Numerical Simulation and Modeling of Hydrogen Gas Evolution on Planar and Microwire Array Electrodes

The E–E Model can be slightly simplified in the situation of interest, because the momentum and continuity equations for the two phases can be combined, while retaining a gas phase transport equation to track the volume fraction of the bubbles.

The momentum equation is:

$$\begin{eqnarray}&&\begin{array}{l}{\varnothing }_{l}{\rho }_{l}\displaystyle \frac{\partial {u}_{l}}{\partial t}+{\varnothing }_{l}{\rho }_{l}{u}_{l}\cdot {\rm{\nabla }}{u}_{l}=-{\rm{\nabla }}p\\ \,\,\,\,\,\,\,\,\,\,+\,{\rm{\nabla }}\cdot \left[{\varnothing }_{l}\left({\mu }_{l}+{\mu }_{T}\right)\left({\rm{\nabla }}{u}_{l}+{\rm{\nabla }}{u}_{l}^{T}-\displaystyle \frac{2}{3}\left({\rm{\nabla }}\cdot {u}_{l}\right)I\right)\right]\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\,{\varnothing }_{l}{\rho }_{l}g+F,\end{array}\end{eqnarray} \tag{ 3 }$$

where u represents the velocity vector, p and ρ are the pressure and the density, respectively, ϕ is the phase volume fraction, μ_l is the dynamic viscosity of the liquid, and μ_T is the turbulent viscosity. The turbulent viscosity is neglected herein because the flows can be assumed to be laminar due to the low Reynolds numbers. The subscripts "l" and "g" denote quantities related to the liquid phase and gas phase, respectively.

The continuity equation is:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left({\rho }_{l}{\phi }_{l}+{\rho }_{g}{\phi }_{g}\right)+{\rm{\nabla }}\cdot \left({\rho }_{l}{\phi }_{l}{u}_{l}+{\rho }_{g}{\phi }_{g}{u}_{g}\right)=0,\end{eqnarray} \tag{ 4 }$$

and the gas-phase transport equation is:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\rho }_{g}{\phi }_{g}}{\partial t}+{\rm{\nabla }}\cdot \left({\rho }_{g}{\phi }_{g}{u}_{g}\right)=-{m}_{gl},\end{eqnarray} \tag{ 5 }$$

where m_gl represents the rate of mass transfer the gas to the liquid. The gas velocity u_g is calculated as the sum of the liquid phase velocity, u_l , the relative velocity between the phases, u_slip , and the drift velocity, u_drift :

$$\begin{eqnarray}&&{u}_{g}={u}_{l}+{u}_{slip}+{u}_{drift}\end{eqnarray} \tag{ 6 }$$

The drift velocity is associated with turbulence and thus was neglected in our treatment.

In the bulk electrolyte, the sum of the forces on the bubbles is composed of gravity, buoyancy, viscous drag force, shear-induced lift force, and virtual mass force. The virtual mass force accounts for the contribution of the changing volume of the bubbles. ¹⁷ A comparison of the size of the different terms allows for a simplifying assumption that the pressure forces on a bubble are balanced by the viscous drag force, f_D , with the resulting simplified treatment commonly designated as the pressure-drag balance model:

$$\begin{eqnarray}&&{\varnothing }_{l}{\rm{\nabla }}p={f}_{D},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{f}_{D}=-{C}_{d}\displaystyle \frac{3}{4}\displaystyle \frac{{\rho }_{l}}{{d}_{b}}\left|{u}_{slip}\right|{u}_{slip},\end{eqnarray} \tag{ 8 }$$

where d_b represents the bubble diameter and C_d represents the viscous drag coefficient.

