Physical Theory of Platinum Nanoparticle ... - ACS Publications

J. Phys. Chem. C 2010, 114, 5773–5785

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Physical Theory of Platinum Nanoparticle Dissolution in Polymer Electrolyte Fuel Cells Steven G. Rinaldo,*,† Ju¨rgen Stumper,‡ and Michael Eikerling† Department of Chemistry, Simon Fraser UniVersity, 8888 UniVersity DriVe, Burnaby, B.C., Canada, and AutomotiVe Fuel Cell Cooperation Corporation, 9000 Glenlyon Parkway, Burnaby, B.C., Canada ReceiVed: October 23, 2009; ReVised Manuscript ReceiVed: January 6, 2010

The loss of electrochemically active surface area (ECSA) causes severe performance degradation over relevant lifetimes of polymer electrolyte fuel cells. Using a simple physical model, we analyze the interrelations between kinetics of platinum nanoparticle dissolution, evolution of the particle size distribution, and ECSA loss with time. The model incorporates the initial particle radius distribution, and it accounts for kinetic processes involving Pt dissolution, Pt-O formation, and Pt-O dissolution. Employing reasonable simplifying assumptions to the governing equations, a full analytical solution was found under potentiostatic conditions. The simplified model predicts the evolution of the particle radius distribution as well as ECSA loss with time, in close agreement with experimental ex situ and in situ studies. The study indicates that the rates of chemical Pt-O dissolution, driven by the particle size dependence of the cohesive energy, may dominate over electrochemical dissolution. Fitting of the model to experimental data provides an effective surface tension and an effective rate constant of Pt-O dissolution. Implications of the model for the development of strategies to reduce ECSA loss are discussed. 1. Introduction Tremendous improvements in power density and cost reduction of polymer electrolyte fuel cells (PEFCs) have been achieved over the last two decades, which have brought PEFC systems close to the benchmarks that are deemed critical for commercialization.1 At present, however, fuel cell development falls short in achieving durability and lifetime goals of targeted applications.2 To address these shortcomings, the focus of research and development in academia and industry has been shifting toward understanding durability issues and mitigating their impact on operation. The performance decline is seen in a loss of power density that the PEFC can provide, as indicated by measurable performance decay rates. For example, under drive cycle testing, single-cell voltage decay rates have been measured to be on the order of 10 µV h-1.3 Using cyclic voltammetry (CV) of CO or H adsorption as an experimental probe, a significant portion of the kinetic performance losses can be associated with the loss of electrochemically active surface area (ECSA) with time.3 In the cathode catalyst layer (CCL), materials that have received foremost attention in the recent literature on degradation phenomena include the Pt nanoparticle catalyst and the carbon support.4-8 Pt nanoparticles at the cathode side have been shown to dissolve in the CCL and re-emerge partly in the membrane as the so-called Pt-in-the-membrane (PITM) band.6,9-12 At potentials above the open-circuit voltage (OCV), the carbon support may also corrode; however, the effects of carbon corrosion below OCV are negligible. It is evident from experimental studies that catalyst surface area loss via Pt dissolution constitutes a principal mechanism of degradation in the cathode catalyst layer of a PEFC. Despite the importance of Pt dissolution, the role of operating conditions and their relation to pertinent mechanisms of Pt * To whom correspondence should be addressed. Tel.: (778) 782-3699. Fax: (778) 782-3765. E-mail: [email protected]. † Simon Fraser University. ‡ Automotive Fuel Cell Cooperation Corp.

