Guidelines for the Rational Design and Engineering of 3D Manufactured Solid Oxide Fuel Cell Composite Electrodes

The model is based on the continuity equation for the ionic current, solved in both the porous composite domain and the dense domain (see Figure 1c), as follows:

The ionic current density i_O is calculated according to the Ohm's law:^33,34

where V_O and σ_O are the potential and ionic conductivity of the ion-conducting phase, respectively. k_eff represents the effective (or relative) conductivity factor of the ion-conducting phase within the composite domain. k_eff is defined as the ratio between the effective conductivity of the ionic phase and its bulk conductivity,^9,46 that is, k_eff = σ_O,eff/σ_O. Sometimes k_eff is defined as the ratio between the volume fraction and the tortuosity factor of the ion-conducting phase in the composite domain,^47,48 which is a definition equivalent to that adopted in this study. This dimensionless factor is always smaller than 1 and depends on the microstructural properties of the composite electrode, in particular on the volume fraction, connectivity, tortuosity and neck size of ion-conducting particles,⁴⁹ while being independent of the bulk conductivity σ_O. Eq. 1 considers another microstructural parameter of the composite electrode domain, which is the TPB length density L_TPB. The TPB density depends on porosity, composition and particle size of the composite domain.^9,12,41

The current density per unit of TPB length i_TPB in Eq. 1 is calculated according to a linear electrochemical kinetic expression:⁵⁰

where the term in brackets represents the activation overpotential while F, R and T are the Faraday constant, the gas constant and the absolute temperature, respectively. i⁰_TPB is the exchange current density per unit of TPB length and represents the kinetic constant of the charge-transfer reaction. More generally, Eq. 3 represents a linear local current-voltage relationship, which is often more appropriate for SOFCs than a non-linear Butler-Volmer kinetic expression.⁵¹

The mathematical problem is closed by boundary conditions. Symmetric boundary conditions (i.e., no flux) are applied at the lateral boundaries. A no flux condition is applied at the top of the composite domain, indicating that the current enters the electrode in electronic form only. A Dirichlet boundary condition V_O = –η_tot is applied at the bottom of the computational domain,²² representing the interface of the electrolyte with the cathode. Such a boundary condition sets the overpotential η_tot applied to the system composed by electrode and electrolyte. This set of equations and boundary conditions therefore implicitly ensures that the electronic current is completely converted into ionic current within the composite electrode domain.

A numerical solution of Eqs. 1–3 is required for complex 2D geometries as those represented in Figure 1b. The total current density i_tot converted within the electrode is numerically calculated by integrating for the ionic current density leaving from the bottom of the computational domain.³⁵ Note that since model equations are linear in V_O, i_tot scales linearly with η_tot.

In a flat electrode (Figure 1a), the ionic potential V_O can be assumed uniform in each planar cross section^48,52,53. In such a case, the mathematical problem represented by Eqs. 1–3 becomes one-dimensional, allowing for an analytical solution of the ionic potential along the thickness of the electrode as reported in other studies, for example, by Costamagna et al.^36,54

The total current density i^flat_tot is determined by considering the electrode resistance R_ed and the electrolyte resistance t_ey/σ_O in series as follows:

where t_ey is the electrolyte thickness.

By solving Eqs. 1–3, the electrode polarisation resistance R_ed is equal to:

where the minimum electrode resistance R^min_ed and the dimensionless Thiele modulus Γ are calculated as follows:

In Eq. 6b, t_ed is the electrode thickness while t*_ed is the characteristic thickness of the coupled reaction/conduction process.⁵⁵ Both the Thiele modulus and the characteristic thickness are typically defined in coupled reaction/diffusion problems in chemical engineering.^36,56,57 In particular, the characteristic thickness t*_ed depends on transport properties (σ_O), kinetic properties (i⁰_TPB) and microstructural characteristics of the composite electrode (L_TPB and k_eff), while being independent of the electrode thickness (note that the electrode thickness t_ed can arbitrarily be smaller, equal or larger than t*_ed). The characteristic thickness t*_ed can be regarded as the decay factor of the electrode resistance R_ed as a function of electrode thickness t_ed according to Eqs. 5 and 6b. More practically, t*_ed is an indicator of the characteristic distance from the electrolyte interface required for electrochemical reaction and ionic conduction to take place in ideal conditions. For an ideal semi-infinite electrode (i.e., t_ed → ∞), which provides the upper bound of total current, the current converted within the characteristic thickness t*_ed is equal to the 63.2% (i.e., 1–e^–1) of the total current converted in the electrode.

