A Porous-Electrode Model To Predict the Quantity ... - ACS Publications

J. Phys. Chem. C 2009, 113, 15433–15443

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O2 Reduction Mechanism on Non-Noble Metal Catalysts for PEM Fuel Cells. Part II: A Porous-Electrode Model To Predict the Quantity of H2O2 Detected by Rotating Ring-Disk Electrode Fre´de´ric Jaouen* INRS E´nergie, Mate´riaux et Te´le´communications, 1650 Bd Lionel Boulet, Varennes (Que´bec) Canada, J3X 1S2 ReceiVed: January 28, 2009; ReVised Manuscript ReceiVed: July 7, 2009

During O2 electroreduction, some O2 molecules are only reduced by two electrons to form H2O2 while the rest is reduced by four electrons to form water. The relative H2O2 production is quantified with the % H2O2.The latter number means that if the reduction of 100 O2 molecules yields, e.g., 20 H2O2 and 160 H2O molecules, then the % H2O2 is 20%. Here, O2 electroreduction in a porous electrode is modeled to calculate the polarization (I-V curve) and % H2O2-voltage curves (%-V curve) for non-noble metal catalysts that are assumed to reduce O2 following either (i) 2e reduction + H2O2 disproportionation or (ii) 2e + 2e electroreduction. Using a set of base-case parameters, a series of I-V and %-V curves has been calculated for case i upon variation of (a) porous electrode thickness, (b) electrode rotation rate, (c) rate constant of H2O2 disproportionation, and (d) rate constant of O2-to-H2O2 electroreduction. For case ii typical I-V and %-V curves have also been modeled for one electrode thickness, one rotation rate, and the experimental rate constants of O2-to-H2O2 and H2O2-to-H2O electroreductions obtained in part I of this paper (DOI 10.1021/jp900837e) for the Fe/N/C catalyst. Analytical solutions for the % H2O2 are also given when the porous electrode is under either kinetic or diffusion control. It is concluded that the experimental I-V and %-V curves of the Fe/N/C catalyst of part I (DOI 10.1021/jp900837e) cannot be explained by mechanisms i and ii and that the 4e direct electroreduction of O2 to H2O is the main path occurring on the Fe/N/C catalyst investigated in part I (DOI 10.1021/jp900837e). 1. Introduction The object of this work was introduced in part I (DOI 10.1021/jp900837e). It was recently unveiled for some catalysts that the relative amounts of the two possible products of O2 reduction, namely, H2O and H2O2, depend on the catalyst loading.1-5 The relative quantity of H2O2 produced during O2 reduction was measured with the rotating ring-disk electrode (RRDE). For a given experiment, the relative importance of H2O2 as a product of the O2 reduction is quantified by the % H2O2. The latter number means that if the reduction of 100 O2 molecules yields, e.g., 20 H2O2 and 160 H2O molecules then the % H2O2 is 20%. The effect of the catalyst loading on the % H2O2, with various magnitudes, has been observed in acidic medium on a non-noble metal catalyst (NNMC),2 on Pt catalysts,1,4,5 and on a Ru-Se catalyst.3 To better understand this, the development of a mathematical model is required. The present model focuses on NNMC that have the particularity to show linear H2O2 electroreduction kinetics (part I of this work, DOI 10.1021/jp900837e). Models that address O2 electroreduction to H2O2 or H2O, electroreduction of H2O2 to H2O, as well as H2O2 disproportionation have been developed for a smooth electrode.6-11 Dong et al. assumed a Tafel kinetics for H2O2 electroreduction.10 This assumption is valid for Pt but not for NNMC in acid medium (part I of this work, DOI 10.1021/jp900837e). The main findings of these models for a smooth electrode are as follows.6,8,9,11 (i) For a totally indirect oxygen reduction reaction (ORR) pathway (direct 4e reduction blocked) or partly indirect pathway (2e reduction to H2O2 and 4e reduction to H2O coexist), the % H2O2 * To whom correspondence should be addressed. Phone: 450 929 8176. Fax: 450 929 8102. E-mail: [email protected].

measured by RRDE depends on the rotation rate. (ii) At infinite rotation rate, the % H2O2 converges to 100% for a totally indirect pathway only while at zero rotation rate the % H2O2 always converges to 0%. (iii) At a fixed potential, a plot of the ratio of the disk-to-ring currents, |Id/Ir|, vs the inverse of the square root of the rotation rate, ω-1/2, always yields a linear relationship. The previously mentioned models were developed for smooth electrodes. However, in electrochemistry, practical catalysts are powders of high surface areas of, e.g., metal, oxide nanoparticles, carbon, or nanoparticles of an expensive metal dispersed on carbon powders. Powder catalysts imply that the electrode deposited on the disk is not smooth. It is a porous electrode with a thickness commensurate with that of the diffusion layer in the electrolyte. Thin electrodes of 0.1 µm of powder catalysts may be considered smooth electrodes. However, the structural entities forming a catalyst powder are usually >0.1 µm. For example, Pt is usually supported on carbon blacks that have an agglomerate size of 0.1-0.4 µm.12 As a result, incomplete coverage of the disk by the catalyst is a risk for thin electrodes of 0.1 µm.13 Incomplete coverage renders the classical theory developed for RDE (Levich and Koutecky-Levich equations) obsolete because mixed planar and spherical diffusion occurs and the limiting current is smaller than that predicted by Levich.14-17 Incomplete coverage must therefore be avoided by ensuring practical thicknesses of porous electrodes in the order of 1-10 µm. For the ORR with H2O2 as a possible intermediate, porous and smooth electrodes may thus result in completely different % H2O2 detected using the RRDE method. In a porous electrode, an H2O2 molecule produced at one site has a high probability to be electroreduced or disproportionated at other sites before