dissolution are not well understood. In what follows, we will present first a brief review of the literature on Pt dissolution in PEFCs. Thereafter, we will develop a physical model of Pt dissolution and discuss expedient assumptions that lead to a full analytical solution of the simplified model equations. Results and discussion sections focus on application of the model to the analysis of experimental degradation data, highlighting its capabilities for parameter extraction. 2. Pt Dissolution in Cathode Catalyst Layers of PEFCs 2.1. Experimental Insights. PEFC degradation models in the literature suggest that Pt mass loss primarily occurs through an electrochemical dissolution pathway.13-16 However, none of these studies have tried to discern the difference in rates between chemical dissolution of Pt-O and electrochemical dissolution of bare Pt. Further complicating the challenge, nanoscale phenomena render comparisons between dissolution of bulk and nanoparticle Pt difficult. To reach conclusions on the prevailing mechanisms, or to make simplifications in modeling efforts, one needs to understand the implications of Pt size and of relevant electrochemical conditions at the catalyst surface on different ECSA loss mechanisms. Conditions that affect the electrocatalytic activity as well as the apparent rates of ECSA loss include electrode potential, composition of the reactant gas and of the electrolyte, water distribution, pH, and temperature. Structural parameters that affect degradation include Pt particle radius distribution (PRD), porous morphology and surface structure of the support, and the morphology of ionomer phase domains. Earlier works have examined the effect of electrode potential on Pt dissolution in phosphoric acid fuel cells (PAFC). Bindra et al.17 correlated the measured equilibrium solubility of bulk Pt with calculated equilibrium Pt2+ concentrations using the Nernst equation and equilibrium potentials determined by Pourbaix.18 The soluble ionic species of platinum found in solution is debatable. We assume for simplicity that the predominant species is Pt(II), although Pt(IV) has been reported

10.1021/jp9101509  2010 American Chemical Society Published on Web 03/04/2010

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elsewhere.22 Honji et al.19 found that from CCL potentials of 0 to 0.8 V vs RHE the mass of Pt in a Pt/C electrode remained constant but dropped to 1/3 of the initial value in approximately 100 h at 0.9 V vs RHE. These results indicate that Pt is soluble under highly acidic conditions and that the loss of Pt due to dissolution becomes pronounced at potentials above 0.8 V vs RHE. Wang and co-workers examined equilibrium concentrations of Pt ions resulting from the dissolution of a polycrystalline Pt wire as a function of electrode potential.20 For potentials from 0.6 to 1.2 V vs SHE, the equilibrium ion concentrations that they found were less dependent on potential than expected for a typical 2-electron process, and completely independent of potential above 1.3 V vs SHE. The authors interpreted the measured equilibrium Pt2+ concentrations in the potential invariant region as evidence for chemical dissolution of Pt-O. A similar study for carbon-supported Pt nanoparticles (Pt/C) indicated that the equilibrium concentrations of Pt ions were independent of potential from approximately 1.0 to 1.3 V vs SHE.21 Ferreira and co-workers identified a weaker than expected potential dependence of equilibrium concentrations of Pt2+ ions for Pt/C from 0.9 to 1.0 V vs RHE.10 Yet, the potential dependence was not evident from 1 to 1.05 V vs RHE. Although some controversy remains, the majority of experimental findings nurture the idea that Pt-O dissolution contributes significantly to overall Pt dissolution and hence catalyst layer degradation at sufficiently high electrode potentials. However, the extent of this contribution and the effect on equilibrium Pt2+ concentrations in solution are currently unresolved. A recent paper by Mitsushima et al.22 studied the solubility of a platinum black catalyst in acidic media. In sulphuric acid electrolyte and without an externally applied potential, the solubility of unsupported Pt nanoparticles increased with decreasing pH, indicative of an acidic dissolution mechanism. The solubility was higher in pure oxygen as compared to air, and it decreased dramatically under a nitrogen atmosphere. The soluble Pt species detected was Pt(IV), and the solubility increased with increasing temperature. The authors concluded that the mechanism was exclusively based on chemical dissolution of Pt-O/Pt-OH/Pt-O2. Umeda et al. argued that, in concentrated sulphuric acid, the amount of precursor formed (oxygenated surface species) is proportional to the amount of Pt lost through dissolution.23 They concluded that Pt dissolution occurs via a chemical (Pt(OH)2) or electrochemical (Pt(OH)4) reduction reaction. Potentiostatic experiments on conventional CCLs in PEFC and PAFC have indicated that ECSA loss does not depend strongly on potential under relevant operating conditions.10,24 Bett et al. showed that, for PAFC, ECSA loss was independent of potential from 0.3 to 1.2 V vs SHE.24 Ex situ studies on the durability of Pt/C (Vulcan XC-72) catalysts at 0.9 V vs SHE (@ 60 °C) and 1.2 V vs RHE (@ 20 °C) in 0.5 M H2SO4 electrolyte had similar ECSA decay rates.25,26 Dam et al. studied Pt dissolution and ECSA loss decay rates at 1.05, 1.15, and 1.25 V vs RHE (@ 80 °C) and found a modest dependence on potential.27 The degradation study of Ferreira and co-workers demonstrated that the ECSA decay rates at 0.75 and 0.95 V vs RHE and overall voltage decay rates were similar.10 If the prevailing mechanism of ECSA loss were to follow the electrochemical path of Pt dissolution, the initial ECSA decay rates at 0.75 and 0.95 V vs RHE in Ferreira et al.’s study should differ by several orders of magnitude. In summary, the main inferences from the reviewed works on potentiostatic Pt dissolution and ECSA loss are: (i) data to

