# H2 bubble motion reversals during water electrolysis

A. Bashkatov111E-mail<EMAIL_ADDRESS>Institute of Fluid Dynamics,
Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstrasse 400, Dresden,
01328, Germany  Institute of Process Engineering and Environmental Technology,
Technische Universität Dresden, Dresden, 01062, Germany  Hydrogen Lab, School
of Engineering, Technische Universität Dresden, Dresden, 01062 Germany  A.
Babich  Institute of Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf,
Bautzner Landstrasse 400, Dresden, 01328, Germany  S. S. Hossain  Institute of
Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstrasse
400, Dresden, 01328, Germany
X. Yang  Institute of Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf,
Bautzner Landstrasse 400, Dresden, 01328, Germany  G. Mutschke  Institute of
Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstrasse
400, Dresden, 01328, Germany  K. Eckert222E-mail<EMAIL_ADDRESS>Institute
of Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstrasse
400, Dresden, 01328, Germany  Institute of Process Engineering and
Environmental Technology, Technische Universität Dresden, Dresden, 01062,
Germany  Hydrogen Lab, School of Engineering, Technische Universität Dresden,
Dresden, 01062 Germany

###### Abstract

The dynamics of hydrogen bubbles produced by water electrolysis in an acidic
electrolyte are studied using electrochemical and optical methods. A defined
cyclic modulation of the electric potential is applied at a microelectrode to
produce pairs of interacting H2 bubbles in a controlled manner. Three
scenarios of interactions are identified and systematically studied. The most
prominent one consists in a sudden reversal in the motion of the first
detached bubble, its return to the electrode and finally its coalescence with
the second bubble. Attested by Toepler’s schlieren technique, an explanation
of contactless motion reversal is provided by the competition between buoyancy
and thermocapillary effects.

##  1 Introduction

The growth and detachment of nano- and micrometer gas bubbles are omnipresent
phenomena in nature and engineering, e.g. see the review by [1]. The growth of
the gas bubbles in alkaline water electrolysis is a particularly interesting
problem of high practical relevance. Although alkaline water electrolyzers are
the most mature technology, they still suffer from low efficiency ([2]) as a
considerable part of the losses are caused by generated gas bubbles that block
electrocatalytic sites and also raise the Ohmic cell resistance ([3]). Thus,
the rapid and efficient detachment of the bubbles from the electrodes is
important; it is closely linked to a better understanding of the balance of
forces acting on the bubble, the concept of which is comprehensively covered
by [4]. Recently, progress has been made in identifying and quantifying the
forces of attraction on H2 bubbles that counteract their buoyancy.
Investigating the growth of H2 bubbles under extreme cathodic potentials in
acidic electrolytes, [5] and [6] attributed the positional oscillations of H2
bubbles prior to detachment to the action of two forces. First, the electric
force $F_{e}$, given by

$\overrightarrow{F_{e}}=\int_{\mathcal{S}}\sigma E_{z}\,\mathrm{d}A$ (1)

where $E_{z}$ is the vertical ($z$) component of the external electric field,
directed from the anode to the cathode, see Fig. 1(a). $\mathcal{S}$ is the
interface between the bubble and the electrolyte, and $\sigma$ is the
corresponding surface charge density, which is positive for gas bubbles in
acidic electrolytes below the iso-electric point at pH $<$ 2…3 ([7]). The
second important force is the hydrodynamic force $F_{h}$, given by ([8, 6])

$\overrightarrow{F}_{h}=\overrightarrow{F}_{M}+\overrightarrow{F}_{n}=\int_{\mathcal{S}}\overrightarrow{\tau_{h}}\,\mathrm{d}A=-\int_{\mathcal{S}}\overrightarrow{\tau_{M}}\,\mathrm{d}A+\int_{\mathcal{S}}\overrightarrow{\tau_{h,n}}\,\mathrm{d}A$
(2)