Hadamard and Rybczynski have proposed a model to calculate the drag coefficient for small spherical bubbles with a diameter less than 2 mm and with a bubble Reynolds number (R_e,b) less than 1:

$$\begin{eqnarray}&&{C}_{d}=\displaystyle \frac{16}{{R}_{e,b}},\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{R}_{e,b}=\displaystyle \frac{{d}_{b}{\rho }_{l}\left|{u}_{slip}\right|}{{\mu }_{l}}.\end{eqnarray} \tag{ 10 }$$

The gas density ρ_g is obtained using the ideal gas law:

$$\begin{eqnarray}&&{\rho }_{g}=\displaystyle \frac{\left(p+{p}_{ref}\right)M}{RT},\end{eqnarray} \tag{ 11 }$$

where M is the molecular weight of the gas, R is the ideal gas constant (8.314 J mol⁻¹ K⁻¹), p_ref is a reference pressure, which in our work is 1 atm, and T is the absolute temperature. In this work, room temperature was taken to be T = 298.15 K.

For two-phase flow,

$$\begin{eqnarray}&&{\varnothing }_{l}=1-{\varnothing }_{g},\end{eqnarray} \tag{ 12 }$$

The mass transport rate from the gas phase to the liquid phase, m_gl , can also be specified through two-film theory as:

$$\begin{eqnarray}&&{m}_{gl}=k\left({c}^{* }-c\right)Ma,\end{eqnarray} \tag{ 13 }$$

where k represents the mass transfer coefficient, a is the interfacial area per unit volume, c is the local dissolved gas concentration, and c* is the equilibrium concentration of the gas dissolved in liquid, as given by:

$$\begin{eqnarray}&&{c}^{* }=\displaystyle \frac{p+{p}_{ref}}{H},\end{eqnarray} \tag{ 14 }$$

where H is the Henry's law constant.

The interfacial area per volume can be calculated through the number density, n, and the volume fraction of gas, ϕ_g :

$$\begin{eqnarray}&&a={(4n\pi )}^{1/3}{(3{\varnothing }_{g})}^{2/3}.\end{eqnarray} \tag{ 15 }$$

Simulation of the transport of electrolyte species was performed using the Nernst-Plank equation:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {c}_{i}}{\partial t}+{\rm{\nabla }}\cdot \left(-{D}_{i}{\rm{\nabla }}{c}_{i}-{z}_{i}{u}_{m,i}F{c}_{i}{\rm{\nabla }}V\right)+{u}_{l}\cdot {\rm{\nabla }}{c}_{i}={R}_{i},\end{eqnarray} \tag{ 16 }$$

where D_i , c_i , z_i and u_m,i are the diffusion coefficient, concentration, charge number, and mobility, respectively, of species i within the electrolyte. Because the Euler-Euler approach considers both phases as interoperating liquids and uses a continuum approach to the gas bubble and liquid phases, the diffusion coefficient of the dissolved hydrogen was decreased in the simulations to account for the impact of the existing bubbles. Specifically, the diffusion coefficient was multiplied by the volume fraction of bubbles in the solution. R_i indicates the reaction and mass transfer rate of species i in the electrolyte. The species of concern are dissolved hydrogen (H₂), H⁺ , and OH⁻. For H⁺ and OH⁻, the reaction rate is calculated through water dissociation. For dissolved hydrogen, although no chemical reaction is involved, mass transfer rates are important due to the difference between the equilibrium concentration and the local dissolved gas concentration. The mass transfer rate is calculated through Eq. 13.

The value of u_m,i can be calculated by Nernst-Einstein relation:

$$\begin{eqnarray}&&{u}_{m,i}=\displaystyle \frac{{D}_{i}}{RT}.\end{eqnarray} \tag{ 17 }$$

The only chemical reaction considered herein was water dissociation:

$$\begin{eqnarray}&&{H}_{2}O\underset{\mathop{\unicode{x027F5}}\limits_{{k}_{1b}}}{\overset{{k}_{1f}}{\longrightarrow }}{H}^{+}+O{H}^{-}\end{eqnarray} \tag{ 18 }$$

The dissolved gas has an additional rate, which is related to the mass transfer rate, m_gl :

$$\begin{eqnarray}&&{R}_{g}=-{m}_{gl}/M.\end{eqnarray} \tag{ 19 }$$

The electrolyte was assumed to be 0.1 M to 1.0 M H₂SO₄ that was saturated with dissolved hydrogen. The initial pressure in the solution was set as ρ_l gh + p_ref , where h is the distance from the outer boundary layer. The other parameters were the same as those specified by Glas and Westwater (Table I).