Rinaldo et al. directly compare rates of Pt dissolution and Pt-O dissolution are a scarce commodity; (ii) Pt ion concentrations exhibit only a weak dependence on electrode potential above a certain value of potential, which is lower for carbon supported Pt nanoparticles (∼1.0 V; hereafter, unless indicated otherwise, potentials will be specified vs SHE) than for polycrystalline Pt (1.3 V); (iii) the apparent rate of Pt nanoparticle dissolution depends on pH, oxygen partial pressure, and temperature; and (iv) potentiostatic ECSA loss studies on PAFC and PEFC have indicated weak potential dependencies, which seem difficult to reconcile with an electrochemical dissolution mechanism. There has been a large body of work done to examine the character of the oxide species on bulk Pt. However, this contentious issue continues to fuel vivid scientific discussions.28-30,30-32 Consequently, the nature and abundance of different oxide species such as Pt-O, Pt-OH, Pt-OOH, and Pt-O2 on Pt nanoparticles are difficult to ascertain. Irrespective of this uncertainty, it is well-known that the onset potential of oxide formation shifts to lower values for Pt/C as compared to bulk Pt. The importance of chemical dissolution of oxide-covered Pt in relation to electrochemical dissolution of bare Pt has remained, thus, an unsettled issue in the literature. It is without justification that several recently presented Pt dissolution models have set the rate of chemical dissolution of oxide-covered Pt to arbitrarily low levels or ignored it completely.14-16 2.2. Particle Size Effects. Regardless of the detailed mechanism of the dissolution process, experimental data suggest that the initial Pt PRD in the CCL plays an important role in determining the net rate of ECSA loss. In general, a significant surface energy contribution to the cohesive energy destabilizes Pt nanoparticles in comparison to bulk Pt, irrespective of the detailed adsorbate structure. The particle size effect on cohesive energy is caused mainly by the undercoordination of surface atoms, which becomes pivotal for particles with size 0.8 V) there is an inherent loss in Pt activity due to the blockage of surface sites via the adsorption of oxide species.51 Thus it is reasonable to assume that, above a certain critical constant potential of Ec ∼ 0.9 V, oxide coverage is 1. Thus, the electrode potential determines which dissolution mechanism prevails. For E < Ec, ECSA loss will be influenced by the rates of all three reactions, eqs 6-9. For E > Ec, catalyst dissolution can be assumed to be dominated by eq 9, that is, the chemical dissolution route. The assumption of an almost complete oxide monolayer at sufficiently high electrode potentials, thus, eliminates eqs 6 and 8. If, furthermore, Pt ion concentration in the CCL is low and constant values of potential, proton concentration, and oxygen concentration are assumed, the problem of catalyst dissolution becomes spatially invariant, i.e., independent of variations in local reaction conditions. In this Article, we only examine the route via dissolution of Pt-O, assuming full monolayer coverage. It should also be noted that at potentials much higher than Ec, the dominating surface species may change from Pt-O to Pt-O2. Thus the upper potential limit

∆G∞3 cH+ cO2 θPtO ref RT cH+ cOref2

)

y3

(15)

Implementing simplifications discussed above, the set of governing equations reduce to a single continuity equation, which can be written in a parameter-free form,

∂f(ξ, τ) ∂f(ξ, τ) + ξ2 exp(ξ) ) -ξ2 exp(ξ)f(ξ, τ) ∂τ ∂ξ

(16) in nondimensional variables

τ)