10.1021/jp900838x CCC: $40.75  2009 American Chemical Society Published on Web 08/03/2009

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J. Phys. Chem. C, Vol. 113, No. 34, 2009

it may reach the electrolyte. In a smooth electrode, that probability is much lower. A first attempt to develop a model for ORR taking into account the various reactions involving O2 and H2O2 (Figure 1 in part I, DOI 10.1021/jp900837e) combined with the porous character of the electrode was made by Appleby and Savy.18 This work is however unclear in that the electrode thickness L never appears in the derived equations. Also, the diffusion and reaction equations (eqs 3 and 4 in the present work) that are the foundation of the problem do not seem to have been solved numerically. Since then, this problem has received very little attention from a modeling viewpoint. The porous character of the electrode has been considered in models assuming only a direct 4e ORR pathway to water.19,20 The new results concerning RRDE measurements of NNMC prompted us to further investigate the ORR mechanism on such catalysts (part I of this work, DOI 10.1021/jp900837e). Some NNMC show a strong variation of % H2O2 with loading,2 while others show a weak variation (ref 21 and part I of this work, DOI 10.1021/jp900837e). Also, the NNMC are peculiar compared to Pt-based catalysts in that they do not show a Tafel law for H2O2 electroreduction but a linear law (Figure 2 in part I, DOI 10.1021/jp900837e). Thus, the development of an RRDE model that takes into account the porous character of the electrode as well as the linear H2O2 electroreduction kinetic shown by NNMC is needed. This work investigates the effect of transport parameters and kinetic parameters on the % H2O2. Analytical solutions for % H2O2 are derived for two limiting cases: reaction under kinetic control and reaction under diffusion-limited control. By simply changing the boundary conditions, the model also predicts polarization curves in a deaerated but H2O2-containing electrolyte. The model shows that for hypothetical catalysts with high activity for either H2O2 electroreduction or H2O2 disproportionation, the RRDE method may not be able to reach a sufficiently high mass transport rate in order to reveal high % H2O2. Therefore, the RRDE method may not be able to discriminate between apparent 4e ORR (i.e., 2e + 2e or 2e + H2O2 disproportionation) and true 4e ORR for such catalysts. On the other hand, the RRDE method can discriminate between apparent 4e ORR and true 4e ORR for catalysts with low or moderate activity for H2O2 electroreduction and H2O2 disproportionation. This is the case for non-noble metal catalysts. 2. Model for O2 Reduction Mechanism The model is at steady state and one-dimensional, i.e., the coordinate in the porous electrode. The direct 4e electroreduction of O2 is assumed to be impossible. The porous electrode comprises a powder catalyst, a proton-conducting ionomer (Nafion), and pores that are filled by the liquid electrolyte. 2.1. Rate Expressions, Mass Balance, and Boundary Conditions.

O2 + 2H+ + 2e f H2O2 Reaction 1; V1 ) k1c1 H2O2 f H2O + 1/2O2 Reaction 2; V2 ) k2c2 H2O2 + 2H+ + 2e f 2H2O Reaction 3; V3 ) k3c2 where Vi is the consumption rate of reactants in reaction i in a small volume of the porous electrode (mol s-1 m-3), ki is the forward apparent rate constant (s-1), and c1 and c2 are the O2

Jaouen and H2O2 concentrations, respectively. It must be noted that the apparent rate constant k2 presently defined for the porous electrode is not identical to the apparent constant k2 defined in part I (eqs 2 and 3 in part I, DOI 10.1021/jp900837e) because the systems are different (electrode * electrolyte), and thus, the values for parameter “A” (m2 catalyst per m3 of the system, eqs 4 and 5 in part I, DOI 10.1021/jp900837e) differ by orders of magnitude. In contrast, the definition of the specific rate constant k2s (eq 4 in part I, DOI 10.1021/jp900837e) is the same in parts I (DOI 10.1021/jp900837e) and II, and its value is catalyst specific. The forward reactions (eqs 1-3) are assumed to be of order 1 with the concentration of O2 or H2O2. The rate constant k1 is assumed to obey a Tafel law

k1 ) k01 exp(RF(E0 - E)/(RT))

(1)

where superscript 0 corresponds to any fixed potential E0 for which the k1 value is given and R is the transfer coefficient. The rate constant k2 is assumed independent of the electrode potential. The rate constant k3 is assumed to increase linearly with decreasing potential as observed experimentally on NNMC (part I (DOI 10.1021/jp900837e), Figures 2 and 4)

k3 ) k*3 (E* - E)

(2)

where the superscript asterisk (*) corresponds to any potential (E* - 1) at which the k3 value is given. Reaction and diffusion of O2 and H2O2 in the volume of the porous electrode are considered. Mass balance at steady state yields the following equations

D1,eff

D2,eff

d2c1 dx

d2c2 dx2

2

- k1c1 +

k2c2 )0 2

+ k1c1 - (k2 + k3)c2 ) 0

(3)

(4)

where subscript 1 after c (concentration) or D (diffusion coefficient) is for O2, subscript 2 for H2O2, and subscript eff for the effective diffusion coefficient in the electrode. x is the coordinate in the electrode, oriented from the current collector toward the electrolyte. Depending on the boundary conditions, the model can predict the current and % H2O2 during ORR (O2-saturated electrolyte) or the current in an H2O2-containing electrolyte (O2 free). Boundary Conditions in Common. Zero diffusion flux at the glassy-carbon disk

dci dx

|

x)0

)0

(5)

