of the microchannel, which could induce surfactant Marangoni
stresses. The cleaning and experimental preparation were performed using lab
coats and thoroughly washed nitrile gloves (Fisherbrand) (we note that
standard laboratory gloves have traces of chemicals on their surface which
induce surfactant Marangoni stresses). The preparation of the
polydimethylsiloxane (PDMS, Sylgard® 184) microchannels was done in a clean
room. In this protocol, apart from the PDMS, only materials that could be
cleaned thoroughly were used for the surfaces that were in contact with water
during the experiments. In particular, all common plastic materials were
avoided as they tend to release chemical traces with surfactant effect when in
contact with water. All tubings were made of FEP ($0.5\text{\,}\mathrm{mm}$
internal diameter, The Dolomite Centre Ltd), connectors were made of stainless
steel, the syringes (Gastight Hamilton) used to handle water and the micro-
bead suspension were made of glass and PTFE, and fitted with stainless steel
needles (24 G injection needles, Carl Roth GmbH), the inlet and outlet
reservoirs were made of glass. All the tubings, connectors, needles and
reservoirs undergone five washing, rinsing, and curing cycles over a 10-day
period prior to the experiments. The curing containers were large glass
beakers that had been cleaned in an acid rinse dishwasher, and further rinsed
for 5 minutes with purified water. During the curing process, all the beakers
were covered to reduce contamination from the air. On the day of the
experiments, all the tubings and reservoirs were washed again with purified
water. Washed metal tweezers were used to handle tubings and connectors in
order to avoid touching surfaces that could be in contact with the water
flowing through the microchannel. The fluorescent micro-beads
(LifeTechnologies FluoSphere® carboxylate
$0.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter yellow/green 505/515)
used to perform $\mu$-PIV were washed and rinsed ten times with purified water
to dilute significantly any potential surfactant contamination traces. The
cover slip forming the base of the microchannel was washed with abundant
purified water and then air dried.

This strict cleaning protocol was only followed when conducting some steady
forcing experiments. The results, similar to those presented in Fig. 3D
(conducted following the normal cleaning protocol described below), showed no
or little slip in comparison to theoretical and numerical predictions with
surfactant-free flows. As we show in Fig. 2D, the level of contamination
necessary to induce surfactant Marangoni stresses is extremely small, of the
order of ${10}^{-4}\text{\,}\mathrm{m\textsc{m}}$ for the SDS surfactant.
Moreover, as we show in Fig. 5, SDS is not the strongest surfactant and can be
considered as inducing mild Marangoni stresses (see Tables 1 and 3 in 39, for
a comparison of a broad range of surfactants). Therefore, it is very likely
that, even following this strict cleaning protocol, sufficient traces of
chemicals with a surfactant effect contaminated our experiments. We believe
that the most likely source of contamination in our experiments is the PDMS
and its associated impurities. Uncrosslinked PDMS chains or impurities trapped
in the PDMS could have a surfactant effect as has been observed in (45).
Contamination could also come from other sources, which might simply be
unavoidable in normal laboratory conditions.

As surfactant contaminations were simply unavoidable in our experiments, a
less time-consuming cleaning protocol was used for all the experimental
results presented in this study. We used flexible tubing (Tygon® ND-100-80)
instead of the FEP tubing, which were more difficult to handle due to their
rigidity. The outlet and inlet reservoirs were replaced by polypropylene tubes
(Eppendorf) or plastic syringes (BD Plastipack™). All the other elements of
the apparatus, preparation tools and materials were the same. Furthermore,
cleaning of the tools, materials, tubings, connectors and reservoirs was
performed on the day of the experiment. They were all washed with plenty of
purified water: typically with at least ten times the volume they can contain.
The fluorescent micro-beads were washed at least three times.