R0 t and ξ ) T0 r

(17)

where characteristic time and radius are defined by

T0 )

2βγPt RTk∞3

and R0 )

j Pt 2βγPtV RT

(18)

One clear advantage of the parameter-free form of the governing equation is that the formulation of the problem via electrochemical dissolution from eq 6 or chemical dissolution from eq 9 simply involves a rescaling of characteristic time and radius. Dimensionless ECSA loss curves would be the same for both electrochemical and chemical dissolution. Equation 16 can be solved readily with the method of characteristics as outlined in Appendix B. The implicit solution is defined in terms of two equations,

f(ξ, τ) ) f0(ξ0) exp(-(ξ - ξ0))

(19)

and

-

exp(-ξ0) exp(-ξ) + Ei(1, ξ0) + Ei(1, ξ) ) τ ξ ξ0 (20)

where ξ0 is a parametric variable, and f0(ξ0) is the initial particle radius distribution parametrized by ξ0; Ei(1,ξ0) is the exponential integral of ξ0. The parametric solution does not allow the PRD to be determined directly in terms of the variables ξ and τ. An initial distribution function f0(r) needs to be converted to a nondimensional initial distribution function, f0(ξ). Equation 20 can then be solved numerically to find ξ0 at given τ and ξ. The time evolution of the PRD in dimensionless particle space can be calculated from eq 19, converted back to dimensional variables, and numerically integrated to obtain SECSA(t). 4. Results 4.1. Parametric Study. For this simplified case, the main parameter controlling PRD evolution and ECSA loss is the

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Rinaldo et al.

Figure 1. ECSA loss curves for various values of βγPt. An initial lognormal PRD with r0 ) 1.45 nm and σSD ) 0.3 was used. For model calculations a temperature of 80 °C and characteristic time of T0 ) 2000 h was assumed for the calculations.

surface tension, which appears in the model as a product with the transfer coefficient, βγPt; we refer to this combined parameter as an effective surface tension. Figure 1 shows ECSA loss curves for various values of βγPt using an initial log-normal PRD, employed to represent typical commercially available CCLs. The mathematical form of initial distribution functions used in the forthcoming analysis is given in Appendix C. The initial PRD was found by comparing normalized TEM particle size counts from a range of literature sources and fitting the corresponding counts with a log-normal distribution.10,25,26 The distribution had a mean radius of r0 ) 1.45 nm and a standard deviation of σSD ) 0.3. A temperature of 80 °C and characteristic time of T0 ) 2000 h were assumed. From the analysis, it is clear that the effective surface tension strongly influences the curvature and time scale of ECSA loss curves. With an effective surface tension of βγPt ) 0.2 J m-2, almost complete Pt loss occurs within approximately 2000 h; in comparison, this span is only 100 h for an effective surface tension of βγPt ) 1.2 J m-2. The effects of mean radius, r0, and PRD width, σSD, on ECSA loss are shown in Figure 2 for an initial Gaussian distribution with an effective surface tension, temperature, and characteristic time of βγPt ) 0.6 J m-2, 80 °C, and T0 ) 4 × 104 h, respectively. For comparative purposes, a Gaussian distribution was chosen because r0 and σSD can be varied independently. The same comparisons are difficult to make with a log-normal distribution. Increasing the mean radius (or conversely changing the surface tension, cf., eqs 17 and 18) significantly reduces the rate of ECSA loss, due to the exponential dependence of the dissolution rate constant on r-1. However, the ECSA loss rate is invariant to the width of the distribution until approximately 20% of the initial ECSA is lost. The standard deviation, σSD, determines the tailing of ECSA loss curves at large times. Varying the effective surface tension exerts a strong effect on the evolution of the PRD, as suggested by the corresponding changes in ECSA loss behavior. Figure 3 shows different modes of PRD evolution. As the effective surface tension increases from Figure 3a to d, a noticeable drift of the PRD to larger mean particle radii is observed. Qualitatively, a large effective surface tension represents a large variation of dissolution rates

Figure 2. The effects of varying mean radius (a) and PRD width (b) of an initial Gaussian PRD on ECSA loss. In (a) a constant value of 0.2 was assumed for the standard deviation and in (b) a constant value of 1.5 nm was assumed for the mean radius. For the calculations, an effective surface tension, temperature and characteristic time of βγPt ) 0.6 J m-2, 80 °C and T0 ) 40000 h were assumed.