which is obtained by integration of the stress tensor,
$\overrightarrow{\tau_{h}}=-p_{h}\overrightarrow{n}+\mu\frac{\partial\overrightarrow{u}}{\partial
n}+\mu\nabla u_{n}$ over the bubble surface. $p_{h}$ is the hydrodynamic
pressure, $\mu$ is the dynamic viscosity of the electrolyte,
$\overrightarrow{u}$ is the electrolyte velocity vector, $\overrightarrow{n}$
is the surface-normal unit vector, and
$u_{n}=\overrightarrow{u}\cdot\overrightarrow{n}$. $F_{h}$ originates in the
fact that the surface tension ($\gamma$) of the gas-electrolyte interface
depends on the temperature $T$ and/or species concentration $c$. Any gradient
in $T$ or $c$ causes a gradient in $\gamma$. This gradient in $\gamma$
generates an imbalance in the shear stress that causes bubble surface
elements, and the nearby electrolyte, to move from high to low $\gamma$
regions (see [9, 10, 11, 12]).
$F_{M}=-\int_{\mathcal{S}}\overrightarrow{\tau_{M}}\,\mathrm{d}A$ and
$F_{n}=\int_{\mathcal{S}}\overrightarrow{\tau_{h,n}}\,\mathrm{d}A$ capture the
contribution from the tangential stress that leads to the Marangoni
convection, and the contribution from the normal stress (see [6]). Previous
work by [13] provided evidence that the dominant source of the Marangoni
convection observed at the bubble foot is thermocapillary rather than
solutocapillary (see [14, 8, 12]).

Despite this progress, a number of unresolved phenomena remain, such as the
bubble jump-off after the coalescence of two bubbles and its subsequent
reattachment to the electrode (see [15, 16, 17]). There has been speculation
as to whether electrostatic attraction, Marangoni effects ([18]) or
coalescence, as recently proposed by [17], are behind the physics of the
reattachment.

The focus of the present work is to achieve a better understanding of such
bubble interaction phenomena. As will be shown, this can be achieved by more
closely examining the phenomenon of thermocapillary migration. [19] and [20]
were the first to show that an air bubble subjected to a vertical temperature
gradient can move downward against the direction of buoyancy if the liquid is
heated from the bottom. Experiments performed on a NASA Space Shuttle in orbit
([21]) reported a migration velocity of $\sim 0.3$ mm/s in a 1 K/mm
temperature gradient for air bubbles of $\sim$7…10 mm in diameter, in good
agreement with the theory developed by [19]. A further type of bubble motion
against buoyancy is the periodical bouncing of a plasmonic bubble in a binary
liquid ([22]) as a result of competition between soluto- and thermocapillary
effects. By interrupting continuous laser irradiation during the bubble growth
on photocatalytic surfaces [23, 24] recently succeeded in forcing bubbles to
take on a bouncing motion, during which they detach and reattach at the
photocatalyst’s surfaces. The re-attachment has been attributed to a thermal
Marangoni effect.

In this work we add to previous studies by [5] by including modulation cycles
of the cathodic potential. This modulation enables consecutive pairs of
hydrogen bubbles of a well-defined size to be produced, which further allows
the forces $F_{e}$ and $F_{h}$ to be varied. In this way we are able to
systematically study the phenomenon of bubbles returning towards the
electrode. By using Toepler’s schlieren technique alongside shadowgraphy and
PTV, we were able to identify the origin of the H2 bubble motion reversal as a
kind of self-organized thermocapillary migration provoked by the interaction
of two H2 bubbles.

Figure 1: (a) Schematic of the pair of H2 bubble produced by the current
density j (dashed lines of red arrows) together with the forces acting on the
bubbles (cf. Section 1). The contour lines on the right represent the
isotherms, which decay from red to violet. (b) Scheme of the three-electrode
electrochemical cell and the optical systems. $d_{1}=2R_{1}$ and
$d_{2}=2R_{2}$ are the diameters of the first and second bubbles, and $S$
denotes the distance between the center of the first bubble and the electrode
surface.

##  2 Experimental setup and procedure

Consecutive single H2 bubbles were generated during water electrolysis in 0.5
M H2SO4 at a $\diameter$100 $\mu$m Pt microelectrode acting as a cathode, see
Fig. 1(b). Two Pt wires served as the anode and pseudo reference electrode,
respectively. The cathodic potential, $E$, is modulated over time as shown in
Figure 2(a) to study the bubble-bubble interaction. The modulation cycle of
$E$ consists of three phases. In phase 1, the potential $E_{1}$ is applied for
a short time $t_{1}$ to produce the first bubble, which grows up to a radius
$R_{1}$. In the following phase ”0” with a duration of $t_{0}$, the potential
is switched off, i.e., $E=0$. As this leads to the decay of the retarding
forces $F_{e}$ and $F_{h}$, it allows the first bubble to detach from the
electrode and to rise over the distance $S$. In the subsequent phase 2, the
cathodic potential is switched on again at a larger value $E_{2}$ for a time
period $t_{2}$. In this phase, a second bubble quickly grows, thereby possibly
interacting with the first bubble if it is still close enough. Finally, the
potential is again set to $E=0$ over a longer timespan $t_{w}$ to allow the
bubbles that are produced to detach and the resulting electrolyte flow to
decay. After this, the next cycle is initiated. A large number of such cycles,
e.g. 105 in Fig. 3(d) and 133 in Fig. 3(e), have been studied to ensure that
the statistics for the results reported are robust.