The analytical solution for a flat electrode reported in Eqs. 4–6 can be applied to estimate the resistance in the limit case of semi-infinite electrode thickness even for electrodes with mesoscale structural modification.

The geometric ratio ω is defined as:

where W is the half-width of pillars and w is the half-distance between pillar walls, see Figure 1c. In the limit of t_ed → ∞, in 2D ω is equal to the volume fraction of pillars, while in 3D the volume fraction of pillars is equal to ω = W²/(W + w)². In this condition the electrode can be regarded as a homogeneous continuum, with average microstructural properties, identified with a bar sign, evaluated as follows:

Eq. 8a indicates that the average TPB density is weighted for the volume fraction of composite electrode domain 1–ω, that is, the volume fraction of electrode wherein the electrochemical reaction can take place. Similarly, the average effective conductivity factor of the ion-conducting phase is calculated according to the Wiener equation^58,59, by averaging for the effective conductivity factor of the dense pillars, equal to 1, and the effective conductivity factor of the composite electrode domain, equal to k_eff.

These average microstructural properties are used to calculate approximate values of the electrode resistance and current density with mesoscale structural modification in the limit of semi-infinite thickness, as follows:

Eqs. 9a and 9b are formally equal to Eqs. 6a and 4, respectively, derived previously for a flat electrode, however with the use of the average microstructural properties and as defined in Eq. 8.

The values obtained from the application of Eqs. 7–9 are an approximate solution of the 2D model reported in the Model equations section. However, Eqs. 7–9 allow for a quick estimation of the current density by analytical means, providing also an accurate assessment of the upper bound of performance as discussed later.

It is worth analysing the electrochemical performance of a flat electrode because any improvement provided by a mesoscale structural modification must be evaluated in comparison with the current density that a benchmark flat electrode would exhibit in the same conditions.

In this study, unless otherwise specified, the reference conditions for all the simulations are those reported in Table I. These conditions refer to the best porous composite Ni/ScSZ anode produced and tested by Somalu et al.,¹⁰ which showed the lowest polarisation resistance in a series of flat electrodes fabricated with different Ni volume fractions and porosities. The choice of materials, i.e., Ni and ScSZ, is particularly indicated for operation at intermediate temperature,^60,61 which is set to 973 K. The thickness of the electrolyte, made of ScSZ, is set to 10 μm, which is typical for anode-supported planar SOFCs.^2,32,62,63 Material-specific and microstructural properties were evaluated by Bertei et al.³⁹ Although the absolute value of the results depends on the choice of this specific set of parameters, the emergent trends and the scaling laws discussed afterwards are general and highly applicable to a wide range of conditions and materials.

Figure 2 shows the current density predicted by Eqs. 4–6 as a function of electrode thickness for a flat configuration. The current density monotonously increases as the thickness increases, approaching an asymptotic value i^{flat, asympt}_tot in the limit of semi-infinite electrode thickness (t_ed → ∞).³⁵ The Figure also reports the characteristic thickness t*_ed, equal to 20.4 μm. Note that in a hypothetical electrode showing the same minimum electrode resistance R^min_ed with halved k_eff and doubled L_TPB, the characteristic thickness t*_ed is smaller by a factor 2 and equal to 10.2 μm, see the dashed line in Figure 2. Indeed, the minimum electrode resistance R^min_ed scales with while the characteristic thickness t*_ed scales with according to Eqs. 6a and 6c, respectively.

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It is also interesting to note that, in the asymptotic limit, the electrolyte area specific resistance is t_ey/σ_O = 2.07·10⁻⁶ Ω m² while the electrode polarisation resistance is equal to R^min_ed = 2.35·10⁻⁵ Ω m². The ratio of the two resistances, equal to the fuel cell Biot number Bi_FC = (t_ey/σ_O)/R^min_ed as defined by Konno et al.,²⁴ is equal to 0.088. This means that the dominant resistance is from the electrode, thus the whole cell performance would be significantly enhanced by focussing on the reduction of the electrode polarisation resistance, such as through a mesoscale structural modification.

Any improvements produced by a mesoscale structural modification of the electrode must be compared with a benchmark current density of the flat electrode evaluated in the same conditions. The asymptotic current density i^{flat, asympt}_tot is a well-defined and unambiguous benchmark. Thus, the performance indicator proposed in this study to quantify the improvement due to mesoscale structural modification is the relative increase in current density δi_tot, defined as follows:

where i_tot is the total current density produced in the electrode with mesoscale modification. Note that defining δi_tot in this way allows one to quantify any improvements against the maximum current density that a flat electrode can theoretically yield.