Boundary Conditions for O2-Saturated Electrolyte. In the electrolyte, at a distance δ from the electrode-electrolyte interface, the O2 concentration is c*1 while that of H2O2 is zero. δ is the diffusion layer thickness for an RDE, predicted from the Levich equation. The expression of δ vs rotation rate is given in Appendix 1. Thus, c1(x ) L + δ) ) c*1 and c2(x ) L + δ) ) 0, where L is the electrode thickness. The boundary condition at L + δ is transformed into a condition at x ) L- by stating

Non-Noble Metal Catalysts for PEM Fuel Cells

J. Phys. Chem. C, Vol. 113, No. 34, 2009 15435

the conservation of flux at x ) L- and x ) L+ (L- and L+ are the lower and upper limits of L, respectively)

dc1 dx

|

x)L-

dc2 dx

|

)

(c*1 - c1(L)) D1 δ D1,eff

(6)

Id ) -4F(ΦO2 - ΦH2O2) - 2FΦH2O2 ) -4FΦO2 + 2FΦH2O2

(9) The % H2O2 leaving the electrode is now defined by

% H2O2 ) 100ΦH2O2 /ΦO2 x)L-

)

-c2(L) D2 δ D2,eff

(7)

where D is the diffusion coefficient in the electrolyte outside the porous electrode while Deff is the effective diffusion coefficient inside the porous electrode. The volumetric fraction of electrolyte in that medium represents only a fraction of the volume, and therefore, Deff < D. Boundary Conditions for H2O2-Containing Electrolyte (O2-free). At a distance δ from the electrode-electrolyte interface, the H2O2 concentration is c*2 and that of O2 is zero. c1(x ) L+ δ) ) 0, and c2(x ) L + δ) ) c*2 . The same transformation as above (eqs 6 and 7) can be done again (not shown, deductible from eqs 6 and 7 by analogy). Numerical Resolution. The position x in the electrode is discretized in N nodes with N ) 200-500 in practice. The O2 and H2O2 concentration profiles are solved for a given set of kinetic parameters and for a given electrode potential. The starting guess for the O2 concentration profile is a straight line with cO2(0) ) 0 and cO2(L) ) c*O2. The starting guess for the H2O2 concentration profile is a straight line with cH2O2(0) ) 0.01 and cH2O2(L) ) 0. The diffusion eqs 3 and 4 are discretized to form a linear system of equations. This system is solved with the Newton-Raphson method. Calculation of Current and Percent H2O2. When the O2 and H2O2 concentration profiles across the electrode have been numerically calculated for a given electrode potential, the current produced by the electrode on the disk, Id, and the % H2O2 released by that electrode can be calculated according to the following reasoning. The O2 flux entering the electrode, ΦO2, and the H2O2 flux leaving the electrode, ΦH2O2, can be calculated from Fick’s diffusion law (ΦO2 and ΦH2O2 are defined positive and are the absolute values of Di dci/dx at x ) L+). The flux ΦH2O2 leaving the electrode represents an equal flux of O2 molecules, Φ2e O2, entering the electrode and that are reduced only by two electrons. The complementary of ΦO2 (ΦO2 - Φ2e O2) therefore corresponds to a flux of O2 molecules that is reduced by four electrons. Thus

ΦO2 ) ΦH2O2 + ΦO4e2 ) ΦO2e2 + ΦO4e2

(10)

(8)

Thus, the total current (defined 0, H2O2 electrooxidation to O2) assuming that collection efficiency, N, is 1. If not, Ir must be replaced by Ir/N. In the present model, if the entire H2O2 flux leaving the electrode is reoxidized to O2 according to backward reaction 1, it will create a current Ir ) 2FΦH2O2. Combining the latter expression with eq 9, eq 10 indeed results in % H2O2 ) 200 × Ir/(Ir - Id). In the case of an electrolyte containing H2O2 and free of O2, the current can also be calculated from eq 9. 2.2. Asymptotic Solutions. In order to predict the full curve of % H2O2 vs potential, the numerical resolution is required. However, under control of the ORR either by diffusion (low potential) or kinetics (high potential) the % H2O2 numerically obtained can also be found analytically. The various cases are now presented, and Table 1 gives a summary of all analytical solutions and their conditions of applicability. 2.2.1. No H2O2 Disproportionation or Electroreduction: k2 ) k3 ) 0. The % H2O2 is of course 100% since direct 4e ORR is assumed not to occur. Equation 3, simplified since k2 ) 0, can now be integrated. The analytical expression of c2(x) (given in the Supporting Information for section 2.2.1) shows that the maximum concentration of H2O2 in the electrode is found at x ) 0 (glassy carbon-porous electrode interface) under diffusionlimited current. The expression of the maximum H2O2 concentration is c*1 D1/D2. At room temperature, c*1 at pH 1 is 1.3 mM and D1 and D2 are 1.9 × 10-9 and 1.3 × 10-9 m2 s-1, respectively.22-24 This gives c2(max) ) 1.9 mM. 2.2.2. Electrode under Diffusion Control (Low Potential): k1 . k2 and k1 . k3. At low potential, the constant k1 (exponential increase with decreasing E) becomes much larger than k2 (potential independent) and much larger than k3 for NNMC (linear increase with decreasing E, see part I, DOI 10.1021/jp900837e). Then, the entire O2 flux diffusing from the electrolyte is transformed into H2O2 by reaction 1 at the electrode-electrolyte interface (x ) L) and H2O2 produced at x ) L diffuses either in the electrode or in the electrolyte. Under diffusion control the O2 concentration at x ) L is almost zero and the limiting O2 flux, D1c*1 /δ is transformed at x ) L in two fluxes of H2O2, one entering the electrode (second term in eq 11) and the other leaving it (third term in eq 11)

TABLE 1: Analytical Solutions of Percent H2O2 When the Electrode Is under Kinetic or Diffusion Control a