Table I. Kinematic and operating parameters used in the modeling.

Diffusion coefficient of H2, ${D}_{{H}_{2}}$	7.38 × 10−9 m2 s−1
Diffusion coefficient of H+, ${D}_{{H}^{+}}$	9.31 × 10−9 m2 s−1
Diffusion coefficient of OH−, ${D}_{O{H}^{-}}$	5.26 × 10−9 m2 s−1
Surface tension, σ	0.075 N m−1
Solution density, ρl	1000 kg m−3
Hydrogen gas density, ρg	0.09 kg m−3
Saturation concentration of H2 in solution, ${c}_{{H}_{2}}^{0}$	0.78 mM
Operating temperature, T	298.15 K
Operating pressure, p	1 atm

Two-phase fluid model

At the electrolyte-electrode interface, a slip condition was assumed for the liquid phase, whereas the gas mass flux and the number density flux were fixed:

$$\begin{eqnarray}&&{u}_{l}\cdot n=0.\end{eqnarray} \tag{ 20 }$$

A slip condition was assumed for the liquid phase at the boundary, allowing for the (initial) tangential motion of the liquid but restricting the (initial) normal velocity to zero. At the other end of the bulk electrolyte layer, no slip condition was assumed for the liquid phase, and the gaseous phase was considered as the outlet. At the vertical wall, no slip condition was assumed for the liquid phase, whereas no flux was set for the dispersed phase. The no slip condition means that the fluid will have zero velocity relative to the boundary.

The break-off radius of the bubbles, R_r , is required to deduce the mass flux and the number density flux of the gas. Many factors determine R_r during electrolysis, including the electrolyte; surface roughness and wettability; applied current density; operating temperature and pressure; gas and liquid density; dissolved gas diffusion coefficient in the solution; saturation concentration of dissolved gas in the solution; and the surface tension between the gas and the electrode surface as well as between the gas and the solution. The value of R_r is also affected by disturbances due to adjacent bubbles, which can result in premature bubble departure from the electrode surface. Experimental observations indicate that the bubble break-off radius is primarily influenced by the applied nominal current densities and the roughness of the electrode. Data and analysis have been presented by Vogt describing the change in the break-off bubble radius as a function of the nominal current density for different electrode roughness factors. ¹⁸ In our work, this relationship was used to provide approximate parameterized values for R_r (Fig. 2).

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The bubble break-off radius can be related to the fractional bubble coverage on the electrode surface:

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{r}}{{R}_{f}}={(1+1200{\rm{\Theta }})}^{-0.5},\end{eqnarray} \tag{ 21 }$$

where R_f denotes the bubble radius at zero current as calculated in the Fritz equation, ¹⁸

$$\begin{eqnarray}&&{R}_{f}={\left(\displaystyle \frac{3}{2}\displaystyle \frac{\sigma {R}_{site}}{\left({\rho }_{l}-{\rho }_{g}\right)g}\right)}^{1/3},\end{eqnarray} \tag{ 22 }$$

The following two assumptions are inherent to this treatment:

• 1)  
  Bubbles nucleate as a gas pocket in the cavities on the electrode, the electrode surface contains surface defects in the form of cavities. These cavities can be considered to be shaped geometrically as a conical void with a rounded bottom near the electrode surface that narrows to a sharp tip within the material. R_site = 20 μm (radius of the flat surface of the conical void) represents a rougher surface than R_site = 4 μm. Figure S5 (available online at stacks.iop.org/JES/169/066510/mmedia) presents a schematic illustration of the mechanism of gas bubble evolution at the electrode surface. Molecular diffusion governs the bubble growth, therefore, even if a conical cavity is close by, another bubble will not necessarily be generated and thus the number of the cavities on the electrode surface is not an important factor. According to Vogt, ¹⁸ the fractional bubble coverage can be estimated by parameterization using the bubble break-off radius and the bubble radius at zero current. This relationship is restricted to slow motion of the electrolyte liquid, which is in accord with the assumptions made herein. Scratches and pits on the electrode surface are the active nucleation sites for oxygen or hydrogen bubbles during the electrolysis process. ¹⁹
• 2)  
  A force balance between buoyancy and surface tension is achieved when bubbles on the electrode reach the break-off radius.

The total amount of product gas transferred into the gaseous phase adhering to the electrode surface, f_G , is determined by the fractional bubble coverage, Θ, of the electrode. At very small values of the current density (<1 mA cm⁻²) with nearly no bubbles adhering to the electrode surface, Θ → 0, and all the generated product gas crosses the electrolyte-electrode interface as dissolved gas, f_G → 0. When the electrode surface is overcrowded by adhering bubbles, Θ → 1, and almost 100% of the dissolved gas is transferred into the gaseous phase, f_G → 1. This relationship has been studied using various mathematical models, the outcomes of which do not differ substantially and can approximately be described by: ^20–22

$$\begin{eqnarray}&&{f}_{G}=0.55{{\rm{\Theta }}}^{0.1}+0.45{{\rm{\Theta }}}^{8}.\end{eqnarray} \tag{ 23 }$$

The gas mass flux and number density flux are then given through f_G :

$$\begin{eqnarray}&&{m}_{gas,in}=\displaystyle \frac{| J| {f}_{G}M}{zF},\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&nd=\displaystyle \frac{| J| {f}_{G}M}{zF{\rho }_{g}{V}_{g}},\end{eqnarray} \tag{ 25 }$$

where J represents the nominal cathodic current density and V_g represents the volume of a single releasing bubble from the electrode surface.

Nucleation was treated in accord with conventional approaches used in modeling of bubbles on electrode surfaces. Observations of gas evolution on the microscopic scale reveal that the electrode surface is the site of frequent nucleation, growth, and detachment of bubbles. The total amount of the produced hydrogen through the electrochemical reaction transferred into the gaseous phase is determined by the fractional bubble coverage of the electrode, which can be estimated through the bubble break-off radius and the cavity/cone radius. ¹⁸ The effects of the bubbles on the overpotential of the cathode were determined by assessing the ohmic losses associated with physical obstruction of bubbles within the electrolyte, as well as the concentration overpotential and kinetics effects associated with increases the effective current density due to bubbles masking the electrode surface. ⁹

Transport of electrolyte species model

At the outer bulk electrolyte layer, constant concentrations of all electrolyte species at the initial conditions were assumed, due to the high convective fluxes beyond the boundary layer. At the electrode-electrolyte interface, the inward flux of H⁺ as well as dissolved hydrogen across the boundary can be specified as:

$$\begin{eqnarray}&&{n}_{{H}^{+}}=\displaystyle \frac{J}{F},\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{n}_{dissolved,in}=\,\displaystyle \frac{\left|J\right|\left(1-{f}_{G}\right)}{zF}.\end{eqnarray} \tag{ 27 }$$

On the vertical wall, no flux for each species was assumed in the simulations.

In our work, electroneutrality has been used due to the high ionic strengths assumed. The electrode-electrolyte interface was taken as the reference boundary for the electrolyte potential.