over the range of radii. In other words, small particles dissolve much faster than large particles, and the PRD drifts to a larger mean radius over time. Conversely, a small effective surface tension represents a more uniform dispersion of dissolution rates, and the mean radius of the PRD will remain unaffected or drift to smaller values. Thus, the qualitative behavior of the PRD evolution bears information on the effective surface tension. Experimentally, a drift to larger mean radii is consistently observed in degraded CCLs, suggesting that the effective surface tension is relatively high. Yet, a recent paper has shown that the mean radius could as well decrease with ECSA loss.27 It is observed that increases of mean particle radii in degraded PEFC or Pt/C systems cannot be unambiguously ascribed to redeposition and/or agglomeration mechanisms, as is often done in the literature; they could as well be caused by a mere dissolution process if the surface tension is sufficiently large. Figure 4 is a plot of mean radii versus fractional ECSA loss. It reveals two distinct trends that are caused by the variation of the surface tension, as discussed above. The mean particle radius will initially decrease, when the effective surface tension is sufficiently low, i.e., βγPt < 0.7 J m-2. For larger values of βγPt, the mean particle radius exhibits a monotonous increase with ECSA loss. We will explore in our future work whether this relation between mean radius and ECSA loss could be used to differentiate agglomeration, ripening, and dissolution mechanisms. 4.2. Model Validation. Good correlations between model and experiment are shown for available potentiostatic in situ and ex situ ECSA loss studies conducted at electrode potentials above Ec ) 0.9 V, which included information on initial TEM PRDs. Figures

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Figure 3. Different modes of PRD evolution for varying values of βγPt. An initial log-normal PRD with r0 ) 1.45 nm and σSD ) 0.3 was used. For model calculations a temperature of 80 °C and characteristic time of T0 ) 2000 h was assumed for the calculations.

Figure 4. Mean radii versus ECSA loss for various values of βγPt. An initial log-normal PRD with r0 ) 1.45 nm and σSD ) 0.3 was used. For model calculations a temperature of 80 °C and characteristic time of T0 ) 2000 h was assumed for the calculations.

Figure 5. Comparison of predicted ECSA loss behavior with experimental data from Wang et al.25 Input parameters and experimental conditions are given in Table 1. An initial log-normal distribution with r0 ) 1.35 nm and σSD ) 0.3 was used.

5, 7, and 9 show the comparison between model predictions and constant voltage ECSA loss experiments. The parameters used for

fitting and the experimental conditions are listed in Table 1. Ex situ experiments, from which data for Figures 5, 6, 7, and 8 were

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Figure 6. Comparison of initial (a) and final (b) TEM data from Wang et al.25 Input parameters and experimental conditions are given in Table 1. An initial log-normal distribution with r0 ) 1.35 nm and σSD ) 0.3 was used.

Figure 7. Comparison of predicted ECSA loss behavior with experimental data from Shao et al.26 Input parameters and experimental conditions are given in Table 1. An initial log-normal distribution with r0 ) 1.5 nm and σSD ) 0.45 was used.

extracted, were performed with the same electrolyte and similar catalyst materials.25,26 Both studies exhibited similar ECSA decay rates for Pt supported on Vulcan XC-72. The effective surface tensions (βγPt), obtained by fitting the ECSA loss data, exhibit a variation of 15%. If used for the data by Wang et al.,25 ECSA loss curves (Figure 5) and end-of-life (EOL) particle radius distributions (Figure 6) show the weakest correspondence between experimental results and model calculations. This could be ascribed to the neglect of agglomeration/ redeposition processes in the model. On the other hand, the

Rinaldo et al.

Figure 8. Comparison of initial (a) and final (b) particle radius distributions obtained from TEM micrographs in Shao et al.26 Input parameters and experimental conditions are given in Table 1. An initial log-normal distribution with r0 ) 1.5 nm and σSD ) 0.45 was used.

