Potentials Near a Curved Anode Edge in a PEM Fuel Cell : Analytical Solution for Placing a Reference Electrode

We consider a PEM fuel cell with concentric circular electrodes: the small anode and the large cathode. A model for in-plane distributions of the cathode overpotential ηc and the membrane potential in the anode-free region of the cell is developed. Mathematically, the problem reduces to the axially symmetric Poisson–Boltzmann equation for ηc . An approximate analytical solution shows that |ηc| exhibits rapid decay to zero with the radius, while | | grows to the value of |η 0 c |, the cathode overpotential in the working domain of the cell. For typical ηc , the radial shape of ηc far from the anode edge only weakly depends on η 0 c ; this effect is analogous to Debye screening in plasmas. The smaller the anode radius, the faster approaches ηc with the distance from the anode. It follows, that a reference electrode for measuring the cathode overpotential in the working area can be placed at a small distance from the curved anode edge. © The Author(s) 2015. Published by ECS. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is properly cited. [DOI: 10.1149/2.0331510jes] All rights reserved.

Performance of a polymer electrolyte membrane fuel cell (PEMFC) is determined by overpotentials driving the electrochemical reactions on either side of the cell.One of the most useful techniques for measuring the half-cell overpotentials is a method of reference electrode (RE).A typical schematic of a cell with the RE is depicted in Figure 1a.The hydrogen-fed RE is located at a certain distance L gap from the aligned anode and cathode edges.Neglecting the potential loss for the hydrogen oxidation/evolution reactions, the potential of the RE is equal to the membrane potential at the point of the RE location.If the distance L gap is large enough, the measured corresponds to the membrane potential at some point along the y-axis between the anode and cathode in the working cell area (Figure 1a).Measuring and the electrode potentials allows one to separate the anode and cathode overpotentials in the cell. 1,2][5][6] The problem with the system in Figure 1a is that even a small misalignment of the anode and cathode edges may strongly distort at the RE location.[5]7 Recently, a design free from this drawback has been suggested 8 (Figure 1b).Here, the cathode is continuous and the RE is located at a distance L gap from the straight anode edge.The absence of the cathode edge eliminates the misalignment problem.It has been shown, that for the design in Figure 1b, L gap must satisfy to the condition 8 L gap 3  σ m b ox l m 2 j 0 ox [1]   Here, σ m is the membrane proton conductivity, b ox is the Tafel slope of the oxygen reduction reaction (ORR), l m is the membrane thickness, j 0 ox is the superficial exchange current density (A cm −2 )o f the cathode in the working cell area.With typical cell parameters (Table I), L gap appears to be on the order of several centimeters.For a typical laboratory-size cell of about 10 × 10 cm 2 , this value of L gap is quite large.Below, we show that a curved anode edge enables significant reduction of L gap not sacrificing the accuracy of measurements.
We report a model for the radial distribution of the membrane potential and the cathode overpotential η c in the anode-free area of a cell with the small circular anode and the large cathode.An approximate analytical solution of the problem is derived.It shows that far from the anode edge, the radial shape of the cathode overpotential η c (r ) only weakly depends of the value η 0 c at the edge, provided that |η 0 c | is sufficiently large.This effect is quite analogous to Debye screening in plasmas.Further, analytical solution allows us to calculate the width of the gap between the curved anode edge and the RE for measuring the cathode overpotential in the working domain of the cell.In a certain range of anode radii, the smaller the radius of the anode edge, the closer to the edge can be located the RE.

