## 1 Introduction
### Bubble Growth Dynamics  
Bubble dynamics is characterized by three key stages: nucleation, growth, and detachment [31]. Additionally, a fourth stage, i.e., coalescence of bubbles, can be evaluated as one of the growth stages. When two bubbles come into mutual proximity, they can merge via coalescence into a single bubble, resulting in a decrease in the surface energy from the reduction in the total area of the gas-liquid interface [32]. Nucleation occurs due to the elevated chemical potential of dissolved gas molecules near the electrode. Subsequent bubble growth proceeds through distinct regimes dominated by specific mechanisms. Ingeneral, the growth phase of bubbles can be represented by a power-law dependence of bubble radius \(r_{b}\) versus \(t\):  
\[r_{b}(t)=bt^{x}, \tag{1}\]  
where \(b\) is the growth coefficient and \(x\) is the growth exponent [33]. As growth progresses, the exponent reflects the dominant mechanism. The initial stage is governed by inertia from the surrounding liquid, with bubble radius (\(r_{b}\)) scaling linearly with time:  
\[r_{b}=bt. \tag{2}\]  
Experimental evidence of bubble growth during the initial stage [6, 33, 34, 35] is usually limited by the spatiotemporal resolution of the acquisition system. The second stage is a diffusion-controlled regime, where bubble growth roughly approaches a square root dependence, as described by [33, 35, 36, 37, 38, 39]  
\[r_{b}=bt^{\frac{1}{2}}. \tag{3}\]  
This occurs as the surrounding liquid becomes locally saturated with gas, and diffusion from the bulk electrolyte toward the bubble drives its growth. [PERSON] et al. [40] also noted that a possible deviation from this diffusion-controlled law is due to the non-uniform distribution of gas concentration that decreases with the distance from the electrode. Some studies have further shown that the mass transfer of the dissolved hydrogen is not the limiting factor of bubble growth rate [6, 36, 39, 41], particularly for high current densities. A third stage of bubble growth is reaction-limited, where bubble growth is constrained by the electrochemical reaction rate at the electrode surface. The bubble radius now scales as  
\[r_{b}=bt^{\frac{1}{3}}, \tag{4}\]  
resulting in a constant gas generation rate. Bubble growth is driven by the direct injection of gas at the bubble foot [33, 42]. In studies on micro- and nanoelectrodes, the bubble is usually larger compared to the size of the electrode during most of the bubble growth time, resulting in the typical presence of the direct injection [40]. [PERSON] et al. [6] have shown that the reaction-limited regime (i.e., smaller bubbles coalescing at the foot of the bigger bubble) is an important mechanism of the bubble growth, despite previous certainty regarding the validity of the growth laws being restricted to cases where bubbles do not mutually interfere. Bubble detachment represents the final phase, when the bubble unpins from the electrode surface. The process of detachment is often triggered by the breaking of the bubble neck [31], or by the loss of contact with an underlying \"carpet\" of microbubbles [43]. The bubble detachment can be enhanced via several methods and technologies, including hydrophilic surfaces, and the use of magnetic, supergravity, and ultrasonic fields [44]. The theoretical maximum radius of a bubble detaching from an upward-facing electrode in stagnant electrolyte can be described by [PERSON]'s formula [45, 46]:  
\[r_{d}^{*}=\left(\frac{3\,r_{0}\text{sin}(\theta)}{2\,\Delta\rho g}\right)^{ \frac{1}{3}}, \tag{5}\]  
where \(r_{0}\) represents the radius of the contact area between the bubble and the electrode, \(\sigma\) the surface tension, \(\theta\) the contact angle, \(\Delta\rho\) represents the density difference between the liquid and the gas phase, and \(g\) is the gravitational acceleration. In gas-evolving electrochemical processes, bubbles significantly influence system performance, particularly through overpotential losses.