% H2O2 at high potential % H2O2 at low potentialb

L/Lc < 1/8

any L and ω values

L/Lc > 1

L/Lc > 1 and ω . ωc

eq 20 eq 17 with tanh(2L/Lc) ≈ 2L/Lc

eq 19 eq 17

eq 19 with tanh(2L/Lc) ≈ 1 eq 17 with tanh(2L/Lc) ≈ 1

% ) 100/(1/2 + 2L/Lc) % ) 100

Analytical solutions obtained at high potential are valid only when k3 ) 0. nonzero. a

b

At low potential, analytical solutions valid even if k3 is

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D1c*1 dc2 ) D2,eff δ dx

|

x)L-

+

D2c2(L) δ

Jaouen

(11)

Thus, from eq 10, the % H2O2 in that case is

% H2O2 ) 100

D2 c2(L) D1 c*1

(12)

Now, c2(L) must be determined. Numerically, it is observed that under those conditions, k1c1 ≈ k2c2/2 at any position x in the electrode, except very close to x ) L. Equation 4 then simplifies to D2,eff d2c2/dx2 - (k2/2 + k3)c2 ) 0. This can be integrated. The constant of integrations are found using the boundary condition dc2/dx(x ) 0) ) 0 and using eq 11. After defining a critical electrode thickness Lc and a constant β according to the following

Lc ) 2

2D2,eff k2 + 2k3

(13)

Di,eff ) βDi

D1c*1 D2

1 δ L 2β tanh 2 +1 Lc Lc

(

100

% H2O2 )

( )

1 L ω + 4β 2 Lc ωc

( ) )

100 % H2O2 ) δ L 1 + 2β tanh 2 Lc Lc

(

( ))

(15)

% H2O2 )

(16)

The % H2O2 decreases with an increase in either δ/Lc or L/Lc. δ is the diffusion layer thickness in the electrolyte and L the electrode thickness. Practically, Lc is the maximum electrode depth that can be efficiently utilized to perform H2O2 disproportionation. Because from an experimental standpoint the rotation rate, ω, is more meaningful than the diffusion layer thickness δ, eq 16 is rewritten with ω as the variable

(

100

( )

1 + 2β c

-1/2

( ))

tanh 2

L Lc

(17)

with ωc (rad s-1) defined so that δ ) Lc when ω ) ωc. From Appendix 1 one obtains

c )

√2 +

L Lc

( )

tanh 2√2

L Lc

(19)

with Lc defined by eq 13 but k3 ) 0. As in eq 17, the term δ/Lc has been replaced by (ω/ωc)-1/2, where ω is the rotation rate (rad s-1) and ωc is defined by eq 18. Moreover, if L/Lc is sufficiently small ( 1, tanh(2L/ Lc) converges to 1 and eq 17 is simplified. Thus, for L > Lc, the % H2O2 at low potential is not a function of L. On the other

( )

(20)

-1/2

This is the same expression as that obtained at low potential (section 2.2.2) and when L/Lc < 1/4, i.e., a particular case of eq 17. This means that for a sufficiently thin electrode, % H2O2 is independent of the electrode potential. On the other hand, eq 19 simplifies when L/Lc > 2/2 because tanh(22 · L/Lc) then converges to 1. In contrast with diffusion control (% H2O2 becomes independent of L at large L), under kinetic control the % H2O2 decreases continuously with increasing L. Moreover, if on top of the condition L/Lc > 2/2 one also has 4β(ω/ωc)-1/2 , 2 (high rotation rate), then eq 19 simplifies to % H2O2 ) 100/(1/2 + 2 · L/Lc). This percentage is independent of the rotation rate. It represents the highest amount of % H2O2 that is predicted at high potential and for a thick electrode when the rotation rate is infinitely high. 2.2.5. Electrode under Kinetic Control and with k2 ) 0 (no H2O2 disproportionation). For ORR catalysts that do not catalyze H2O2 disproportionation when the electrode is polarized (k2 ) 0 or k2 , k3) the expression of % H2O2 (see Supporting Information for section 2.2.3) can be rearranged to yield

% H2O2 )

(18)

100 L ω 1 + 4β Lc ωc

( )

L ω 4β Lc ωc

-1/2

100 +2

L Lc

(21)

1

( )

tanh 2

L Lc

where Lc is defined by eq 13 with k2 ) 0 and ωc is defined by eq 18. Equation 21 is very similar to eq 19. The fundamental



TABLE 2: Base-Case Parameters for the Modela A unit value

-1

m 2.5 × 108

0 k1s

R

-1

ms 10-9

k2s -1

0.7

ms 10-7

k3s m s-1 0

a Superscript 0 means that E ) E0 ) 0.9 V vs a reversible hydrogen electrode (RHE).