The standard finite element method (FEM) solver in the COMSOL multi-physics package was used to model the two-phase flow as well as the electrochemical behavior. The maximum element size, the maximum element growth rate and curvature factor for this 2-D axisymmetric model were 1.3 μm, 1.08 and 0.25, respectively. For all simulations, a relative tolerance of the corresponding variable of 0.001 was applied as the convergence criterion.

Based on the data in Fig. 2, when the radius of the surface cavity decreases, the break-off bubble radius decreases as well, resulting in an increase in the bubble volume fraction in the bulk electrolyte domain. For a planar electrode with a relatively rough surface R_site = 20 μm, the generating bubble volume fraction within the 100 μm boundary layer varied from 0.74% to 12.7% as the nominal current density was varied from 100 to 300 mA cm⁻² (Fig. 4). In contrast, for a planar electrode with surface cavity radius of 4 μm, the H₂ gas volume fraction is 1.9% at a current density of 100 mA cm⁻² but the generating bubble volume fraction increases to 57.8% when the current density increases to 300 mA cm⁻² (Fig. 4a).

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Hydrogen bubbles block some of the pathways of ion transport within the solution, so the conductivity of the bulk electrolyte decreases as the bubble volume fraction increases. Maxwell's equation describes the relationship between the ratio of the conductance with dispersed a phase present to the conductance in the absence of the dispersed phase, K_m , and the void fraction, f_void , ²⁵

$$\begin{eqnarray}&&{K}_{m}=\left(1-{f}_{void}\right)/\left(1+\displaystyle \frac{{f}_{void}}{2}\right).\end{eqnarray} \tag{ 29 }$$

Zhao et al. ²⁵ summarized the numerical method to calculate the potential drop within the electrolyte. Maxwell's equation has been shown to fit the experimental data well when the volumetric fraction of gas adhering to the electrode surface bubble monolayer is less than 50%. ²⁶ For a planar electrode with a surface cavity radius equal to 20 μm, K_m decreases from 0.99 to 0.82 when the nominal current density is increased from 100 to 300 mA cm⁻². Under such conditions, bubbles therefore do not have a substantial impact on the conductance of the electrolyte. In contrast, if the cavity radius on the electrode surface is 4 μm, K_m is > 0.97 at a current density of 100 mA cm⁻², but due to the high volume fraction of the hydrogen gas bubbles in the solution, K_m decreases to 0.33 as the nominal current density increases to 300 mA cm⁻².

Based on the calculated value of K_m , Eq. 29 can then be applied to evaluate the additional ohmic drop during gas evolution ${\rm{\Delta }}{E}_{ohm}:$ ⁹

$$\begin{eqnarray}&&{\rm{\Delta }}{E}_{ohm}={\rm{\Delta }}{E}_{ohm}^{{\prime} }+\displaystyle \frac{\left|J\right|L}{\kappa },\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{E}_{ohm}^{{\prime} }=\displaystyle \frac{\left|J\right|L(1-{K}_{m})}{\kappa {K}_{m}},\end{eqnarray} \tag{ 31 }$$

where the second term on the right-hand side of Eq. 30 represents the ohmic drop in the absence of gas bubbles. ${\rm{\Delta }}{E}_{ohm}^{{\prime} }$represents the remaining ohmic drop after removal of this term, and κ is the solution conductivity, which can be defined as: ²⁷

$$\begin{eqnarray}&&\kappa =F\displaystyle \sum _{i}\left|{z}_{i}\right|{u}_{m,i}{c}_{i}.\end{eqnarray} \tag{ 32 }$$

For a planar electrode with a surface cavity radius of 20 μm, ${\rm{\Delta }}{E}_{ohm}^{{\prime} }$ is 0.036 mV at a nominal current density of 100 mA cm⁻², with ${\rm{\Delta }}{E}_{ohm}^{{\prime} }$ increasing slightly to 2.12 mV at a current density of 300 mA cm⁻² (Fig. 4b). In contrast, when the surface cavity radius is reduced to 4 μm, ${\rm{\Delta }}{E}_{ohm}^{{\prime} }$ is 0.095 mV at a current density of 100 mA cm⁻², but ${\rm{\Delta }}{E}_{ohm}^{{\prime} }$ increases to 19.99 mV at a current density of 300 mA cm⁻², primarily due to the substantial reduction in the conductance of the solution.