It is worth mentioning that such an asymptotic value i^{flat, asympt}_tot is approached if electronic ohmic losses and gas transport limitations are neglected,^9,35,36 as assumed in this study (see Modelling section). However, as the thickness increases, these assumptions are no longer met in real electrodes. This is not a problem for the practical application of Eq. 10: instead of the asymptotic current density i^{flat, asympt}_tot, the 90% of its value might have been considered as a benchmark for the definition of the performance indicator δi_tot. As shown in Figure 2, the 90% of i^{flat, asympt}_tot is met for relatively thin electrodes, where both electronic ohmic losses and gas transport limitations are typically negligible.^36,64,37,39 Thus, the applicability of Eq. 10 can be extended by considering any desired benchmark value for the flat electrode, provided that such a benchmark is well-defined and kept constant. In this study, the asymptotic current density i^{flat, asympt}_tot is used to quantify the relative improvement δi_tot according to Eq. 10 because, in such a case, δi_tot shows the improvement over the maximum current density that a flat electrode can ideally exhibit.

The first mesoscale structural modification analyzed in this study is the insertion of rectangular pillars as in the first sketch in Figure 1b. Rectangular pillars have been tested in a few experimental studies.^15,30 Figure 3a shows the computational domain in greater detail: t_h is the pillar thickness, W is the half-width and w is the half-distance between pillar walls.

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Figure 3b shows the relative improvement in current density δi_tot as a function of the geometric ratio ω defined in Eq. 7 for different values of W in the limit of semi-infinite electrode thickness (t_ed → ∞) and semi-infinite pillar thickness (t_h → ∞) as obtained from the numerical solution of Eqs. 1–3. The Figure shows that δi_tot exhibits a parabolic behavior with a defined maximum as a function of ω. This is not surprising because as ω approaches 0 the electrode becomes flat, while as ω approaches 1 the composite electrode volume drops to zero and no electrochemical reaction occurs. The maximum improvement in current density over the flat electrode performance is 27% for ω ≈ 0.4.

As W increases, the general improvement given by the mesoscale structural modification drops. This is explained by analysing the inset in Figure 3b, which shows that the ionic current is not evenly distributed within the pillar width for large values of W: since the inner region of the pillar is no longer optimally exploited for ionic conduction, δi_tot decreases. Accordingly, the maximum of δi_tot shifts to larger values of ω as W increases (see dashed line in Figure 3b) to compensate for the lower utilisation of the pillar width. For W > 60 μm, pillars are detrimental for performance as δi_tot ≤ 0 in the whole range of ω, thus such structures should be avoided under the conditions used for this study.

A dimensionless factor Λ_W can be defined as:

Such a dimensionless factor, which compares the pillar half-width with the characteristic thickness of the composite electrode domain, is useful to identify two regimes of ionic conduction within the pillar width. For Λ_W << 1 the whole width of the pillar is utilised for conduction and the ionic current density distribution is uniform, while for Λ_W >> 1 the inner region of the pillar is not optimally utilised. Interestingly, this means that the maximum improvement in current density is achieved for Λ_W << 1 while for Λ_W >> 1 the mesoscale structural modification can in fact be detrimental, in agreement with the numerical results in Figure 3b. This outcome is supported by published studies, for example by Nagato et al.,³⁰ who proved experimentally that pillars with smaller width result in higher power density than wider pillars. In addition, similar predictions were reached by Tanner et al.³⁵

Interestingly, the dot-dashed line in Figure 3b shows that the average approximation reported in Eqs. 8 and 9 reproduces remarkably well the trend of δi_tot as a function of ω for Λ_W ≤ 0.1. In addition, results of a 3D simulation in the limit of W → 0, reported with open diamonds, lie on the curve predicted with the analytical average approximation, confirming that 2D simulations are representative of the general trends of a three-dimensional structural modification. Thus, the upper bound of performance and the optimal ω (i.e., the optimal pillar spacing) can be analytically predicted by using Eqs. 8–9 as a function of electrode microstructural and material properties. This is a useful result for a rapid short-cut assessment and design of mesoscale structural modifications.