difference is that k3 is potential dependent, and thus, Lc and ωc in eq 21 are potential dependent too while they are not in eq 19. 2.3. Model Parameters. A set of standard kinetic parameters (base case) is used to show model predictions. Table 2 gives the specific rate constants, k01s and k2s. The apparent rate constants 0 and A · k2s k10 and k2 in the porous electrode are equal to A · k1s respectively, where A is the surface of catalyst per volume of electrode (value of A in Table 2, see calculation in Appendix 2). Note that for every 83 mV decrease of the electrode potential, k1 will increase by a decade. With base-case parameters, k1 becomes larger than k2 at E < 733 mV against a reversible hydrogen electrode (RHE). A Tafel slope of 83 mV dec-1 was assumed and corresponds to an exchange coefficient R ) 0.7 (Table 2). A slope of 70-80 mV dec-1 is usually observed for Fe-based nonnoble catalysts.25-27 H2O2 electroreduction is not considered in the base case and for calculations of section 3 (k3s ) 0). Model parameters kept constant throughout this work are as follows: the electrolyte viscosity at pH 1, ν ) 10-6 m2 s-1;28 the O2 concentration in the O2-saturated electrolyte, c*1 ) 1.3 mol m-3;23 the diffusion coefficients D1 ) 1.9 × 10-9 m2 s-1 and D2 ) 1.3 × 10-9 m2 s-1.23,24 Next, the parameter β is defined by Deff ) β · D (eq 14). Assuming a Bruggeman law Deff ) D · (pore fraction)3/2 and a pore fraction of 0.5 in the electrode (pores filled by electrolyte), then β ) 0.35. The value 1/3 was chosen for simplicity. With base-case parameters, Lc ) 11.8 µm (eq 13 with k3 ) 0) and ωc ) 2425 rpm (eq 18). 3. Model Results In this section, reaction 3 is not considered in order to simplify the model response. Omitting reaction 3 however does not change the nature of the problem. It only retrieves the electricpotential-dependent consumption rate of H2O2 associated with reaction 3 but the fundamental equations (eqs 3 and 4) remain the same. Only if reaction 3 had a rate constant of the same order of magnitude as that of reaction 1 would the problem be fundamentally changed. This is not the case with NNMC, which show a sluggish H2O2 electroreduction with linear kinetics (Figure 2 in part I, DOI 10.1021/jp900837e). 3.1. In-Depth Analysis with Base-Case Parameters. The base-case parameters (Table 2) are used, and the rotation rate is 200 rpm. Major differences in the results of % H2O2 are exemplified for two extremes of electrode thickness. 3.1.1. Thin Electrode: L , Lc (0.1 µm , 11.8 µm). Figure 1 presents O2 and H2O2 concentration profiles in an electrode of thickness L ) 0.1 µm and for four different electrode potentials. As will be seen in the polarization curve (Figure 3) corresponding to Figure 1, at 800 mV the reaction is under kinetic control, while at 600 mV it is under kinetic and masstransport control, and at 400 and 200 mV it is under diffusion control. Due to the thinness of the electrode, the concentration of either O2 or H2O2 is uniform in the electrode (Figure 1) from high potential down to 500 mV vs RHE (500 mV curve not shown).

Figure 1. Calculated O2 (thick line) and H2O2 concentration (dashed line) vs electrode position for a thin electrode. Base-case parameters, rotation rate 200 rpm, L ) 0.1 µm. Calculations for four electrode potentials (in mV against a reversible hydrogen electrode, abbreviated as RHE) are indicated on the left for O2-concentration profiles and on the right for H2O2 profiles.

Figure 2. Calculated H2O2 concentration vs electrode position. Same calculations as in Figure 1 but at electrode potentials close to 300 mV vs RHE.

In Figure 1, all O2 concentration profiles are increasing functions of x, the position in the electrode, meaning that O2 diffuses from the bulk electrolyte (on the right) toward the electrode and further diffuses inside the electrode. For H2O2, the situation is more complex. As long as the electrode potential is >320 mV, the profiles of H2O2 concentration are decreasing functions of x, meaning that H2O2 diffuses outside of the electrode toward the electrolyte. However, at E < 320 mV, the profiles of H2O2 concentration show a maximum (Figure 2), meaning that part of the H2O2 flux diffuses inside the electrode. The flux is proportional to the derivative dc/dx; therefore, even the shallow maximum seen in Figure 2 means much in terms of flux. This phenomenon becomes more and more important as the electrode potential is lowered. The reason for this is that the electrode part that is used to perform the electrochemical reaction 1 shifts to a zone confined very close to x ) L as the potential is decreased (very high values of k1). This, in turn, means that H2O2 is produced in the same confined zone close to x ) L. In that case, the H2O2 molecules can choose two paths: the bulk electrolyte or the interior of the electrode where they can react according to reaction 2. The relative importance of these two H2O2 fluxes will depend on the driving forces for these two paths. For diffusion in the electrolyte, the driving force is the rotation rate, and for diffusion in the electrode the driving

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Jaouen

Figure 4. Calculated O2 (thick line) and H2O2 concentration (dashed line) vs electrode position for a thick electrode. Base-case parameters, rotation rate 200 rpm, L ) 20 µm. Calculations for four electrode potentials in mV vs RHE indicated on the left for O2-concentration profiles and on the right for H2O2 profiles. NB: H2O2 profiles at 600 (or 450 mV) and 750 mV cross each other at position x ) 11.5 µm. Figure 3. Calculated polarization curve and % H2O2 for a thin electrode. Base-case parameters, rotation rate 200 rpm and L ) 0.1 µm. (Bottom) Dashed lines are the diffusion-limited currents for 0 and 100% H2O2. The % H2O2 at points A and B can be analytically calculated with eqs 17 and 20, respectively.