Model
The model below is close in spirit to the model 8 for the straight anode edge above the infinite cathode (Figure 1b).Here, however, we consider a system of concentric circular electrodes depicted in Figure 2. Let the radii of the anode and cathode electrodes be R a and R c , respectively; we will assume that R c →∞.In the following, the Figure 1.Schematic of the fuel cell with the reference electrode.(a) A conventional scheme; note that a small misalignment δ of the working anode and cathode edges strongly distort the membrane potential at the reference electrode location.(b) A scheme. 8 Here, the reference electrode is located at a large distance L gap from the anode edge, while the cathode is continuous.cell domains r ≤ R a and R a < r < ∞ will be referred to as the working and anode-free domains, respectively (Figure 2).Our goal is understanding the distribution of currents and potentials in this system.The main variable in this problem is the membrane potential , which obeys to the Poisson equation 1 r Infinite cathode means that the radial extent of the problem R c is by several orders of magnitude larger, than the membrane thickness (the exact criterium is given below).This allows us to approximate the second derivative along z in Eq. 2 by the difference of proton currents coming in and out of the membrane, 9 which yields 1 r Here j a and j c are the proton current densities at the anode and the cathode side of the membrane, respectively.In a well-humidified membrane, variation of along z is close to linear, and hence the quasi-2D approximation of Eq. 3 works well everywhere, except a small region in the vicinity of the anode edge.Close to the anode edge, a fully 2D distribution of forms.Numerical calculations of Adler et al. 4 show that in the system with the straight anode edge, this 2D-domain extends to the distance on the order of l m from the edge.In the axially symmetric system, we may expect even smaller 2D domain.Thus, the radial distributions reported below may be not accurate in a ring of a width of l m just outside the anode edge.However, of largest interest is the behavior of the potentials outside this ring, on a large radial distance from the anode edge.
Further, j a and j c are assumed to obey the approximate Butler-Volmer kinetics where j hy and j ox are the superficial exchange current densities of the anode catalyst layer (ACL) and the cathode catalyst layer (CCL), respectively, η a and η c are the local anode and cathode overpotentials, and b hy and b ox are the corresponding Tafel slopes.The approximate form 5 of the Butler-Volmer equation allows us to analyze a smooth transition from large to small current densities in the anode-free domain, which is a feature of this problem.Eq. 5 reduces to the standard Butler-Volmer equation if the transfer coefficient α of the ORR ratedetermining step is 1/2.With another values of α,Eq.5 is approximate in the region of η c ≃ b ox , and it reduces to the exact asymptotics for η c ≪ b ox and η c ≫ b ox .Indeed, for η c ≫ b ox , the reverse reaction is negligible, while for η c ≪ b ox , the ORR kinetics is linear, and hence in both these limits, the Buler-Volmer equation and Eq. 5 lead to the same form of the ORR rate.Note that we assume that the transport losses in the electrodes are small and hence the dependencies on reactant concentrations are included in j hy and j ox .The half-cell overpotentials are given by where φ a and φ c are the anode and the cathode potentials, and E eq HOR = 0andE eq ORR = 1.23 V are the equilibrium potentials of the respective half-cell reaction.We will assume that the anode is grounded (φ a = 0) and, hence, φ c is the cell potential.
Substituting Eqs.4-7 into Eq. 3 and introducing dimensionless variables Here, the superscript 0 indicates the value in the center of the working domain (r = 0) and H is the Heaviside function, which equals 1 in the working domain and zero in the anode-free domain.The absence of the anode catalyst outside the working domain is modeled as zero exchange current density of the hydrogen oxidation reaction (HOR).Figure 3 shows the anode 4 and the cathode 5 current densities resulting from numerical solution of the problem 9 with the boundary conditions ˜ (0) = ˜ 0 , ∂ ˜ /∂ r r =∞ = 0.At the anode edge, a current double layer arises 10 (Figure 3b), which is analogous to a charged double layer at the surface of a solid charged particle in plasmas.The current double layer determines the shape of the cathode overpotential in the anode-free domain, as discussed below.
In the remainder of this work we will be interested in the distributions of and η c in the anode-free domain.In this region, production of current on the anode side vanishes, and Eq. 9 simplifies to It is convenient to rewrite this equation in terms of the cathode overpotential 7, which in the dimensionless variables is ηc = φc − ˜ − Ẽeq ORR [11]   Substituting this into Eq.10,weget where ox [13]   is the inverse dimensionless Debye length: κ = λ−1 D .Indimensional form, the Debye length in this problem is given by The physical meaning of λ D is discussed in Debye screening section.
Figure 3.The anode j a and cathode j c current densities at the edge of the circular anode of the radius Ra = 20.For numerical calculation, the Heaviside function in Eq. 9 was smoothed, which is equivalent to the effective anode radius of Ra ≃ 20.2.Parameters for the calculation are listed in Table I.Note a large peak of the anode current at the edge of the electrode.Note also a "double layer" structure of the difference j a − j c , which appears on the right side of Eq. 9.
Formally, Eq. 12 is a Poisson-Boltzmann (PB) equation.Quite analogous equations arise in the theory of a diffuse double layer at the metal/electrolyte interface (Gouy-Chapman model 11,12 ), and in the physics of charged macroscopic solid particles in plasmas. 13The ORR rate in the Butler-Volmer term on the right side of Eq. 12 plays a role of a positive charge and the rate of the reverse reaction of water electrolysis in this term is equivalent to a negative charge in plasmas.
The main difference with the Gouy-Chapman problem for the planar diffuse double layer is axial symmetry of Eq. 12. Dyachkov 13 published analytical solution to Eq. 12 in the form of an infinite series.Zholkovskij et al. 14 derived matched asymptotic solution for the PB equation with spherical symmetry.Below, we employ a simple matching technique which leads to an accurate approximate analytical result.