The experiments were performed at $E_{1}=-2...-6$ V and $E_{2}=-8$ V applied
for $t_{1}$ = 1…5 ms and $t_{2}$ = 40…200 ms, respectively, while interruption
times $t_{0}$ of 120…200 ms were applied for the detachment and rise of the
first bubble. The waiting time between subsequent cycles was chosen as $t_{w}$
= 500 ms. This modulation scheme allows (first) bubbles of a very defined
radius of e.g. $R_{1}=(66\pm 1)\,\mu$m to be produced, as in Fig. 2. These
travel over a distance $S_{E2}$ before the second bubble is produced.

A high-speed shadowgraphy system (resolution: 1000 pix/mm, frame rate: 5 kHz)
was used to visualize the bubble dynamics, as already described in [25].
Monodisperse polystyrene particles ($\diameter$5 $\mu$m, $\rho_{ps}=1.05$
g/cm3) were seeded into the electrolyte to study the electrolyte flow by means
of particle tracking velocimetry (PTV). For that purpose, the particle’s path,
acquired over 16 images per time instant (corresponding to 3.2 ms at 5 kHz)
was additionally averaged over the 105 cycles. The optics are further
complemented by a Toepler’s schlieren system ([26]) consisting of an aperture
stop, two lenses (focal length $f$ = 100 mm) and a horizontally installed
knife edge to map the vertical refractive index gradients $\partial n/\partial
z=dn/dT\cdot\Delta T/\Delta z+dn/dc\cdot\Delta c/\Delta z$ accompanying the
evolution of electrogenerated H2 bubbles. To enhance the contrast and signal-
to-noise ratio, each schlieren image is divided pixel by pixel by a plain
schlieren image without bubbles ([27]). As $dn/dT\cdot\Delta T/\Delta z\sim
10^{-3}$ (water, $\Delta T\sim 10$ K) ([28]) while $dn/dc\cdot\Delta c/\Delta
z\sim 0.5\times 10^{-5}$ (air-saturated vs degassed water) ([29]), the
refractive index gradient maps the temperature rather than the concentration
gradient.

##  3 Results

###  3.1 Spontaneous H2 bubble motion reversal opposite to buoyancy – Scenario
I

Figure 2: (a) The cathodic potential $E$ modulated over time and the modulus
of the resulting electric current $|I|$. (b) Distance $S$ between the center
of the first bubble and the electrode over time. (c) Velocity $V$ of the first
bubble versus distance $S$. The dashed line marks $V=0$, the dotted one
corresponds to a continuously rising bubble for unmodulated, constant $E$. (d)
Snapshots of the bubbles’ behavior at time instants labeled in subfigure (b).

Figure 2 describes the basic phenomenon studied in terms of $E$ and $I$ (a),
the distance to the electrode $S$ (b), the velocity of the first bubble (c)
and snapshots of the H2 bubble(s) (d) during the modulation cycle of the
potential $E$. In phase 1 (potential $E_{1}=-2$ V), the first bubble is
produced. It already has a radius of around $R=54\>\mu$m after 2.6 ms
(snapshot 1), and reaches a final size of around $R=66\pm 1\>\mu$m after
$t_{1}=5$ ms. In phase ”0”, where the potential is set to $E=0$, the hydrogen
evolution reaction stops. The bubble resides at the electrode for a short time
before detaching, depicted by snapshot 2 at 65 ms. After detaching, the bubble
performs a free rise (snapshots 3–4). When phase 2 begins, the potential is
switched to $E_{2}=-8$ V (snapshot 5) and the second bubble is produced. From
now on, a completely unexpected process sets in. After the initial
acceleration, the first bubble starts to decelerate (snapshots 5–6). At a
distance $S_{rev}$ (snapshot 7), it finally reverses its direction of motion.
Without any external influence, the bubble henceforth moves against buoyancy
towards the second bubble, and coalesces with it (snapshots 8–9).