forces are the apparent rate constant of H2O2 disproportionation, k2, and the electrode thickness L. The thicker is the electrode, the larger the catalyst area available to the H2O2 molecules to react according to reaction 2. However, owing to diffusion limitation, the H2O2 molecules cannot efficiently use the electrode interior that is situated at a distance > Lc (eq 13) from the electrode-electrolyte interface. Figure 3 shows that the reaction is under kinetic control down to 0.7 V vs RHE, then under mixed kinetic and diffusion control, and then under diffusion control at E < 0.5 V vs RHE. The polarization curve shows a limiting current. The % H2O2 is 96% and independent of electrode potential. The % H2O2 at point A in Figure 3, where O2 reduction is under diffusion control, can be analytically calculated with eq 17. The % H2O2 at point B, where O2 reduction is under kinetic control, can be calculated with eq 20. The independence of % H2O2 with electrode potential is due to the fact that the tanh term in eq 17 can be approximated by 2L/Lc when L, Lc (section 2.2.4). In the present situation, electrode thinness impedes H2O2 disproportionation inside the electrode. 3.1.2. Thick Electrode: L > Lc (20 µm > 11.8 µm). Figure 4 presents O2 and H2O2 concentration profiles for an electrode of thickness L ) 20 µm. As will be seen in the corresponding polarization curve (Figure 5), the reaction is under kinetic control at 900 mV, while at 750 mV it is under kinetic and diffusion control, and at E < 600 mV it is under diffusion control. Due to the thickness of the electrode, the concentration of either O2 or H2O2 is nonuniform when the electrode is not under kinetic control. Also, in contrast with the thin electrode, the H2O2 profiles at E < 750 mV show steeper slopes. This means that a larger flux of H2O2 diffuses inside the electrode where it disproportionates. Figure 5 shows the polarization curve and % H2O2. Compared to Figure 3 (L ) 0.1 µm) there are three major changes: (i) the horizontal translation to the right of the E-I curve, expected because of increased catalyst area in the electrode; (ii) the drastic decrease of % H2O2 (10-30%) compared to that for the thin electrode (96%); (iii) the % H2O2 now depends on the electrode potential (Figure 5). The increase of the % H2O2 from 10% to

Figure 5. Calculated polarization curve and % H2O2 for a thick electrode. Base-case parameters, rotation rate 200 rpm, and L ) 20 µm. (Bottom) Dashed lines are the diffusion-limited currents for 0 and 100% H2O2. The % H2O2 at points A and B can be analytically calculated with eqs 17 and 20, respectively.

30% when the potential is lowered from 800 to 500 mV coincides with the shift from kinetic to diffusion control. Under diffusion control, O2 is electroreduced in a zone confined to x ) L; therefore, H2O2 is produced in that zone too, and this favors a higher fraction of H2O2 to diffuse in the electrolyte as compared to the kinetic control. The % H2O2 at point A in Figure 5 can be analytically calculated with eq 17 and with the tanh term approximated by 1 since L > Lc. The % H2O2 at point B can be calculated with eq 19 with the tanh term approximated by 1 as well. 3.2. Driving Forces Determining the Percent H2O2. Masstransport conditions (electrode thickness, rotation rate) or kinetic 0 ) may affect the apparent % H2O2 predicted. parameters (k2s, k1s The effect of these parameters is now investigated. 3.2.1. Effect of Electrode Thickness, L. At low potential, the % H2O2 decreases with increasing electrode thickness until it reaches an asymptotic lower limit that is thickness independent (Figure 6). This was predicted by the analytical expressions when L/Lc > 1 (section 2.2.2 after eq 18). The % H2O2 decreases


Figure 6. Effect of electrode thickness on polarization curve and % H2O2. Base-case parameters, rotation rate 200 rpm. Electrode thickness of 0.1, 1, 2, 10, 20, and 50 µm. The dashed lines are the diffusionlimited currents for 0 and 100% H2O2. The % H2O2 at points A and B can be analytically calculated with eqs 17 and 20, respectively.

fast with L until the ratio L/Lc approaches 1 (Lc ) 11.8 µm with base-case parameters). At high potential, the % H2O2 decreases down to near 0% with increasing L value, as predicted by analytical solutions (section 2.2.4). The % H2O2 at point A can be calculated with eq 17 and at point B with eq 19. The ORR onset potential increases with increasing thickness, expectedly. For the thickest electrode (50 µm), the less sigmoidal-shaped I-E curve at 800-900 mV is due to nonuniform O2 concentration in the electrode even at small current densities. This occurs when the electrode thickness approaches the value of the diffusion layer thickness in the electrolyte, δ (41 µm at 200 rpm). Experimentally, the minor effect of catalyst loading (i.e., electrode thickness) on the % H2O2 for the catalyst Fe/N/C (Figure 5 in part I, DOI 10.1021/jp900837e) is proof of a direct 4-electron ORR mechanism. However, other Fe-based catalysts show a loading effect on their % H2O2.2 3.2.2. Effect of Rotation Rate, ω. The electrode thickness is 2 µm, i.e., lower than Lc (11.8 µm with base-case parameters). The rotation rate is varied from 50 to 3200 rpm. The % H2O2 increases with increased rotation rate (Figure 7). This was expected since the rotation rate is a driving factor for forcing H2O2 to diffuse toward the electrolyte. The % H2O2 is almost potential independent, as expected since L/Lc , 1. The % H2O2 at point A can be calculated with eq 17 with the tanh term approximated by 2L/Lc (since L/Lc < 1/4). The % H2O2 at point B can be calculated with eq 20. Experimentally, the absence of effect of rotation rate on the % H2O2 measured on the Fe/N/C catalyst (Figure 6 in part I, DOI 10.1021/jp900837e) is proof of a direct 4e ORR mechanism. However ORR on other catalysts or at other pHs have shown a % H2O2 dependency on rotation rate: Pt in alkaline medium,7 heat-treated Fe-porphyrin on carbon in 0.5 M H2SO4,29 glassy carbon in alkaline medium,30 gold in alkaline medium.8 3.2.3. Effect of the Specific Rate Constant of H2O2 Disproportionation, k2s. The electrode thickness is 2 µm and the rotation rate 800 rpm. The parameter A is unmodified (2.5 × 108 m-1), and k2 ) A · k2s. The effect of increased k2s value is drastic (Figure 8). A higher k2 value means a lower Lc (eq 13). Thus, for a fixed L, the ratio L/Lc increases as k2 increases. Thus, expectedly, there is a shift from a potential-independent % H2O2 (low k2s value) to a potential-dependent % H2O2 (high k2s value).