Analytical Solution
The problem for the straight anode edge above an infinite cathode leads to Eq. 12 with the second derivative in Cartesian coordinates d 2 ηc /dx 2 on the left side. 8It has been shown, that in PEM fuel cells, due to a very high exchange current density of the HOR, ηc at the straight anode edge is very close to its value in the "bulk" area of the working domain. 8Thus, in axial geometry, we may also expect that ηc ( Ra ) ≃ η0 c ,w h e r e η0 c ≡ ηc (0) is the ORR overpotential at the axis of symmetry; numerical tests confirm this assumption.This allows us to exclude the working domain from consideration by fixing ηc ( Ra ) = η0 c at the anode edge.With this, the boundary conditions to Eq. 12 are The second equation means zero in-plane proton current through the cathode edge.Validity of the infinite cathode approximation is guar- Note, however, that Eq. 16 is redundant; a more accurate condition is discussed below.The idea of the solution technique used here is as following.Close to the anode edge, the ORR rate dominates, which allows us to replace the sinh-function in Eq. 12 by the leading ORR exponent.The domain where this simplification works will be referred to as the ORR-dominated domain, as for | ηc | > 2, the cathode reaction is shifted to oxidation and the rate of the reverse reaction is small.Mathematically, this situation is equivalent to the dominance of positively charged species in the plasma close to the surface of a negatively charged macroscopic rod.Far from the anode, the rates of the forward and reverse reaction are nearly equal, and the sinh-function in Eq. 12 reduces to the linear dependence.Formally, this domain is equivalent to a quasineutral region in plasmas.Matching of the ORR-dominated and "quasineutral" solutions is performed by extending these solutions to the point, where | ηc |=1.
Let the current density in the working domain be sufficiently large, i.e, | η0 c |≥2.In the dimensional form this means that |η 0 c |≥2b ox ≃ 100 mV, a condition, which in PEMFCs holds already at the cell current density of about 0.01 mA cm −2 .Then, in the vicinity of the anode edge, we can retain only the leading exponent in the expression for sinh-function in Eq. 12. Noting that ηc < 0, Eq. 12 in the ORRdominated domain simplifies to where A and B are constants to be determined from the boundary and matching conditions.Note that the subscript S marks the short-range solution, valid in the vicinity of the anode edge.Far from the anode edge, we have η+ c ≪ 1, and the sinh-function on the right side of 12 can be expanded in Taylor series.With this, Eq. 12 simplifies to where the boundary condition expresses zero in-plane proton current at the cathode edge.Solution to Eq. 20 is where K is the modified Bessel function of the second kind and C is the constant to be determined from the matching conditions.The subscript L marks the long-range solution, valid far from the anode edge.
To fill the "gap" in the range of η+ c ≃ 1, we extend Eq. 19 down to η+ c = 1, and Eq.21 up to η+ c = 1.Let the radius where η+ c = 1be R1 : η+ c ( R1 ) = 1 Setting r = R1 in Eq.21, we get a relation for the constant C; expressing C from this relation and substituting the result to 21,w e obtain The solutions 19 and 22 contain the three constants A, B and R1 to be determined.The first relation between these constants is obtained by setting r = R1 in Eq. 19: Another relation results from continuity of the first derivatives of 19 and 22 at r = R1 The last relation gives the boundary condition for the overpotential 19 at r = Ra The Equations 23-25 determine the constants A, B and R1 ; these three parameters, in turn, fully determine the solution of the problem: [26] Eqs.23-25 have multiple roots, as they contain periodic functions.The physical solution provides the set {A, B, R1 } with the minimal positive A; the other sets lead to unphysical local peaks of η+ c in the anode-free domain.Note that the case of the cathode of a finite radius Rc leads to somewhat more complicated solutions listed in Appendix A.