Figure 2(b) analyses this phenomenon in terms of the distance $S$ between the
center of the first bubble and the electrode surface. As long as the bubble is
attached to the electrode, $S=R_{1}$. After it detaches and during the short
acceleration phase of the first bubble, $S$ increases nearly linearly with
time between snapshots 4 and 5. When the second bubble appears (see red
curve), upon switching to $E_{2}$ at $S=S_{E2}$, $S$ increases at a higher
rate until a maximum $S_{rev}$ is attained. Thereafter, the bubble motion
occurs in the reverse direction, and $S$ decreases until the two bubbles
coalesce (9).

The velocity $V=dS/dt$ of the first bubble during these stages is plotted in
Figure 2(c). Until the point of motion reversal at $S_{rev}$, the velocity of
the first bubble is positive, and negative afterwards. When $E_{2}$ is
switched on, the quickly growing second bubble displaces the electrolyte. This
accelerates the first bubble upwards, and its velocity increases from $V_{E2}$
to a maximum of $V_{p.1}$ = 18.9 mm/s (black square, attained at snapshot 6).
After that point, the interaction between the two bubbles’ first forces the
velocity of the first bubble to decrease to $V=0$ at $S_{rev}$. Afterwards,
the first bubble is accelerated against buoyancy towards the second bubble.
Hence, a second peak in the velocity, $V_{p.2}$ = -24.3 mm/s, the magnitude of
which is larger than $V_{p.1}$, is attained shortly before coalescence with
the second bubble.

###  3.2 Full parameter space: Scenarios I–III

The occurrence of scenario I, described above, depends crucially on the first
bubble’s distance $S$ at which $E_{2}$ is switched on. Figure 3 supplements
Figure 2 by plotting the three possible scenarios in terms of the position of
the first bubble over time $S(t)$ (a) and its velocity $V(S)$ (b). Each line,
labeled from 1 to 6, represents the trajectory of the first bubble for six
different cycles. The experiments were performed at $E_{1}=-6$ V, applied for
$t_{1}=5$ ms, and $E_{2}=-8$ V, applied for $t_{2}=40$ ms (1st to 5th bubble)
and 45 ms (6th bubble).

Figure 3: Overview of the three scenarios in terms of the position $S$ of the
first bubble over time (a) and its velocity $V(S)$ (b). Lines 1 to 3 are
scenario I, lines 4 and 5 are scenario II, and line 6 is scenario III. The
insets in (a) show $S(t)$ shortly after the bubble nucleation (left) and close
to the time $E_{2}$ is switched on (right). (c) Bubble snapshots belonging to
line 4 (scenario II). (d) Velocity magnitude contour and vectors in the
electrolyte during the growth of the second bubble (left) and at the point of
reversal of the first bubble at $S_{rev}$. (e) Velocity peak values $V_{p.1}$
and $V_{p.2}$ versus distance $S_{E2}$ for three different bubble radii
(different colors). Triangles, circles and squares relate to scenarios I, II
and III, respectively.

The radius of the first bubble is determined by $E_{1}$ and $t_{1}$. As both
values were identical for all 6 cycles, the bubbles have nearly the same
radius $R=66\pm 1$ $\mu$m. However, the tiny variations in $R$ already cause
slight deviations in the detachment process, see left inset in (a), and Table
1. As such, the distances $S_{E2}$ at the onset of $E_{2}$ vary, as seen in
the right inset in subfigure (a), where the different bubble paths are also
labeled by numbers.

The scatter in $S_{E2}$, although being smaller than 30 $\mu$m in size, plays
a crucial role in deciding which scenario the bubble follows later on. For
small distances $S_{E2}<S_{E2.crit}$, the first bubble follows scenario I
(lines 1…3) as described in Section 3.1. For larger distances $S_{E2.crit}\leq
S_{E2}<S_{E2.6}$, a different scenario II is found (lines 4 and 5), which is
visualized in subfigure (c). Here, a reversal of the motion of the first
bubble at a distance $S_{rev}$ is also observed. However, the values of
$S_{rev}$ are shifted to somewhat higher values compared to scenario I. The
subsequent downward acceleration of the first bubble towards the second one is
comparably weaker and not sufficient to force the two to coalesce. Hence, upon
approaching the second bubble, the first bubble is repelled and continues its
rise. Line 5 corresponds to a case of even slower downward motion.