Figure 7. Effect of rotation rate on polarization curve and % H2O2. Base-case parameters, L ) 2 µm. Rotation rates of 50, 200, 800, 1600, and 3200 rpm. The % H2O2 at points A and B can be analytically calculated with eqs 17 and 20, respectively.

Figure 8. Effect of rate constant of H2O2 disproportionation, k2,s, on the polarization curve and % H2O2. Base-case parameters (except for k2s), L ) 2 µm, and rotation rate 800 rpm. k2s values of 10-8, 10-7, 2 × 10-7, 5 × 10-7, 10-6, and 10-5 m s-1 were used.

Experimentally, k2s for the Fe/N/C catalyst was found to be 1.6 × 10-8 m s-1. Thus, if the Fe/N/C catalyst worked under the 2e + H2O2 mechanism, according to Figure 8 it would show a % H2O2 > 80% for a 2 µm thick electrode (corresponding to a loading of 125 µg cm-2). However, experimentally, only about 10% H2O2 is measured, even at 54 µg cm-2 (Figure 5 in part I, DOI 10.1021/jp900837e). Thus, the conclusion is that the ORR mechanism on the Fe/N/C catalyst investigated in part I (DOI 10.1021/jp900837e) is not 2e + H2O2 disproportionation. 3.2.4. Effect of the Specific Rate Constant of O2 Electrore0 . Electrode thicknesses of 2 and 30 µm are duction to H2O2, k1s considered. The rotation rate is 200 rpm, and all parameters 0 . have base-case values except k1s For the thin electrode, the % H2O2 is independent of k01s (figure not shown). For the thick electrode, the curve % H2O2 vs E is simply translated horizontally because the potential at which the electrode shifts from a kinetic control to a diffusion control 0 by five is changed (figure not shown). Thus, the change of k1s

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Figure 9. Chart predicting the % H2O2 released by a porous electrode that electroreduces O2 to H2O2 followed by H2O2 disproportionation. L/Lc and ω/ωc are two dimensionless parameters representing the effects of electrode thickness and rotation rate, respectively. Thick lines: Equipercentage lines of H2O2 at low potential (diffusion control). Dashed lines: Equipercentage lines of H2O2 at high potential (kinetic control). The calculations leading to this chart assumed no H2O2 electroreduction (assumed k3 ) 0).

decades has no influence on the % H2O2 for catalysts that perform only O2 electroreduction to H2O2 and H2O2 disproportionation. 0 In practice, the change of k1s by several decades is obtained by tuning the metal content used to prepare NNMC.26 In that work, a similar low % H2O2 of 1-4% at pH 1 was measured by us on Fe- or Co-based catalysts with various metal contents, spanning 2.5 orders of magnitude in ORR activity. 3.2.5. Chart for Predicting the Percent H2O2 as a Function of L/Lc and ω/ωc. For an ORR catalyst working under the mechanism of either 2e + 2e or 2e + H2O2 disproportionation, a sufficiently low value of L/Lc combined with a sufficiently large value of ω/ωc will theoretically yield a % H2O2 close to 100%. Then, this 100% value correctly reflects the totally indirect nature of the assumed ORR mechanism (no direct 4e ORR assumed). However, the present model predicts that the % H2O2 approaches 0% under slow mass transport conditions even if the ORR mechanism is totally indirect. Experimentally, one desires to have fast mass transport in order to detect the highest possible % H2O2 and reveal whether the ORR mechanism is direct or indirect. However, depending on the values of the apparent rate constants k2 and k3, the RRDE system may not be able to reach sufficiently rapid mass transport in order to reveal the indirect nature of the ORR mechanism. It is shown in this section that, knowing the apparent rate constants k2 and k3 for a given catalyst, it is then possible to define the required values of electrode thickness and rotation rate. With these required values, the RRDE experiment should then show close to 100% H2O2 if the ORR mechanism on that catalyst is totally indirect. A chart showing the % H2O2 predicted by the present model is drawn, with L/Lc as the x axis and ω/ωc as the y axis, and the chart is drawn in log-log scale (Figure 9). To draw the chart in Figure 9, it was assumed that k3 ) 0 (no H2O2 electroreduction). The reasoning and results would however not be fundamentally changed if k3 were assumed nonzero, and the reader could replot the chart with the appropriate analytical equations given in the paper when k3 is nonzero. When k3 ) 0, the asymptotic solutions (eqs 17 and 19 for low and high potential, respectively) enable us to draw the equipercentage lines of H2O2 in a log-log plane [L/Lc, ω/ωc]. Figure 9 can be used practically in the following manner. Knowing k2 for a given catalyst in a porous electrode, the