For a planar electrode system with a boundary layer of 100 μm and a surface cavity radius equal to 4 μm, the dissolved H₂ concentration at the electrode-electrolyte interface decreases from 33.76 mM to 9 mM when the nominal current density increases from 100 mA cm⁻² to 300 mA cm⁻² (Fig. 5). At a surface cavity radius of 20 μm, the dissolved H₂ concentration at the electrode-electrolyte interface initially increases as the nominal current density increases to 200–250 mA cm⁻² and then deceases as the current density increases further. The Euler-Euler approach utilized considers both liquid and gas phases as interoperating liquids and uses a continuum approach to both phases, therefore, a single value was used to represent the surface concentration for the planar electrode configuration (average concentration on the surface).

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An accumulation of dissolved H₂ at the electrode-electrolyte interface will shift the local reversible hydrogen electrode potential in accord with the Nernst equation:

$$\begin{eqnarray}&&{\eta }_{C}=\displaystyle \frac{2.3RT}{zF}lo{g}_{10}\displaystyle \frac{{c}_{{H}_{2}| \,at\,the\,interface}}{{c}_{{H}_{2}}^{0}}.\end{eqnarray} \tag{ 33 }$$

The value of η_C is calculated based on the dissolved H₂ concentration at the electrode-electrolyte interface (Fig. 5). With a surface cavity radius equal to 4 μm, η_C decreases from 48 mV to 31 mV as the nominal current density increases from 10 to 300 mA cm⁻². When the surface cavity radius increases to 20 μm, η_C reaches 49 mV at a current density of 100 mA cm⁻², and η_C increases with the nominal current density to 55 mV at 250 mA cm⁻² and then decreases as the applied current density increases further. The values of η_C and η_Tafel have mutually the same order of magnitude and therefore η_C is a non-negligible term with regard to the total overpotential in the system.

When electrode surfaces are covered with adhering hydrogen bubbles, an additional overpotential is produced due to an increase in effective current density at sites that are not covered by bubbles. This additional overpotential is given by:

$$\begin{eqnarray}&&{\eta }_{h}=b\,lo{g}_{10}\displaystyle \frac{A}{{A}^{^{\prime} }},\end{eqnarray} \tag{ 34 }$$

where η_h represents the hyperpolarization. The value of η_h is determined by the Tafel behavior as well as by the ratio of the surface area A to the remaining active area A'. The value of A/A' is related to the fractional bubble coverage on the electrode surface, Θ (Fig. 6). The value of Θ increases as the nominal current density increases. When the surface cavity radius is 20 μm, at a nominal current density of 100 mA cm⁻², Θ is 12.9%, but Θ is 36.8% at a current density of 300 mA cm⁻². The fractional bubble coverage is not substantially affected by the surface cavity radius at the electrode, because the bubble departure radius and the Fritz radius are correlated. Hence, the surface roughness of the electrode does not substantially impact the hyperpolarization. The value of η_h varies slightly, from 1.4 mV to 5.9 mV, under nominal current densities between 100 mA cm⁻² and 300 mA cm⁻². The value of η_h is one order of magnitude less than η_Tafel , so a change in η_h does not substantially affect the total overpotential in the cathodic side of the cell.