The transition between scenario I (line 3) and scenario II (line 4) occurs at
approximately $S_{E2.crit}\approx(S_{E2.3}+S_{E2.4})/2=346$ $\mu$m during
which the difference $\Delta S_{E2}=S_{E2.4}-S_{E2.3}$ amounts to 4 $\mu$m
only. At larger distances $S_{E2}\geq S_{E2.6}\sim 388\>\mu$m, the motion of
the first bubble is only affected by displaced electrolyte during the fast
growth of the second bubble. However, deceleration does not occur that leads
to a reversal of the direction of motion. This corresponds to scenario III,
represented by line 6.

Two snapshots of the velocity field, obtained by PTV (cf. Section 2), are
documented in Figure 3(d). Images of the bubbles, drawn to scale, are
schematically superimposed. White arrows indicate the direction of the
expansion and the motion of the bubbles, respectively. The velocity by which
the growing second bubble displaces the surrounding electrolyte scales with
the bubble’s growth rate and decays with distance $S$. Thus, (first) bubbles
at a higher $S_{E2}$ experience a smaller advection by the displacement flow.
As the bubble’s growth rate decreases with time, the velocity of the
displacement flow also decreases, see the differences between left and right
images in Figure 3(d). Although this flow is still directed upward in the
right image, which shows the bubble at $S_{rev}$, the bubble starts to reverse
the direction of its movement, and is accelerated towards the electrode. We
further note that the high magnitude of the velocity visible at the foot of
the second bubble (Figure 3(d), right image) is caused by the temperature
gradient along the bubble surface arising from Joule heating due to the high
current density at the rim of the microelectrode ([14]). The resulting
Marangoni convection is further enhanced by the bubble expansion.

The characteristic velocity maxima, $V_{p.1}$ and $V_{p.2}$, attained by the
first bubbles in Figure 3(a) are summarized in Table 1 for the three different
scenarios represented by lines 1 to 6. The velocity $V_{E2}$ of the first
bubbles at the onset of potential $E_{2}$ is also included. It is interesting
to compare all three quantities to the terminal velocity $V_{t}$ of a freely
rising bubble ([30]). For the present case ($R=66\>\mu m$, Reynolds number Re
$\approx$ 5, Eötvös number Eo $\ll$ 1), $V_{t}$ can be estimated as amounting
to $V_{t}=\frac{2R^{2}\Delta\rho g}{9\mu}\sim 8.8$ mm/s where $\Delta\rho$ and
$\mu$ denote the density difference and the dynamic viscosity. Inspecting
Table 1, we note that all $V_{E2}$ values are $7...16\%$ smaller than $V_{t}$.
The main reason is that the bubbles have not yet finished the initial
acceleration phase at the comparatively short distances of $S<400\,\mu$m
reached before $E_{2}$ is switched on. For that reason, $V_{E2}$ rather than
$V_{t}$ is used as a reference velocity to non-dimensionalize $V_{p,1}$ (cf.
Table 1). On examining $V_{p.1}/V_{E2}$, we see that $V_{p.1}$ exceeds the
velocity of the freely rising bubble by a factor of 1.7 to 2.4. The reason is
that the bubble’s velocity with respect to a non-moving frame results from the
superposition of free rise and advection by the ”bow wave” of the displaced
electrolyte. The high magnitudes of $|V_{p.2}|$ attained by the bubble during
its reverse motion against buoyancy are even more noticeable. They may exceed
$V_{p.1}$ by up to a factor of 1.5 in scenario I.