###  C.2 Steady forcing experiments

For steady forcing experiments, the chamber was first filled with the
suspension of microbeads, taking great care to avoid trapping any air bubble
in the tubing or in the chamber. In order to obtain a low level of flow rate
in the chamber while allowing accurate positioning of the inlet reservoir with
respect to the outlet reservoir, a constriction was introduced on the
hydraulic line by mounting a Gauge 30 polypropylene syringe tip (Adhesive
Dispensing Ltd) on the inlet reservoir (syringe from BD Plastipack™). Using a
manual linear $150\text{\,}\mathrm{m}\mathrm{m}$ motorized stage (Thorlabs
NRT150/M) driven by a precision controller (Thorlabs APT BSC201, with
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ precision, maximum acceleration:
$50\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-2}$, maximum speed:
$50\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$), the inlet reservoir was
moved vertically after initial filling of the chamber until no flow could be
observed in the middle of the chamber. This corresponded to the level of zero
pressure gradient along the microfluidic line. The inlet position was then
shifted by $$5\text{\,}\mathrm{mm}$\pm$10\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ using the linear manual stage. Imaging was then performed in the central
longitudinal portion of the gratings. Stacks of 30 successive images were
taken at approximately 10 fps at different $z$ positions in the chamber and at
different time points for each experiment. Once a microchannel was
successfully prepared to conduct a series of experiments, most of the
plastrons of the SHS gratings remained stable for approximately two hours.

The details of the parameters for the experiments presented in Figure 3D-E are
described in Table 2. The mean forcing speed $U$ was calculated from the
experimental velocity profile by fitting a parabolic profile
$u(z)=4u_{\mathrm{max}}z(H-z)$ to the data.

Table 2: Parameters for steady-forcing experiments. Parameter | Symbol | Value | Unit
---|---|---|---
|  | Fig. 3D | Fig. 3E |
Grating length | $g$ | $2\text{\times}{10}^{3}$ | $3\text{\times}{10}^{4}$ | $\mathrm{\SIUnitSymbolMicro m}$
Grating width | $w$ | $4\text{\times}{10}^{1}$ | $4\text{\times}{10}^{1}$ | $\mathrm{\SIUnitSymbolMicro m}$
Chamber height | $H$ | $1.3\text{\times}{10}^{2}$ | $9.9\text{\times}{10}^{1}$ | $\mathrm{\SIUnitSymbolMicro m}$
Ridge width | $r$ | $2\text{\times}{10}^{1}$ | $2\text{\times}{10}^{1}$ | $\mathrm{\SIUnitSymbolMicro m}$
Max. forcing speed | $u_{\mathrm{max}}$ | $1.1\text{\times}{10}^{2}$ | $1.3\text{\times}{10}^{2}$ | $\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-1}$
Mean forcing speed | $U$ | $7.3\text{\times}{10}^{1}$ | $8.7\text{\times}{10}^{1}$ | $\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-1}$
Temperature | $T$ | $23$ | $23$ | $\mathrm{\SIUnitSymbolCelsius}$

###  C.3 Pressure-relaxation experiments

The protocol for each experiment had two phases: an initial loading phase with
strong background flow and then a second phase without background flow to
measure the surfactant Marangoni driven backflow. During the initial loading
phase the flow was driven at very high background pressure gradient, in order
to transport any surfactant along the air–water interface to the downstream
stagnation end of a grating. This phase lasts for four minutes, during which
images of the flow field were taken at different heights in the channel in
order to obtain the vertical distribution of the streamwise velocity profile.
A typical velocity profile measured during the loading phase of the experiment
shown in Fig. 4A is presented in Fig. 6. As the exposure time on the camera
was smaller but of the same order of magnitude than the typical time for the
fluorescent particles to cross the field of view, particles were seen as
streaks of dots on each image. The dots forming the streaks originate from the
imaging by the spinning disk used with the microscope. The velocity profile
was obtained by measuring the mean length of the fluorescent particle streaks
on the image and dividing by the exposure time. As we can see in Fig. 6, the
velocity profile measured in a vertical plane centered in the middle of a
grating does not show any significant slip velocity at the air–water
interface. We note that the spatial resolution and the technique to measure
the velocity were not designed to measure accurately the velocity close to the
plastron, as was done in the steady forcing experiments (see previous section
and Fig. 3). The aim here was to obtain an estimate of the mean flow field in
the microchannel. The profile follows a classical two-dimensional Poiseuille
profile, from which we computed the mean loading speed $U$ plotted in Fig. 4C.