Results and Discussion
Shapes of overpotential for different anode radii.-Numericalsolution to the system 23-25 for the constants A, B and R1 is obtained as following.Solving Eq. 24 for B we get Substituting 27 into Eqs.23 and 25, we get a system of two equations for the parameters A and R1 .This system can be solved by standard numerical procedures utilizing Newton's method, taking A in the range of 0.25 to 1 and R1 in the range of 1/κ to 2/κ as the initial guess.The parameter B is then calculated with Eq. 27.Note that a more accurate initial guess for A provides a function The solution 26 is valid outside the working cell area, in the anodefree domain r ≥ Ra .Comparison of Eq. 26 to the numerical solution of the full problem, Eq. 9 for 0 ≤ r < ∞ is depicted in Figure 4.As can be seen, the analytical curves are very close to the numerical results (points).Note rapid decay of the overpotential close to the anode surface (Figure 4).
For the anode radii Ra < 10, the numerical solution is difficult to obtain due to a very steep gradient of η+ c near the anode edge.A key feature of this problem is that this gradient increases with the decrease of the anode radius.A more detailed view of the η + c shapes, which illustrates this behavior is depicted in Figure 5a. Figure 5b shows the analytical curves in Figure 5a shifted along the r -axis in such a way, that the anode edges coincide.Note a faster decay of η + c near the anode of a smaller radius.Qualitatively, this effect is similar to behavior of the Laplace potential between a charged axially-symmetric metal tip and a plane: the smaller the tip radius, the faster decays the potential  I.
along the symmetry axis of the problem. 15The main practical conclusion from Figure 5b is that in a cell with the smaller anode radius, the reference electrode can be positioned closer to the anode edge.This is discussed in detail in Positioning of the reference electrode section.Debye screening.-Considerthe system of Equations 23, 24 and 25 for the constants A, B and R1 and suppose that η+,0 c →∞.It can be shown that A is bounded (Appendix B): A ≤ M,whereM is some constant.Thus, the tan-function in Eq. 25 must be large; this means, that for η+,0 c →∞the argument of this function tends to π/2, and hence for large η+,0 c Eq. 25 can be replaced by With this, the parameters A, B and R1 can be determined from the reduced system of Equations 23, 24 and 28, which is independent of η+,0 c .It follows, that at large η+,0 c , the long-range radial shape of the overpotential in Eq. 26 only weakly depends on its boundary value η+,0 c .Large overpotential means that η+,0 c ≫ 1.For the typical PEMFC parameters (Table I), this condition holds at very small cell currents (see above) i.e., for the typical cell current densities of 0.1 to 1 A cm −2 , the long-range shape of overpotential weakly depends on the cell current density.To illustrate this effect, Figure 6 shows comparison of the width of the ORR-dominated domain determined from the reduced system 23, 24 and 28, to the exact value resulted from the system 23, 24 and 25.The two curves are very close to each other.Taking into account that R1 is on the order of 1/κ, this result is quite analogous to Debye screening in plasmas: at the distance on the order of λD = 1/κ, the charge of the macroscopic rod-like particle is screened, so that any variation of this charge is not "seen" at larger distances.Here, "screening" makes the behavior of the cathode overpotential at the distances r > 1/κ only weakly dependent of the overpotential (or current) in the working domain of the cell.This effect can also be demonstrated in the case of the straight anode edge above the large cathode active area (Figure 1b).Calculations 8 have shown, that in this cell, the overpotential η+ c decays with the distance x from the anode edge according to In general, the constant G is determined from matching of Eq. 30 and the solution in the working cell area.However, as discussed above, in PEMFC, the cathode overpotentials at the anode edge and in the working domain are nearly the same, and we can determine G from the condition η+ c (0) = η+,0 ,w h e r ex = 0 is located at the straight edge of the anode (Figure 1b).Substituting this G into Eq.30 and solving equation η+ c = 1, we find the distance L1,x where η+ c drops to  I).
Dashed line-the same width calculated from the reduced system of equations 23, 24 and 28, which is independent of the overpotential in the working cell area η+,0 c (i.e., it is independent of the cell current density).(cf. Figure 5b) and the respective shapes of the positive membrane potential + for the indicated anode radii Ra . unity: For η+,0 ≫ 1, this result simplifies to which is independent of η+,0 c , i.e, independent of the cell current.This is a signature of Debye screening.
Positioning of the reference electrode.-Inthe remainder of this work, we will discuss the analytical curves η+ c (r ), as they practically coincide with the numerical solutions.Figure 7 shows the shapes of the positive membrane potential + =− >0 for the same anode radii as in Figure 5b.The shapes of + have been calculated using the dimensional versions of Eqs.11, 26 and the parameters in Table I.As can be seen, at large r , + tends to the limiting value +,∞ = η +,0 c .Indeed, subtracting the relations as in PEMFCs, ˜ +,0 ≃ 0, i.e., the membrane potential in the working domain is close to zero.Here, the superscript ∞ marks the values at r →∞.
