3.1. Spontaneous H$_2$ bubble motion reversal opposite to buoyancy – scenario I

Figure 2 describes the basic phenomenon studied in terms of $E$ and $I$ (figure 2a), the distance to the electrode $H$ (figure 2b), the velocity of the first bubble (figure 2c) and snapshots of the H$_2$ bubble(s) (figure 2d) during the modulation cycle of the potential $E$. In phase 1 (potential $E_1=-2$ V), the first bubble is produced. It already has radius approximately $R = 54\,\mathrm {\mu }$m after 2.6 ms (snapshot 1), and reaches final size approximately $R = 66 \pm 1\,\mathrm {\mu }$m after $t_1=5$ ms. In phase ‘0’, where the potential is set to $E=0$ ($t_0 = 120$ ms), 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 $H_{rev}$ (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 $H$ between the centre of the first bubble and the electrode surface. As long as the bubble is attached to the electrode, $H=R_1$. After it detaches and during the short acceleration phase of the first bubble, $H$ increases nearly linearly with time between snapshots 4 and 5. When the second bubble appears (see red curve), upon switching to $E_2$ at $H=H_{E2}$, $H$ increases at a higher rate until a maximum $H_{rev}$ is attained. Thereafter, the bubble motion occurs in the reverse direction, and $H$ decreases until the two bubbles coalesce (snapshot 9).

The velocity $V={\rm d}H/{\rm d}t$ of the first bubble during these stages is plotted in figure 2(c). Until the point of motion reversal at $H_{rev}$, the velocity of the first bubble is positive; it is 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 $V_{p.1} = 18.9\,{\rm mm}\,{\rm s}^{-1}$ (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 $H_{rev}$. Afterwards, the first bubble is accelerated against buoyancy towards the second bubble. Hence a second peak in the velocity, $V_{p.2} = -24.3\,{\rm mm}\,{\rm s}^{-1}$, 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 $H$ 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 $H(t)$ (figure 3a) and its velocity $V(H)$ (figure 3b). Each line, labelled 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 $t_2=200$ ms. The interruption time $t_0$, denoted in figure 2(a), amounted to 40 ms (1st to 5th bubble) and 45 ms (6th bubble). 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 $R=66 \pm 1\,\mathrm {\mu }$m. However, the tiny variations in $R$ already cause slight deviations in the detachment process; see left inset in figure 3(a), and table 1. As such, the distances $H_{E2}$ at the onset of $E_2$ vary, as seen in the right inset in figure 3(a), where the different bubble paths are also labelled by numbers.

Table 1.

Summary of the characteristic values from figure 3 for the three different scenarios. Here, $V_{E2}$ refers to the free rise velocity of the first bubble at distance $H_{E2}$, and $V_{p.1}$ and $V_{p.2}$ characterize the velocity peaks of the first bubble; see figures 3(b) and 3(e). Also, $\Delta t$ is the time interval between instants of time 5 and 9 as marked in figure 2(b).

The scatter in $H_{E2}$, although smaller than $30\,\mathrm {\mu }$m in size, plays a crucial role in deciding which scenario the bubble follows later on. For small distances $H_{E2}< H_{E2.crit}$, the first bubble follows scenario I (lines 1–3) as described in § 3.1. For larger distances $H_{E2.crit} \leqslant H_{E2} < H_{E2.6}$, a different scenario II is found (lines 4 and 5), which is visualized in figure 3(c). Here, $H_{E2.6}$ refers to $H_{E2}$ belonging to line 6 (orange) from figure 3(a). In scenario II, a reversal of the motion of the first bubble at a distance $H_{rev}$ is also observed. However, the values of $H_{rev}$ 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 bubbles 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 $H_{E2.crit} \approx (H_{E2.3}+H_{E2.4})/2=346\,\mathrm {\mu }$m, during which the difference $\Delta H_{E2}=H_{E2.4}-H_{E2.3}$ amounts to $4\,\mathrm {\mu }$m only. At larger distances $H_{E2} \geqslant H_{E2.6} \sim 388\,\mathrm {\mu }$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.