Figure 6: Vertical distribution of the streamwise velocity profile during the
loading phase of the experiment presented in Fig. 4A. The dashed line shows
the result of a parabolic fit of the velocity profile. The mean speed of the
flow extracted from this fit is
$U\approx$3.9\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$$.

In the second phase, the background pressure gradient was suppressed to stop
the flow. Images of the flow field were recorded at a distance of
$z=$-4\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\pm$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ from the air–water interface.
Images were recorded for 1 or 2 minutes starting approximately five seconds
before the background pressure gradient was suppressed (this corresponds to
$t=0$ in Fig. 4 A, B and D), at a frame rate of around 24 frames per second.
An example is shown in Fig. 7. These images were then analyzed with $\mu$-PIV,
as explained in the Materials and Methods section, to produce the velocity
fields, such as the one displayed in Fig. 4A. Then, the experiment was
repeated but with imposing a negative background pressure gradient in the
microchannel in order to produce an opposite flow. We could thus verify that
the effect observed was independent of the flow direction in the initial
loading phase.

The apparatus and the microchannels were prepared following the initial steps
of the protocol described in the previous section. Then, to control the
background pressure gradient imposed in the microchannel during the loading
phase and ensure a rapid and smooth transition between the two phases, the
inlet reservoir was attached at mid-height onto the linear
$150\text{\,}\mathrm{m}\mathrm{m}$ motorized stage. This mid-height
constituted our zero elevation reference. The outlet reservoir was attached
onto the fixed part of the motorized stage, which was itself attached onto a
millimetric precision vertical ramp. Adjusting the height of the motorized
stage effectively controlled the hydrostatic pressure in the microchannel and
thus the interfacial deflection of the plastron. An initial upward loading
phase was conducted, without recording any images, in order to determine
accurately the level of zero background pressure gradient required for the
second phase. Indeed, during this first loading phase where the inlet
reservoir was raised to a given height ($\Delta H_{r}>0$) compared with the
outlet reservoir, water transferred from the inlet reservoir to the outlet
reservoir. This led to a slight increase in the neutral elevation of the inlet
reservoir corresponding to a zero background pressure required for the second
phase. Then, in order to avoid complete depletion of the inlet reservoir by
having flow in one direction only for all the experiments, the next loading
phase was conducted with a flow in the opposite direction, by lowering the
inlet reservoir to a negative or opposite elevation ($\Delta H_{r}<0$) of the
exact same distance as in the upward loading phase. At the end of this first
downward loading phase, which also last exactly four minutes, the inlet
reservoir could simply be returned to the original zero reference elevation
having transferred back to the inlet reservoir the same amount of water that
was depleted during the upward loading phase. The neutral elevations at the
end of the upward and downward loading phases were found with an accuracy of
$2$ to $3$ microns, which produced a very small flow below the level of
detection of the $\mu$-PIV system. This cycle was then repeated four times,
with images recorded during the loading phase to measure the velocity of the
backflow.

Figure 7: The loading flow was downwards in this image, such that an upward
motion corresponds to a backflow, as shown by the red arrows. This corresponds
to Exp. 1-6 in Table 3.

The experimental parameters of all the pressure-relaxation experiments are
presented in Table 3. Note that by convention the background flow mean speed
$U$ is always considered positive, independently of the flow direction in the
microchannel, so that the backflow is always negative, as shown in Fig. 4 A
and B. To indicate whether the loading phase was conducted upward or downward,
$\Delta H_{r}$ is shown as positive or negative, respectively. As can be
noticed, there is a small asymmetry of approximately $\pm 5$% between the mean
flow speed of an upward loading phase and a downward loading phase. This is
due to a different spatial arrangement of the tubings between the two loading
phases. We also note that not all experiments could be exploited
quantitatively. Due to the strong pressure change at the end of the loading
phase, the plastron of some gratings failed and led to their wetting. Wetting
could either affect the grating under study or an adjacent grating (see notes
in Table 3). With adjacent grating failure, the backflow was still observed on
the remaining plastron, with qualitatively similar magnitude and time scale,
but the backflow velocity was affected through viscous stresses in the water.
These data were therefore not quantitatively accurate and have not been
included in the graphs presented in Fig. 4 C and D. The contrast between the
flow field of a non-wetted grating and a wetted grating is also very clear
when stopping the background pressure gradient. Similar to the flow above a
ridge, the flow in the wetted grating stops immediately at the end of the
loading phase, showing no backflow.