As discussed above, + corresponding to smaller anode radius tends to η +,0 c faster (Figure 7).This effect is illustrated in Figure 8, which shows the radial width of the ORR-dominated domain L1,r , Eq. 29.The width L1,r increases with the anode radius Ra (Figure 8).Due to Debye screening, the parameter L1,r weakly depends of the cell current density and the shapes of η+ c ( Ra ) and of L1,r ( Ra )a r e governed mainly by the Debye parameter κ.
For further estimates we will assume that the reference electrode can be located at the distance L1,r from the anode edge.Figure 7 shows that this assumption provides 10%-accuracy of η+ c determination (the bottom straight dotted line).It is advisable to compare the distance L1,r to the analogous distance L1,x ,Eq.32 for the straight anode edge geometry (Figure 8).Straight long-dashed lines in Figure 8 depict the value of L1,x for the two sets of parameters; L1,x is an asymptote to which the respective L1,r curve tends as Ra →∞ .For small anode radii Ra 10, the distance L1,r between the curved anode edge and the RE is at least twice smaller, than this distance L1,x for the  I).Lower solid curve corresponds to j 0 ox = 6 • 10 −6 Acm −2 , b ox = 0.05 V and to the cell potential of 0.65 V; the other parameters are those indicated in Table I.Open circles-the approximate fitting Equation 34.Long-dashed lines-the distance L1,x , corresponding to the straight anode edge (see text).straight anode edge (Figure 8).Note rapid decay of L1,r for the small anode radii (Figure 8); however, for Ra 1, accuracy of the model decreases, as it ignores three-dimensional effects in a close proximity of the anode edge.
An approximate expression for the dependence L1,r ( Ra )i nt h e range 0 ≤ κ Ra ≤ 1is L1,r ≃ π 2κ ln 67 18(κ Ra ) 7/45 8, open circles).In the dimensional form this equation reads where λ D is given by Eq. 14.E q .35can be used for engineering estimates of the gap distance L gap between the anode tip of a radius R a and the reference electrode.If high accuracy of measurements is needed, taking L gap ≃ 3L 1,r is recommended.Figure 9 shows possible configurations of a fuel cell with the reference electrode.Figure 9a displays the reference electrode in the system with the straight anode edge.In this case, the distance L gap ≃ 3λ D , as reported in. 8Figure 9b exhibits the case of the circular anode; here the distance L gap ≃ 3L 1,r ,whereL 1,r is given by Eq. 35.Note that here L gap is smaller, than in Figure 9a. Figure 9c shows the anode edge with the sharp tip; this tip provides a rapid growth of + with the distance from the tip.Thus, the reference electrode can be positioned closer to the tip not sacrificing the accuracy of measurements.L gap in Figure 9c can be estimated from Eq. 35, taking R a equal to the radius of tip curvature.
From Eq. 35 it is evident, that Eq. 16 is redundant.Eq. 16 describes a minimal size of the anode-free domain for the system with the straight anode edge.In the system with the curved anode edge, the size of the anode-free domain must obey to where L 1,r is given by Eq. 35.Clearly, at the distance on the order of 36, the local cathode overpotential drops to a vanishingly small value, and the model above is applicable.
Discussion.-Eq.12 describes a kind of "electrostatics" for overpotential distribution in the anode-free domain.Thinness of the membrane and its large in-plane size makes this electrostatics almost twodimensional.Numerical solutions to a fully 2D analog of Eq. 12 could give more insights into an optimal shape of the anode tip for positioning the reference electrode.However, due to very large gradients of potentials at the anode edge, fully 2D calculations might require special numerical techniques.
In the anode-free domain, hydrogen is not needed.In the absence of electroosmotic flux of water, the membrane in this domain can be humidified by the backflux of water from the cathode side.If, for technological reasons, a hydrogen flow field "covers" the anode-free domain, the H 2 crossover through the membrane can be blocked by applying some polymer material between the anode flow field and membrane.
In general, in the anode-free domain, the decaying cathode overpotential may accelerate carbon corrosion reaction (CCR) on the cathode side.The overpotential for CCR is η CCR ≃ 0.7V .Inatrue OCV state of a PEMFC, this overpotential is about 0.9 to 1 V, which induces much higher rate of corrosion.In addition, there is no counter electrode in the anode-free domain, and in order to be captured in the ORR, the proton produced in the CCR must travel quite a large distance along the cathode to the point, where the local ORR overpotential is sufficiently large.Due to a relatively large ohmic resistance of the membrane phase in the CCL, this proton transport would lower the overall rate of corrosion.