Three velocity fields around the ${\rm H}_2$ bubbles, obtained by PTV (cf. § 2) over 3.2 ms (16 images) and additionally averaged over 105 cycles, are documented in figure 3(d). The left velocity field refers to the moment when the potential is set to $E_2=-8$ V; the one in the middle is acquired around the point of motion reversal; and the right one captures the period when the first bubble moves towards the electrode. Here, the vectors specify the direction of the electrolyte velocity $V_{el}$, and the colour represents its vertical component $V_{el.z}$. Images of the bubbles, drawn to scale, are superimposed schematically. 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 $H$. Thus (first) bubbles at a higher $H_{E2}$ 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 the left and middle/right images in figure 3(d). Although this flow is still directed upwards in the middle image, which shows the bubble at $H_{rev}$, the bubble starts to reverse the direction of its movement, and is accelerated towards the electrode (also see the right image). We further note that the high magnitude of the velocity visible at the foot of the second bubble (figure 3d) 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 (Yang et al. Reference Yang, Baczyzmalski, Cierpka, Mutschke and Eckert2018). 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(b) are summarized in table 1 for the three different scenarios represented by lines 1–6. 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 (Clift, Grace & Weber Reference Clift, Grace and Weber2005). For the present case ($R = 66\,\mathrm {\mu }{\rm m}$, Reynolds number $Re \approx 5$, Eötvös number $Eo \ll 1$), $V_t$ can be estimated as $V_{t} = {2R^2\,\Delta \rho \,g}/{9 \mu } \sim 8.8\,{\rm mm}\,{\rm s}^{-1}$, 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 $H<400\,\mathrm {\mu }$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 1.7–2.4. 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.

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 functions of the distance $H_{E2}-R_1$, 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 $1\,{\rm ms}$ and 5 ms. As can be seen, all three scenarios are examined. When $H_{E2}-R_1$ increases, $V_{p.1}$ falls monotonically from almost $60\,{\rm mm}\,{\rm s}^{-1}$ at $H_{E2}-R_1\sim 115\,\mathrm {\mu }$m to reach a plateau $V_{p.1} \sim 10\unicode{x2013}20\,{\rm mm}\,{\rm s}^{-1}$ at large $H_{E2}-R_1$ 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 second bubble, the existence of a plateau at $H_{E2}-R_1 \approx 300\,\mathrm {\mu }$m demonstrates that a maximum distance must not be exceeded for an interaction between both bubbles. Furthermore, $V_ {p.1}$ increases, especially at larger distances $H_{E2}-R_1 > 200\,\mathrm {\mu }$m, 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 $(V_{p.1:min}/V_t)_{97\,\mathrm {\mu }{\rm m}} = 0.96$, $(V_{p.1:min}/V_t)_{80\,\mathrm {\mu }{\rm m}} = 1.15$ and $(V_{p.1:min}/V_t)_{60\,\mathrm {\mu }{\rm m}} = 1.7$. As seen in table 1, the terminal velocity is not quite achieved at $H_{E2} < 400\,\mathrm {\mu }$m. Hence in all three cases ($R_{60}, R_{80}, R_{97}$), 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 $V_{p.2}$ behaves differently and shows a parabola-like behaviour at all bubble sizes $R_1$. When $H_{E2}-R_1$ increases, in the case of the smallest bubble $R_1=60\,\mathrm {\mu }$m, $-V_{p.2}$ rises to the maximum value of around $55\,{\rm mm}\,{\rm s}^{-1}$. This is followed by a gradual decline to zero, denoting the end of scenario II if $H_{E2}-R_1$ becomes too large. The local maximum is shifted to higher $H_{E2}-R_1$ values if the radius $R_1$ is increased. The existence of a local maximum of $-V_{p.2}$ 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 $V_{p.2}$ before coalescing with the second bubble.

3.3. Scenario diagrams

Figure 4 summarizes the occurrence of the three scenarios, discussed in figure 3, via the dependence of $H_{E2}-R_1$ on the radius of the first bubble, $R_1$ (figure 4a), and the potential $E_2$ (figure 4b). We recall that $H_{E2}$ marks the distance between the bubble centre and the electrode when the potential $E_2$ is set. Thus $H_{E2}-R_1$ specifies the distance from electrode to bubble nadir (bottom). The experiments in figure 4(a) were performed over 437 cycles at a constant potential $E_2= -8$ V, and in figure 4(b) over 233 cycles at nearly constant radius $\overline {R_1}=68.8\pm 0.5\,\mathrm {\mu }$m. Scenario II, represented by red triangles, acts as a sharp border between scenarios I (orange circles) and III (blue squares). The dotted lines are a linear fit (figure 4a) and a second-degree polynomial fit (figure 4b) of the experimental points of scenario II (red triangles). The fits show that the transition from scenario I to scenario II occurs at smaller distances $H_{E2}-R$ when either the bubble size is increased or the absolute potential is reduced.