Scenario / | $S_{E2}$ | $\Delta t$ | $V_{E2}$ | $V_{p.1}$ | $\mid V_{p.2}\mid$ | $V_{p.1}/V_{E2}$ | $|V_{p.2}|/V_{p.1}$
---|---|---|---|---|---|---|---
Line, `#` | mm | ms | mm/s | mm/s | mm/s |  |
I / 1 | 328 | 13.2 | 7.4 | 17.5 | 26.7 | 2.37 | 1.53
I / 2 | 336 | 17.2 | 7.8 | 16.3 | 23.3 | 2.09 | 1.43
I / 3 | 344 | 25 | 8.0 | 16.0 | 15.9 | 2.00 | 0.99
II / 4 | 348 | — | 7.9 | 16.0 | 12.7 | 2.03 | 0.79
II / 5 | 355 | — | 8.0 | 15.8 | 5.7 | 1.98 | 0.36
III / 6 | 388 | — | 8.2 | 13.9 | — | 1.69 | —
Table 1: Summary of the characteristic values from Figure 3 for the three
different scenarios. $V_{E2}$ refers to the free rise velocity of the first
bubble at distance $S_{E2}$. $\Delta t$ is the time interval between instants
of time 5 and 9 as marked in Fig. 2(b).

To substantiate the features of the different scenarios in terms of the
velocity peak values $V_{p.1}$ and $V_{p.2}$, both are plotted in Figure 3(e)
as a function of the distance $S_{E2}$, which can be increased by choosing
larger values $t_{0}$ of phase ”0”. Furthermore, different radii $R_{1}$ of
the first bubble are studied by varying the duration of $t_{1}$ between 1 ms
$\leq t_{1}\leq$5 ms. As can be seen, all three scenarios are examined.

When $S_{E2}$ increases, $V_{p.1}$ monotonically falls from almost 60 mm/s at
$S_{E2}\sim 175\,\mu$m to reach a plateau of $V_{p.1}\sim 10...20$ mm/s at
large $S_{E2}$ values, at which the first bubble is by now barely affected by
the second one. As $V_{p.1}$ results to a considerable extent from the rapid
displacement of the electrolyte due to growth of the 2nd bubble, the existence
of a plateau at $S_{E2}\approx S_{E2.6}\approx 375\,\mu$m demonstrates that a
maximum distance must not be exceeded for an interaction between both bubbles.
$V_{p.1}$ furthermore increases with the bubble radius $R_{1}$, and hence the
buoyancy, for all the measurements performed. On relating the plateau values
to the corresponding terminal velocity at different $R_{1}$ we obtain:
$(V_{p.1:min}/V_{t})_{97\mu m}$=0.96; $(V_{p.1:min}/V_{t})_{80\mu m}$=1.15 and
$(V_{p.1:min}/V_{t})_{60\mu m}$=1.7. As seen in Table 1, the terminal velocity
is not quite achieved at $S_{E2}<400$ $\mu$m. Hence, in all three cases
($R_{60,80,97}$), the velocity ratios demonstrate the acceleration of the
first bubbles by the displaced electrolyte to velocities close to or higher
than their terminal values.

The second velocity peak $V_{p2}$ behaves differently and shows a parabola-
like behavior at all bubble sizes $R_{1}$. When $S_{E2}$ increases, in the
case of the smallest bubble $R_{1}=60\,\mu$m, $-V_{p2}$ rises to the maximum
value of around 55 mm/s. This is followed by a gradual decline to zero,
denoting the end of scenario II if $S_{E2}$ becomes too large. The local
maximum is shifted to higher $S_{E2}$ values if the radius $R_{1}$ is
increased. The existence of a local maximum of $-V_{p2}$ indicates that the
first bubbles, which are too far away, are only weakly influenced, while
bubbles that are too close do not have the time to develop the local maximum
of $V_{p2}$ before coalescing with the second bubble.

Using schlieren imaging, in Figure 4 we analyze the vertical gradient of the
refractive index, or temperature (cf. Section 2), associated with the
different scenarios I, II and III. When the refractive index is translated
into information on the temperature gradient, taking into account that the
temperature is inversely proportional to the refractive index ([28]), the red
and blue colors in Fig. 4 denote an increase or decrease in the temperature in
the vertical direction (from bottom to top). Although the extent and instant
of their appearance differs, Figure 4 demonstrates the existence of blue
regions on top of both bubbles for all three scenarios. This is reminiscent of
the thermal boundary layer produced during bubble nucleation at the electrode
on top of each bubble due to Joule heating. These boundary layers are advected
during the rise of the first bubble and also during the growth of the second
bubble. As a result, blue regions of decaying temperature are found near the
top of the bubbles. It is also noticeable that the thermal schliere upstream
of the second bubble rises faster due to the displaced electrolyte and the
wake behind the first bubble. As soon as this schliere reaches the bottom of
the first bubble, the deceleration of the latter sets in. For sufficiently
small distances $S_{E2}$, belonging to scenario I, a warmer region is
established between the two bubbles. The resulting temperature gradient around
the first bubble is responsible for its downward acceleration towards the
second bubble (cf. Section 4). As a result of this reverse motion, the thermal
boundary layer on top of the first bubble develops a characteristic shape
resembling a flying bird. If the bubble is further away from the electrode
(scenario II), the schliere still touches the first bubble. However, the
resulting temperature gradient is too weak to provoke an acceleration leading
it to coalesce with the second bubble. In scenario III, the blue zone of
elevated temperature is too far from the first bubble, hence there is no
interaction at all.