Conclusions
A model for the radial distribution of the membrane potential and of the ORR overpotential η c in a PEM fuel cell with the concentric small anode and large cathode is developed.In the anode-free area of the cell, the model reduces to the axially symmetric Poisson-Boltzmann equation for η c .An approximate analytical solution for this equation is constructed.A feature of this problem is weak dependence of the long-range shape of η c (r ) on the value of η 0 c ,w h e r eη 0 c is the ORR overpotential in the working domain of the cell.This effect is quite analogous to Debye screening in plasmas.
The solution shows rapid convergence of to the value of η 0 c with the distance from the anode edge r .Moreover, the smaller the anode radius, the faster tends to η 0 c with r .This result shows that the reference electrode measuring ≃ η 0 c can be located closer to the anode edge, if the latter is curved.

Appendix A: The Case of a Finite Cathode Radius
The model above can be modified for the cathode of the finite radius Rc .In this case, the long-range problem Eq. 12 reads where A, B and R1 are determined by the system of Equations 23, A4 and 25.

Figure 2 .
Figure 2. Schematic of the fuel cell with the concentric circular anode and cathode.

Figure 4 .
Figure 4. Numerical (points) and analytical (lines) shapes of the ORR overpotential for the dimensionless anode radius Ra of (a) 200 and (b) 20.The "infinite" cathode radius is Rc = 2000.The points exhibit the numerical solution to the full problem 9, while the lines show Eq. 26, which is valid outside the anode.Parameters for the calculations are listed in TableI.

Figure 5 .
Figure 5. (a) Zoom of the curves in Figure 4 and the overpotential for the anode radius Ra = 2. (b) The analytical curves shown in (a) shifted in such a way that the anode edges coincide.Indicated are the dimensionless anode radii.Note faster decay of the ORR overpotential η + c near the anode of a smaller radius.The symbol ∞ marks the shape of η + c at the edge of the plane anode. 8

Figure 6 .
Figure 6.Solid line-the exact width of the ORR-dominated domain L1,r = R1 − Ra vs. the anode radius Ra for base-case set of parameters (TableI).Dashed line-the same width calculated from the reduced system of equations 23, 24 and 28, which is independent of the overpotential in the working cell area η+,0

Figure 7 .
Figure 7. Analytical radial shapes of the positive ORR overpotential η + c

Figure 8 .
Figure 8.The width of the ORR-dominated domain L1,r = R1 − Ra vs. the anode radius Ra .At the distance R1 − Ra from the anode edge, the ORR overpotential drops to the value of the Tafel slope b ox , while the membrane potential nearly reaches the value of the cathode overpotential η 0 c in the working domain.Upper solid curve-the base-case set of parameters (TableI).Lower solid curve corresponds to j 0 ox = 6 • 10 −6 Acm −2 , b ox = 0.05 V and to the cell potential of 0.65 V; the other parameters are those indicated in TableI.Open circles-the approximate fitting Equation 34.Long-dashed lines-the distance L1,x , corresponding to the straight anode edge (see text).

Figure 9 .
Figure 9. Schematic of the reference electrode (RE) positioning for different anode geometries.(a) The straight anode edge, (b) the circular anode (c) the anode with the curved tip of a small radius.The smaller the tip radius, the closer to the tip can be located the RE.

c=
φ c − − E eq CCR ,w h e r e E eq CCR ≃ 0.207 V is the CCR equilibrium potential.Far from the anode edge, we have ≃ η 0 c , and thus η CCR c ≃ φ c − η 0 c − E eq CCR .With the typical φ c ≃ 0.6Vandη 0 c ≃−0.3V ,wefindη CCR c