Jaouen constants Lc and ωc can be calculated (eqs 13 and 18 with k3 ) 0). Then, for a porous electrode of thickness L and for a rotation rate ω, the two dimensionless numbers L/Lc and ω/ωc can be calculated. This defines a point in the graph. The % H2O2 at low potential (diffusion control) or high potential (kinetic control) is then about that of the closest equipercent line (solid and dashed lines, respectively). For example, if k2 ) 25 s-1, then Lc ) 11.8 µm and ωc ) 2425 rpm. Thus, if L ) 0.1 µm and ω ) 200 rpm, then L/Lc ) 8.5 10-3 and ω/ωc ) 8.2 × 10-2. It is seen that the % H2O2 predicted is then >90% at any electrode potential (Figure 9). This corresponds to calculations shown in Figure 3. The RRDE system has practical limitations for the rotation rate and electrode thickness. The range of practical electrode thickness is estimated to be 0.9-15 µm. Lower thickness may result in an incomplete coverage of the disk. Larger thickness may result in turbulence at high rotation rate. The range of 0.9-15 µm corresponds to loadings of 54-960 µg cm-2 for carbon-based NNMC (Appendix 3), and they are the loadings experimentally used (part I (DOI 10.1021/jp900837e), Figure 5). The range of practical rotation rate is estimated to be 100-1600 rpm. Lower rotation results in small and noisy ring and disk currents. Higher rotation rate likely produces turbulence. These practical limitations define a window in the log-log plane [L/Lc, ω/ωc]. The position of the window depends on the value of k2 (or k2 and/or k3 in the general case) since the values Lc and ωc depend on k2 (and k3 in the general case), see Figure 9. The accessible window for a catalyst with k2 ) 25 s-1 (basecase value) is drawn in Figure 9. Similar windows are drawn for catalysts with k2 ) 0.25 or 2500 s-1. The usefulness of the chart (Figure 9) is exemplified for the Fe/N/C catalyst of part I (DOI 10.1021/jp900837e). For this catalyst, k2s is 1.6 × 10-8 m s-1. Its BET area is 611 m2 g-1, resulting in an “A” value of 3.8 × 108 m2 per m3 for the porous electrode (Appendix 2). Thus, the value of k2 ) Ak2s is 6 s-1. If the ORR mechanism on that Fe/N/C catalyst were totally indirect (2e + H2O2 disproportionation) one should experimentally observe % H2O2 of 50-90% (Figure 9, window “6 s-1”). This is not what is experimentally observed (maximum of 10% H2O2 at low loading and high rotation rate). In conclusion, the ORR on the Fe/N/C catalyst of part I (DOI 10.1021/jp900837e) cannot proceed according to the 2e + H2O2 disproportionation mechanism. The other possibility, 2e + 2e, is considered in the next section. 4. Comparison ModelsExperiments for the Fe/N/C Catalyst 4.1. Polarization Curve in H2O2-Containing Electrolyte (O2 free). The parameters for the model are now adapted to simulate the Fe/N/C catalyst of part I (DOI 10.1021/jp900837e). The constant k2s was measured in part I (DOI 10.1021/ jp900837e). The parameter A is calculated from the BET area 0 was chosen to of the catalyst (Appendix 2). The constant k1s approximately reproduce the onset potential of the ORR 0 observed with Fe/N/C. The precision on k1s need not be high 0 has no impact on the % H2O2 because it was shown that k1s predicted by the model (section 3.2.4). Last, the specific rate constant k3s is extracted from the experimental results of part I (DOI 10.1021/jp900837e) as explained later in this section. First, it is investigated what the model predicts if reaction 3 is neglected; this gives the three curves that correspond to three different thicknesses (labels b in Figure 10). These curves have the same shape as though oxygen were bubbled in the solution



Figure 10. Calculated polarization curve in H2O2-containing electrolyte (O2 free). Simulation of Fe/N/C catalyst (Figure 4 part I, DOI 10.1021/ jp900837e): c*2 ) 9 mM H2O2, c*1 ) 0, 1500 rpm. L ) 0.64, 1.5, or 6.4 µm (corresponding to loadings of 40, 96, and 400 µg cm-2, respec0 tively). (a) k1s ) 0, k2s ) 0, k*3s ) 2.2 ×10-8 m s-1 (E* ) 0.8 V vs 0 RHE), A ) 3.8 ×108 m2 per m3. (b) k1s ) 5 ×10-10 m s-1 (E0 ) 0.9 V vs RHE), k2s ) 1.6 ×10-8 m s-1, k*3s ) 0, A ) 3.8 ×108 m2 m-3. Dashed lines: kinetic current of reaction 3 (if no H2O2 diffusion limitation). Diffusion-limited current for 2e reduction of H2O2 at 1500 rpm in 9 mM H2O2: 15.0 mA cm-2.

because the current is actually due to O2 electroreduction to H2O2, the only difference being that O2 comes not from the electrolyte but is produced inside the electrode by H2O2 disproportionation. Because the constant k2s of H2O2 disproportionation is potential independent, the current reaches a limit as soon as the electrode potential is low enough so that O2 electroreduction is not a limiting factor. Because the value of k2 is small, in this case the limiting current seen at low potential (Figure 10) is almost not limited by H2O2 diffusion from the electrolyte but is mainly limited by the rate of H2O2 disproportionation in the electrode. It is therefore a chemically limited current. That current increases with increasing rate constant k2 and electrode thickness L. It only slightly increases with rotation rate and above a certain value of rotation rate becomes independent of it. Obviously, the curves predicted by the model when reaction 3 is neglected (curves b in Figure 10) are totally different from the experimental ones (part I (DOI 10.1021/jp900837e), Figure 4). Next, the three solid curves (curves a in Figure 10) in Figure 10 are calculated by assuming k2 ) 0 and considering H2O2 electroreduction with a linear kinetics. Each curve corresponds to a given electrode thickness (or catalyst loading). The values k*3 and E* (eq 2) were determined from the experimental curves (Figure 2, part I, DOI 10.1021/jp900837e) measured on the Fe/ N/C catalyst. First, the extrapolation of the reduction branches to zero current crosses the x axis at 0.8 V vs RHE, which thus gives E* ) 0.8 V vs RHE. This is in agreement with Figure 7 in ref 31 for a NNMC made of Fe-porphyrin and carbon black. Next, the slope of the experimental curve for loading 96 µg cm-2 (after Koutecky-Levich correction) has a value of about 22 A m-2 V-1. Theoretically, the rate of reaction 3 is k3c2 (section 2.1) and the current per volume of electrode is thus -2Fk3c2 (current defined

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