Figure 4: Schlieren images for different stages of scenarios I (a), II (b),
and III (c), shortly after $E_{2}=-8$ V is switched on, taken at $\Delta t=3$
ms (a) and 4 ms (b-c). The intensity of the schlieren images $J_{s}$ is
divided by $J_{b}$, the background schlieren image without bubbles.

##  4 Discussion and conclusions

The key phenomenon discovered in this work is the initially contactless
interaction between two electrogenerated bubbles, forcing a paradoxical
reversal of the bubble motion in the opposite direction to buoyancy. The
origin of this phenomenon needs to be sought in the forces, $F_{e}$ and
$F_{h}$ (Section 1), acting on the bubble. Any influence from the electric
force $F_{e}$ can be excluded for two reasons. (i) [5] showed the non-linear
dependency of $F_{e}$ and its strong decay for electrode distances larger than
approx. $30\,\mu$m. Thus, the electric force is unlikely to play any role at
the much larger bubble-electrode distances of $\approx$ 300 $\mu$m found in
this work. (ii) In another test experiment, a second bubble is produced in the
same way as the first bubble at $E_{2}=-6$ V for 1 ms. Afterwards, $E_{2}$ is
suddenly set to $E_{2}=0$ to force the detachment of the second bubble.
Despite the vanishing potential $E_{2}$, $F_{e}=0$, the first bubble thus
still follows scenario I in a nearly unchanged fashion.

This suggests that $F_{h}$, and in particular the Marangoni force element
$F_{M}$, play a key role. As revealed by the schlieren images in Figure 4, the
H2 bubble motion reversal (scenarios I and II) sets in if the thermal boundary
layer on top of the second bubble is able to touch the bottom of the first
bubble. If this happens, a temperature gradient is built up along the surface
of the first bubbles. As the surface tension $\gamma$ decreases with
increasing temperature, a $\gamma$ gradient is established that pulls the
adjacent electrolyte from the bubble foot towards the equator. As the $\gamma$
gradient mimics the action of the arms of a swimmer, the bubble starts moving
opposite to the buoyancy force. This is in analogy to the classical work by
[19] on thermocapillary migration with the exception that the temperature
gradient is not externally imposed but generated in a self-organized way,
including production by Joule heating and dissipation by thermal diffusion.
According to [31] and [19], the stationary velocity of the “creeping”
thermocapillary migration of a bubble inside a temperature gradient $\partial
T/\partial z$ is given by

$v=\frac{R^{2}\rho g-(3/2)R(\frac{\partial\gamma}{\partial T}\frac{\partial
T}{\partial z})}{3\eta}$ (3)

Taking the second peak velocity $V_{p2}\sim 30$ mm/s as the characteristic
velocity of thermocapillary migration, the required $\partial T/\partial z$
according to Eq. 3 is approx. 90 K/cm. With the characteristic temperature
rise $\Delta T\sim 10$ K at the microelectrodes ([13, 12]) over a bubble
radius $R\sim 100\>\mu$m (0.01 cm), a much larger $\partial T/\partial z\sim
10^{3}\>K/cm$ can easily be achieved. This strongly supports the notion that
(i) thermocapillary migration is at the origin of H2 bubble motion reversal
and (ii) the interaction of electrogenerated bubbles needs to be taken into
account during water electrolysis.

##  5 Acknowledgments

This project is supported by the German Space Agency (DLR) with funds provided
by the Federal Ministry of Economics and Technology (BMWi) due to an enactment
of the German Bundestag under Grant No. DLR 50WM2058 (project MADAGAS II). We
thank J. Boenke for his support with experiments during his student internship
in 2021.

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