Figure 4 shows the effect of the pillar thickness t_h on δi_tot in the limit of t_ed → ∞ for two representative cases: W = 5 μm (i.e., Λ_W < 1, Figure 4a) and W = 30 μm (i.e., Λ_W > 1, Figure 4b). For any values of t_h, δi_tot shows a maximum as a function of ω, as already reported in Figure 3b in the limit of t_h → ∞. More importantly, for both Λ_W < 1 and Λ_W > 1 the improvement over flat electrode δi_tot decreases as the pillar thickness t_h decreases. Similarly to Eq. 11, another dimensionless factor can be defined as follows:

which compares the pillar thickness with the characteristic thickness of the electrode. Figure 4 shows that, for any value of W and ω, Λ_h must be much larger than 1 in order to maximise δi_tot. When both Λ_W and Λ_h are considered together they help determine the maximum performance improvement caused by the mesoscale modification as discussed later.

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The effect of the electrode thickness t_ed on δi_tot is reported in Figure 5 for a representative case W = 5 μm, w = 8 μm and t_h = 20 μm. Figure 5 shows that δi_tot increases as t_ed increases and approaches an asymptotic value in the limit t_ed → ∞. Therefore, given the pillar geometric parameters, while a small electrode thickness can be even detrimental for performance as δi_tot < 0 for t_ed < 40 μm, there is no significant improvement in making the electrode thicker beyond t_ed ≈ 4·t_h.

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In summary, the results reported in Figures 3–5 lead to the following key outcomes informing design guidelines:

• the mesoscale structural modification produces the largest improvement in performance in the limit of thin and long pillars, that is, for Λ_W = W/t*_ed < 0.1 and Λ_h = t_h/t*_ed >> 1. Eventually, this matches the concept of scaffold electrodes,^65–70 that is, electrodes fabricated with a porous ionic backbone which is infiltrated with electron-conducting particles or even a mixture of electron-conducting and ion-conducting particles. In other words, the optimal 3D printed electrode with rectangular pillars results in the production of a scaffold with well-defined geometry;
• the upper bound in performance can be analytically predicted through the approximated model reported in Eqs. 8 and 9. This represents an effective rapid method for the optimisation of the mesoscale structural modification.

The microstructural properties of the composite electrode domain, k_eff and L_TPB, play a significant role in determining the electrochemical performance of a flat electrode, as already discussed above. Similarly, the electrode microstructural properties are crucial also when a mesoscale structural modification is applied. As demonstrated in the previous section, the upper bound of improvement is achieved in the limit of t_ed = t_h → ∞ and is accurately predicted by the average approximation described in Eqs. 8 and 9. Hence, the effect of the microstructure of the composite electrode domain on δi_tot is addressed in this section through the average approximation model.

As already discussed in the previous section, the maximum improvement in performance δi_tot occurs in the limit t_ed = t_h → ∞ for a specific value of ω, that is, for an optimal volume fraction of pillars ω_opt. Such an optimal condition, which represents the upper bound of performance that the mesoscale structural modification can yield, can be calculated by substituting Eq. 8 in Eq. 9a and solving for , thus leading to:

where R^min_ed is the minimum polarisation resistance of the flat electrode (see Eq. 6a) and is the minimum polarisation resistance of the electrode with mesoscale modification corresponding to the upper bound of improvement in δi_tot (i.e., calculated for ω = ω_opt). Note that Eq. 13 is valid only for k_eff ≤ 0.5. For k_eff > 0.5 does not show any minimum as a function of ω and the equation does not have any real solution.

Figure 6 shows ω_opt and the ratio as a function of the effective conductivity factor of the ion-conducting phase in the composite electrode domain k_eff. As k_eff increases from 0 to 0.5, the optimal volume fraction of pillars ω_opt decreases from 0.5 to 0 while the ratio increases from 0 to 1. Note that the lower the ratio , the larger the improvement δi_tot over the flat electrode in accordance with Eqs. 9b and 10.

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Several considerations can therefore be drawn from the analysis of Eq. 13 and Figure 6:

• the optimal volume fraction of pillars ω_opt depends only on k_eff (see Eq. 13a). Other microstructural or material parameters, such as L_TPB, σ_O or i⁰_TPB, do not play any role in determining the optimal volume fraction of the pillars under the current assumptions: the decision of introducing or not a mesoscale modification depends solely on the effective conductivity factor of the ion-conducting phase in the composite electrode domain;
• the mesoscale structural modification is beneficial only for k_eff ≤ 0.5, while it is always detrimental for k_eff > 0.5, because the minimum resistance of the electrode with pillars exceeds the resistance of the benchmark flat electrode R^min_ed (see Figure 6). Thus, for k_eff > 0.5 the improvement of electrode performance must be achieved by different means other than mesoscale structural modification;
• as k_eff decreases, the optimal volume fraction of pillars ω_opt increases (see Figure 6), meaning that pillars are required to compensate for the lower effective ionic conductivity of the composite electrode domain;
• the mesoscale structural modification is particularly beneficial for electrodes with low effective ionic conductivity factor k_eff, since the reduction in electrode resistance , in comparison with the benchmark flat electrode R^min_ed, is relatively more significant (see Figure 6). However, assuming that the other parameters remain constant (in particular L_TPB), since the resistance of the flat electrode R^min_ed scales with (see Eq. 6a), a flat electrode with low effective ionic conductivity is not a good benchmark as it would show poor performance. Thus, a useful framework to improve performance is i) the microstructure of the flat electrode should be improved first, and then ii) an optimal mesoscale structural modification can be applied to further enhance the performance;
• the improvement of the microstructure of the composite electrode domain envisaged in the previous point must be done rationally. Since R^min_ed scales with (see Eq. 6a), the microstructure of the composite electrode domain could be modified in order to reduce k_eff and proportionally increase L_TPB to keep the same R^min_ed. By doing so, the benefits of a mesoscale modification would be amplified, since decreases as k_eff decreases (see Figure 6). This implies that a composite electrode intended for mesoscale structural modification should be optimised for maximum TPB density rather than for maximum k_eff, because the effective ionic conduction is provided by the ion-conducting pillars. As an additional consequence, the microstructure of a composite electrode optimised for flat electrode configuration may not be the best one to be used as a basis for a mesoscale structural modification.

The last two points highlight the importance of a proper design of the microstructure of the composite electrode domain even when a mesoscale structural modification is adopted. In addition, the accurate quantification of the effective conductivity factor is of a paramount concern and appropriate methods,^71,72 based on the tomographic reconstruction of the microstructure, must be employed to avoid any incorrect evaluation.

In the previous sections, rectangular pillars were considered in the analysis. Although this is the pattern currently produced with different techniques,^15,24,30,31 additive manufacturing can potentially print pillars with unique and different shapes,²¹ going beyond the capability of existing manufacturing methods such as screen printing and tape casting.

The introduction of a terrace of width W_t and thickness t_t onto a rectangular pillar, as reported in Figure 7a, is the first shape modification analyzed in this section. Figure 7b shows on the left axis the increase in current density beyond the benchmark flat electrode, as a function of the pillar thickness t_h, for a pillar with terrace of dimensions W_t = t_t = 5 μm (δi^terr_tot, solid line). For comparison, the result for a rectangular pillar without terrace is reported (δi^no terr_tot, dashed line). The difference between the relative improvement given by the terrace beyond an equivalent rectangular pillar, i.e., , is reported on the right axis. The two insets show the distribution of current density converted per unit of TPB i_TPB in the active volume for two cases, t_h = 20 μm and t_h = 40 μm.

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The insets in Figure 7b show that the electrochemical reaction takes place close to the electrode/electrolyte interface, being distributed mainly within a distance roughly equal to the characteristic thickness t*_ed. This current distribution matches what expected in a flat electrode configuration.³⁶ The mesoscale structural modification makes the i_TPB distribution distorted toward the pillar, allowing for more current to be converted in the proximity of the pillar up to a thickness slightly larger than t*_ed. Such an effect, consistent with the work reported by Delloro and Viviani,²² is due to the enhanced conduction that the pillar provides, thus opening a faster ionic conduction path toward the electrolyte.

When the terrace is placed at t_h ≈ t*_ed, such an extra lateral fin boosts additional current conversion, leading to an improvement in current density δi^terr_tot larger than that obtained without terrace δi^no terr_tot, as shown by the right axis in Figure 7b. On the other hand, placing the terrace too far above from the active region does not make any difference with the case with no terrace: decreases and approaches 0 as t_h increases beyond t*_ed in Figure 7b. Similarly, placing the terrace too close to the electrolyte interface (i.e., t_h << t*_ed) produces detrimental effects as becomes negative, meaning that more current is produced when there is no terrace. This detrimental effect is due to the fact that the terrace removes electrochemically active volume from the composite electrode domain, thus placing the terrace too close to the electrolyte interface, exactly where more current i_TPB is produced, reduces the active region wherein current is converted, leading globally to a lower performance.