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The total potential drop between the cathode and reference electrode, after correction for the cell resistance, can be expressed by Eq. 35:

$$\begin{eqnarray}&&{\eta }_{T}={\eta }_{Tafel}+{\eta }_{C}+{\eta }_{h}+{\rm{\Delta }}{E}_{ohm}^{{\prime} }.\end{eqnarray} \tag{ 35 }$$

Figure 7 shows that for a planar electrode with a surface cavity radius of 20 μm, the overpotential related to gas bubbles reaches 51 mV under a nominal current density of 100 mA cm⁻², and under such conditions constitutes 46% of the total potential drop between the cathode and reference electrode. This overpotential increases as the nominal current density increases, and reaches 62 mV at 300 mA cm⁻², but makes a slightly lower relative contribution of 45% to the total potential drop, because η_Tafel increases logarithmically as the current density increases. The effect of the hydrogen bubbles in the bulk on the total potential drop, η_T , is not changed when the surface cavity radius decreases to 4 μm. With smaller cavities, bubbles have a larger influence on the solution conductance, especially at current densities > 200 mA cm⁻², but their impact on the local reversible hydrogen electrode potential is smaller due to the lower dissolved H₂ concentration at the electrode-electrolyte interface.

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The computations described above indicate that for planar electrodes, the main impact of the hydrogen gas on the system resistance is due to the accumulation of dissolved H₂ at the electrode-electrolyte interface. In this section, the behavior of microwire array electrodes is described and η_C is determined under the same nominal current densities as for the planar electrodes. A Pt catalyst was coated over the microwire array electrode surface (Figs. 1b and 1c). Array geometries having 6 μm diameter microwires on a 14 μm center-to-center pitch, μW 6 ∣ 14, and 3 μm diameter microwires on an 11 μm pitch, μW 3 ∣ 11, were investigated.

When the surface cavity radius is set at 4 μm under a nominal current density of 100 mA cm⁻², the volume fraction of the gaseous hydrogen appears to be low near the bottom of the wires, and then increases along the z—axis. The H₂ gas volume fraction in the solution is 1.9% for the planar electrode but is only 0.33% for the μW 6 ∣ 14 electrode. Due to the increased electrode surface area of the microwire array structure, at the same nominal current density, the local current density is lower than for the planar electrode, resulting in smaller break-off bubble radii and less dissolved hydrogen gas transferring into the bubbles. Hence, the H₂ gas volume fraction in the solution is only 0.3% for μW 3 ∣ 11 array electrodes.

Lateral and radial concentrations gradients exist for the non-planar electrode configuration. Figure 8a indicates that the dissolved H₂ concentration varies at different locations on the surface of the microwire array electrode. On the tops of the microwires, the dissolved H₂ concentration is lower than on the planar electrode, leading to a smaller local shift in the reversible hydrogen electrode potential, η_C . On the sides of the microwires, the dissolved H₂ concentration decreases from 39.69 mM near the bottom to 30.45 mM near the top of the microwires, and the calculated η_C consequently decreases from 50.42 mV to 47.02 mV. Under the same conditions, the dissolved H₂ concentration at the electrode-electrolyte interface for the planar electrode is 33.76 mM and η_C is 48.35 mV. On the rest of the surface of a μW 6 ∣ 14 microwire array electrode, the dissolved H₂ concentration is approximately 6 mM higher than for the planar configuration, increasing η_C by 2 mV.

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Figure 9 indicates that the dissolved H₂ at the electrode-electrolyte interface for a μW 3 ∣ 11 microwire array is very similar to that of a μW 6 ∣ 14 microwire array. Use of this microwire array instead of a planar electrode increases the solution conductance due to the lower bubble volume fraction, but does not substantially reduce the local shift in the reversible hydrogen electrode potential.