3.4. Schlieren images of scenarios I–III

To provide evidence on the potential mechanisms underlying scenarios I–III, we analyse in figure 5 the vertical temperature gradient using schlieren imaging. As shown in § 2, the term $J_s/J_b - 1$, in which $J_s$ and $J_b$ refer to the recorded intensities of the schlieren images with and without bubbles, is proportional to $\partial T/\partial z$; cf. (2.4). If $J_s/J_b-1 > 0$, then the temperature gradient in the vertical direction (from bottom to top) is positive; it is negative for $J_s/J_b-1 < 0$. Thus the red and blue colours in figure 5(a) denote an increase (red) or decrease (blue) in the temperature in the vertical direction (from bottom to top). Although the extent and instant of their appearance differ, figure 5(a) 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 $H_{E2}$, belonging to scenario I, a denser schliere, i.e. a stronger gradient, is established close to the nadir of the first (top) bubble. This is illustrated in more detail by the horizontal profiles of $J_s/J_b-1$ shown in figure 5(b) for each scenario. These profiles were taken in the grey boxes between both bubbles, depicted in the first row in figure 5(a) and averaged over 10 pixels ($17.4\,\mathrm {\mu }$m) in the vertical direction. The black, red and blue lines marked as 2, 3 and 4 correspond to the snapshots in columns 2–4 of figure 5(a). On comparing the three scenarios, it becomes obvious that $J_s/J_b-1$ attains its highest (negative) magnitude for scenario I, and reduces constantly towards scenario III. Thus scenario I indeed possesses the highest temperature gradient in the vicinity to the bottom of the first bubble, i.e. from $-$100 to $100\,\mathrm {\mu }$m. As a result, a significant temperature gradient along the interface of the first bubble is created. The resulting shear stress (cf. § 4) is assumed to be responsible for its downward acceleration towards the second bubble. 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), then 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 5.

(a) Schlieren images, represented by the dimensionless intensity $J_s/J_b-1$, for different stages of scenarios I–III (1st–3rd row), shortly after $E_2=-8$ V is switched on, taken at $\Delta t= 3$ ms (1st row) and 4 ms (2nd and 3rd rows). (b) Profile of $J_s/J_b-1$ over the width of the image. Zero position – see the first image in (a) – corresponds to the electrode centre, which nearly coincides with the bubbles’ centres. The data were taken inside the grey boxes shown in (a). The black (2), red (3) and blue (4) lines correspond to the columns of the same number in (a), i.e. different instants of time. Panel (a) is supplemented by movies 5 to 7.

3.5. Multiple cycles

While previously only single cycles, as shown in figure 2, were studied, we analyse in figure 6(a) the position $H$ of the first (red lines) and second (blue lines) bubbles during four cycles, C1–C4, of modulation of the potential (black line) within a single experimental run. The only difference to the foregoing experiments was that the second bubble during C1–C3 is produced in the same way as the first bubble, namely at $E_2=-6$ V for 1 ms. In the last cycle (C4), the potential $E_2$ was set to $-8$ V and applied continuously. Figure 6(b) documents snapshots of the bubbles’ behaviour for each cycle C1–C4 from figure 6(a), with the corresponding time stamps. Each cycle C1–C4 ends by coalescence. As a result, the coalescence radius of the first bubble increases from $R_1 = 70\,\mathrm {\mu }$m (C1) via $88\,\mathrm {\mu }$m (C2) and $100\,\mathrm {\mu }$m (C3) towards $110\,\mathrm {\mu }$m (C4). The important observation occurs when $E_2$ is set from $E_2 = -6$ V to $E_2 = 0$ to force the detachment of the second bubble. Despite the vanishing potential $E_2$, the first bubble thus still follows scenario I in a nearly unchanged way, i.e. it decelerates, moves towards the electrode, and finally coalesces with the second bubble.

Figure 6.

Overview of the four bubble motion reversal cycles performed in succession. Each cycle, denoted by C1–C4, is marked by a different colour. (a) The variable $H$ specifies the distance between the first bubble (red line) and the second bubble (blue line) and the electrode centre over time. The modulation of potential $E$ is shown by the black line. (b) Snapshots of the bubbles’ behaviour. The time stamps correspond to (a).