Figure 7c, which shows the difference as a function of the terrace thickness t_t, provides an additional insight. As t_t increases, the improvement in current density over the pillar without terrace increases and reaches as maximum, beyond which decreases and becomes negative. There is a simple reason behind this trend. For small t_t values, the terrace is too thin to provide an effective conduction path to the main body of the pillar and to collect ionic current from the reacting zone in the composite electrode domain. As t_t increases, the terrace acts as a horizontal conducting fin, effectively providing ionic conduction. However, as the terrace thickness increases, more active volume is removed in the composite domain, especially closer to the electrolyte interface where electrode volume is required by the electrochemical reaction to take place. Thus, the beneficial effect of a more effective lateral ionic conduction is quickly counterbalanced by the removal of active volume in the composite domain.

Hence, the results of Figure 7 provide a clear insight into the role of the terrace and, more generally, on the pillar shape. The terrace is useful when it provides ionic conduction far from the electrolyte interface, especially at a distance comparable with the characteristic thickness t*_ed, in order to collect more ionic current and thus provide a faster conduction pathway toward the electrolyte. Meanwhile, the mesoscale structural modification shall not remove active volume in the composite electrode domain, especially at the electrolyte interface where there is the maximum current conversion rate. In other words, the mesoscale structural modification must provide ionic conduction where needed, that is, farther from the electrolyte interface, without removing active electrode volume where needed, that is, at the electrolyte interface.

The proper balance of these two opposite requirements is a golden rule for the rational design of the pillar shape. In particular, maximising the ionic conduction farther from the electrolyte interface without removing active volume points toward a pillar shape which has to mimic the ohmic overpotential profile along the electrode thickness. In a flat electrode the ohmic overpotential profile is opposite to the activation overpotential distribution,³⁶ being 0 at the electrolyte interface and increasing exponentially along the electrode thickness.^36,54 Such an exponential profile is mathematically mimicked by the following equation:

which is used in Figure 8a to draw the outline of the pillar, resulting in a rounded terrace shape which resembles the ohmic overpotential distribution in a flat configuration.^36,54 In Eq. 14, t_r is a roundness characteristic length: the smaller t_r, the sharper the outline.

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Figure 8b shows the results of a design study on parameters W_t and t_r for W = 5 μm, w = 20 μm and t_h = 25 μm. In particular, given a value W_t, the right axis reports the value of t_r corresponding to the maximum δi_tot, which is reported on the left axis. As W_t increases, δi_tot increases until a broad maximum is reached. Notably, the values of δi_tot obtained for an electrode with rounded terrace are larger than the maximum values provided by rectangular pillars with and without terrace for the same geometric parameters, although the improvement in δi_tot is only minor, in the order of 1% or less. On the other hand, the roundness characteristic length t_r continuously decreases, meaning that the optimal shape becomes sharper as W_t increases as depicted in the insets of Figure 8b.

The results of the rounded pillar shape further confirm the role of the terrace: the mesoscale structural modification must provide conduction farther from the electrolyte without removing active volume at the electrolyte interface, leading to a rather sharp optimal shape which enhances the performance obtained by rectangular pillars. In accordance with this, the triangular shape depicted in the fourth sketch in Figure 1b is expected to be the worst design among the series analyzed, since it removes composite electrode volume at the electrolyte interface without providing any improvement through a conducting fin into the active region within a distance t*_ed from the electrolyte.

Such a rounded terrace shape is an ideal design paradigm, which can be practically mimicked by fabricating dimples according to the method proposed by Dai et al.²⁹ In practice, in a 3D structure the terraces can even touch each other (i.e., W_t = w), provided that the third dimension z (i.e., the through-plane direction in Figure 8a) is exploited to provide gas and electronic current underneath the terrace. In addition, the ability to produce and the mechanical requirements of such a sharp shape are something which needs to be addressed, especially during operation and fabrication if additive manufacturing is to be employed.¹⁵ While clearly of practical importance, these aspects are out of the scope of the current study. The main point of the design analysis on the pillar shape is the underlying rationale that should guide the optimisation of the mesoscale structural modification, which is producing a shape that is capable of providing ionic conduction farther from the electrolyte up to a thickness in the order of t*_ed without removing active volume at the base of the pillar. Hierarchical and fractal-like structures resembling trees or human lungs can be foreseen as well,⁷³ provided the feasibility in producing mechanically stable dense features at the small resolution required by the coupled reaction/conduction process, which has a mesoscale characteristic length equal to t*_ed.