Table 3: Experimental parameters of all the pressure-relaxation experiments.  Exp. |  $\Delta H_{r}$ ($\text{\,}\mathrm{m}\mathrm{m}$) |  $U$ ($\text{\,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{s}$) | Note
---|---|---|---
1-1 | $70$ | $4.4$ |
1-2 | $-70$ | $3.9$ |
1-3 | $70$ | $4.3$ |
1-4 | $-70$ | $3.9$ |
1-5 | $70$ | $4.3$ |
1-6 | $-70$ | $3.8$ |
1-7 | $70$ | $4.3$ |
1-8 | $-70$ | $3.8$ |
2-1 | $100$ | $5.4$ | Adjacent grating failed
2-2 | $100$ | $4.8$ | Adjacent grating failed
2-3 | $100$ | $5.3$ | Adjacent grating failed
2-4 | $100$ | $4.4$ | Adjacent grating failed
3-1 | $-40$ | $2.3$ |
3-2 | $40$ | $2.5$ |
3-3 | $-40$ | $2.3$ |
3-4 | $40$ | $2.5$ |
3-5 | $-40$ | $2.3$ |
3-6 | $40$ | $2.5$ |
3-7 | $-40$ | $2.3$ |
3-8 | $40$ | $2.4$ | Adjacent grating failed
3-9 | $-40$ | $2.2$ | Adjacent grating failed
3-10 | $40$ | $2.4$ | Adjacent grating failed
3-11 | $-40$ | $2.2$ | Grating under study failed
3-12 | $40$ | $2.5$ | Adjacent grating failed
4-1 | $130$ | $7.4$ | Adjacent grating failed

###  C.4 Impact of thermal Marangoni effects in experiments

We assess the potential impact of thermal Marangoni effects in our experiments
to examine whether the backflow observed in the pressure-relaxation
experiments could be due to thermal Marangoni effects. We distinguish steady
temperature gradients from flow-dependent gradients.

Steady temperature gradients can arise due to spatial temperature variations
in the setup close to the microchannel. As the backflow was observed in both
flow directions through the microchannel, within a few minutes’ interval, this
implies that steady spatial temperature gradient did not affect the
experiments.

Temperature gradients could also arise due to heat being advected by the flow
during the loading phase. This heat could come from the laser, although we
note that its power was already very small, less than
$155\text{\,}\mathrm{\SIUnitSymbolMicro W}$, and only a small fraction would
have been absorbed as heat by water. As the microchannel height is only
$H\approx$0.1\text{\,}\mathrm{mm}$$, the timescale for thermal diffusion
across $H$ is of the order of $H^{2}/D_{T}\approx$0.1\text{\,}\mathrm{s}$$,
where
$D_{T}\approx$1.4\text{\times}{10}^{-7}\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$
is the thermal diffusivity of water. Since the bottom of the microchannel is
maintained at a fixed temperature by a Peltier element, as soon as the loading
phase ends the temperature variations in the microchannel should vanish within
approximately $0.1\text{\,}\mathrm{s}$. However, the time scales for the
backflow are typically of the order of one minute. Therefore, thermal
Marangoni effects had negligible impact in our experiments.

##  Appendix D Supporting model for pressure-relaxation experiments

We develop a model that predicts the temporal scale for the rapid backflow
that we observed in the pressure-relaxation experiments (Fig. 4). We consider
the geometry described in Fig. 2A, with the flow in the loading phase in the
positive direction. We assume that the backflow observed at the end of the
loading phase and developing in the negative direction is dominated by
advection, and therefore neglect adsorption/desorption and diffusion along the
interface. We solve the one-dimensional time-dependent advection equation for
the transport of surfactants at the interface. Equation (9) simplifies to

$\frac{\partial\Gamma}{\partial t}+\frac{\partial(u_{I}\Gamma)}{\partial
x}=0.$ (14)