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On the μW 6 ∣ 14 microwire array, when the nominal current density increases to 150 mA cm⁻², the bubble volume fraction in the electrolyte solution increases to 0.82%. At nominally the same current density on the planar electrode, the bubble volume fraction is 7.2%, so compared to the planar electrode the microwire array configuration reduces the bubble volume fraction by 9 times, whereas at a current density of 100 mA cm⁻², the microwire array reduces the bubble volume fraction by 5.7 times. The microwire array configuration thus advantageously decreases losses associated with changes in the solution conductance as the nominal current density increases. Moreover, when the nominal current density increases from 100 mA cm⁻² to 150 mA cm⁻², the dissolved H₂ concentration on the top end of the microwires increases by 12 mM, resulting in a 3 mV increase in η_C . In contrast with the planar electrode, the dissolved H₂ concentration at the electrode-electrolyte interface is reduced very slightly, from 33.76 mM to 33.42 mM. The dissolved H₂ accumulates even more on the other surface regions of the microwire array electrodes, because with a surface cavity radius of 4 μm and a nominal current density of 150 mA cm⁻², the bubble break-off radius is 102 μm for the μW 6∣14 microwire array electrode, whereas the bubble break-off radius is only 48 μm for the planar electrode. This change leads to a much higher mass transfer rate of hydrogen from the liquid phase to the gas phase and substantially less accumulation of dissolved hydrogen at the electrode-electrolyte interface.

The initial H₂SO₄ concentration was 1 M for all the simulations discussed above. Based on Eq. 32, the solution conductivity will decrease linearly as the H₂SO₄ concentration decreases. For a given current density, the ohmic drop in the solution thus increases as the H₂SO₄ concentration decreases. For a planar electrode, the bubble related ohmic drop decreases linearly as the initial H₂SO₄ concentration is increased (Fig. 10). With the same boundary layer thickness, the limiting current density will also linearly decrease as the H₂SO₄ concentration is decreased. For example, with a boundary layer thickness of 100 μm, the limiting current density is 159 mA cm⁻² for 0.5 M H₂SO₄, and is 318 mA cm⁻² for 1 M H₂SO₄.

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At the same nominal current density, the bubble break-off radius is smaller on a smoother electrode surface (Fig. 2), which leads to more dissolved hydrogen converting into the gaseous form and consequently causes an increased gas volume fraction in the solution. Gaseous H₂ will partially block the pathways for ion transport in the electrolyte, and thus result in an increase in the bubble associated ohmic drop. As shown in Fig. 4, the ohmic drop at a smoother electrode surface is six times higher than for a rough electrode surface under a nominal current density of 200 mA cm⁻². Moreover, this difference becomes more pronounced as the nominal current density increases.

As shown in Fig. 5, for rough electrode surfaces, at low current densities the increased amount of dissolved H₂ in the electrolyte is not large enough to compensate for the increasing dissolved H₂ flow from the electrode. As a result, the dissolved H₂ concentration at the electrode-electrolyte interface increases with increases in the nominal current density. In contrast, at current densities that produce a large amount of gaseous H₂ bubbles from the dissolved H₂ gas in the bulk, more dissolved hydrogen flows into the bulk solution, and the dissolved H₂ concentration at the electrode-electrolyte interface decreases accordingly. For smooth electrode surfaces, the amount of dissolved hydrogen in the bulk is always larger than the hydrogen produced by the electrochemical reaction, so the dissolved H₂ concentration at the electrode-electrolyte interface decreases as the nominal current density increases. An accumulation of dissolved H₂ at the electrode-electrolyte interface will shift the local reversible hydrogen electrode potential, and based on Eq. 33, the local reversible hydrogen electrode potential shift follows the same trend as the dissolved H₂ concentration at the electrode-electrolyte interface (Fig. 5).

As compared to a planar electrode, due to the decrease in the local current density, a microwire array electrode will reduce the bubble associated ohmic drop within the bulk solution but is not necessarily beneficial in terms of reducing the shift in the local reversible hydrogen electrode potential. The dissolved hydrogen flow from the bottom of the electrode and from the lateral area of the microwires will accumulate hydrogen, even though the local current density is lower than for the planar configuration. Consequently, the dissolved hydrogen concentration at the electrode-electrolyte interface will still increase, which will increase the local shift in the reversible hydrogen electrode potential.