Since $Re=UH/\nu\ll 1$, viscous spreading across the channel height occurs
very fast compared to surfactant advection, and we approximate $(\partial
u/\partial y)_{I}$ as $u_{I}/H$ in the left-hand side of (12). Given that the
surfactant concentration is very small compared to $\Gamma_{\mathrm{m}}$, the
right-hand side of (12) can be linearized to give

$\mu\frac{u_{I}}{H}=-nRT\,\frac{\partial\Gamma}{\partial x},$ (15)

Substituting $u_{I}$ from (15) into (14) we find the conservation equation for
$\Gamma$:

$\frac{\partial\Gamma}{\partial t}-H\frac{nRT}{\mu}\frac{\partial}{\partial
x}\left(\Gamma\frac{\partial\Gamma}{\partial x}\right)=0.$ (16)

We introduce similarity variables

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{(g+\ell/2-x)}{g}\left(\frac{tU_{\Gamma}}{g}\right)^{-1/3},$
(17) $\displaystyle\quad\frac{\Gamma(x,t)}{\Gamma_{\mathrm{m}}}$
$\displaystyle=$
$\displaystyle\left(\frac{tU_{\Gamma}}{g}\right)^{-1/3}f(\eta),$ (18)

where $U_{\Gamma}=nRT\Gamma_{\mathrm{m}}/\mu$ is a Marangoni-based velocity
scale. Substituting into (16), integrating twice and using the boundary
condition (13), the solution is

$\frac{\Gamma(x,t)}{{\Gamma_{\mathrm{m}}}}=-\frac{(g+\ell/2-x)^{2}}{6H{U_{\Gamma}}t}+C\left(\frac{tU_{\Gamma}}{g}\right)^{-1/3},$
(19)

which is valid for $t$ larger than diffusion time (i.e. at very small $t$
diffusion plays a role) and for time small enough that the front of the
advection is still between the two stagnation points (at $x=g+\ell/2$ and
$x=\ell/2$). For the initial condition, if we assume that the loading phase
had a strong positive background flow $U$ (such as in the pressure-relaxation
experiments), then surfactants have accumulated near the downstream stagnation
point $x=g+\ell/2$ for $t\leq 0$. The exact distribution of the surfactants at
$t=0$ is unknown, but for large loading speed $U$, we can assume that it is
very steep near $x=g+\ell/2$. The constant of integration $C$ is effectively a
measure of the total amount of surfactant on the interface, which is constant
at all time for insoluble surfactants. This is effectively the main unknown in
our experiments. We find

$C=\left(\frac{\int_{x_{f}}^{g}\Gamma/\Gamma_{\mathrm{m}}\,dx}{2/3\sqrt{6Hg}}\right)^{2/3},$
(20)

where $x_{f}(t)$ is the front of the surfactant, i.e. where $\Gamma$ vanishes:

$x_{f}=g+\ell/2-\sqrt{6HgC}\left(\frac{tU_{\Gamma}}{g}\right)^{1/3},$ (21)

which is effectively valid in our finite-length geometry until the front
reaches the stagnation point at $x=\ell/2$, as the model assumes a semi-
infinite lane $x\leq g+\ell/2$.

From (15) and (19), the interfacial speed due to surfactant gradients is
therefore,

$u_{I}=-\frac{g+\ell/2-x}{3t}.$ (22)

Thus, $u_{I}$ is negative, corresponding to the backflow observed in our
experiments (Fig. 4 A and B) for $t>0$. Once the loading phase ends and the
background flow stops, the surfactants travel back along the interface,
driving a Marangoni backflow, to eventually redistribute uniformly along the
air–water interface, $\ell/2\leq x\leq g+\ell/2$. This result also shows that
the backflow velocity should decrease in time as $1/t$. We compare this
scaling prediction with our experimental results in Fig. 4D. As mentioned
previously, this result is not valid at very small time, where diffusion
processes are important and the initial distribution of the surfactant is
unknown. Hence, it cannot inform us about the dependence of the peak backflow
velocity, measured at $t\approx 0$, with $U$ (Fig. 4C). It only informs us
about the trend of the backflow at intermediate times, as shown in Fig. 4D,
until the opposite stagnation point at $x=\ell/2$ starts playing a role.