Figure 9. Comparison of predicted ECSA loss behavior with experimental data from Ferreira et al.10 Input parameters and experimental conditions are given in Table 1. An initial log-normal distribution with r0 ) 1.45 nm and σSD ) 0.3 was used.

parametric study has indicated that the initial ECSA loss rate is rather insensitive to changes in PRD broadness. Thus, omitting larger particles from the initial distribution would have a small effect on ECSA loss predictions but would influence the EOL TEM comparisons. Upon comparison with the data of Shao et al.,26 there is a closer agreement between model predictions and experimental data (ECSA loss in Figure 7 and EOL PRD in Figure 8b). In this study, the BOL distribution (Figure 8a) was much broader. The predicted EOL PRD encompasses the range of particles that were proposed by Shao et al. to undergo growth via



TABLE 1: Experimental Conditions and Model Input Parameters for Analysis in Figures 5-10 experimental conditions ex situ, 0.9 V, 60 °C, 30 wt %, pH ) 1 ex situ, 1.2 V, 20 °C, 20 wt %, pH ) 1 in situ, 0.75 V, 80 °C in situ, 0.95 V, 80 °C

k∞3 (mol m-2 s-1) -12

7.8 × 10 3.4 × 10-12 2.2 × 10-15 8.5 × 10-13

redeposition or agglomeration. This comparison highlights the ambiguity of mechanistic interpretations associated with EOL TEM. The strong correlation with Shao et al.’s study suggests that the model can predict the evolution of the PRD correctly on the basis of ECSA loss data or vice versa ECSA loss on the basis of PRD data. Interestingly, the results obtained in our model are exclusively based on Pt-O dissolution as the origin of changes in PRD and ECSA with lifetime; they query the necessity for including effects of Pt ion migration, redeposition, or agglomeration. Ferreira et al.’s 0.75 and 0.95 V studies (Figures 9 and 10) on ECSA loss allow direct comparisons between the same fuel cell operated at different potentials. Errors in the fits can be estimated to be less than the experimental error involved in calculating ECSA loss from CV data. Data sets in Figure 9 correspond to different Pt-O surface coverages, which affects both the effective rate constant given in eq 15 and the effective surface tension. Row 3 in Table 1 indicates that, due to the lower oxide coverage at ∼0.75 V, the effective rate constant of Pt-O dissolution is reduced; the effective surface tension for this case is, however, comparatively high. At a high cathode potential (∼0.95 V), the surface should be almost entirely covered by Pt-O; the fitting at this potential gives a significantly lower effective surface tension, yet a higher effective rate constant as shown in row 4 of Table 1. Figure 10 shows a slightly increased mean radius of the predicted EOL PRD. Unfortunately, experimental data for EOL PRD of these potentiostatic studies were not available for comparison. The effective surface tension is somewhat of an elusive parameter in PEFC degradation modeling. A value of approximately 2.4 J m-2 was used in the models of Darling and

βγPt (J m-2)

reference

0.67 0.79 1.90 0.71

Wang et al.25 Shao et al.26 Ferreira et al.10 Ferreira et al.10

Meyers,14 Bi and Fuller,15 and Holby et al.,16 for bare Pt. This value of surface tension corresponds to an experimentally determined value at 1310 °C.52 As we have shown, the effective surface tension is a crucial factor for determining the kinetics of dissolution and ECSA loss. Therefore, using a correct value is imperative for accurate modeling approaches. To the best of the authors’ knowledge, there are no independent absolute values for surface tension at, or near, room temperature in an electrochemical environment. Nonetheless, there are techniques to determine relative changes in surface tension with potential. In work by Barker and Young,53,54 a unitless friction coefficient, µf, was determined for a polycrystalline Pt electrode in 0.1 M H2SO4 as a function of potential.55 It had also been suggested that the coefficient of friction was proportional to surface tension.55 Thus, an experimental ratio for friction coefficients at 0.75-0.95 V can be compared to the predicted effective surface tension ratio obtained from our model calculations, cf., rows 3 and 4 in Table 1. The experimental ratio was 0.4, while our model results, shown in Table 1, predict βγPt (Ferreira et al. @0.95 V)/βγPt(Fereira et al. @0.75 V) ) 0.37. Furthermore, Table 1 shows that, as expected, all ECSA loss studies performed above Ec have similar values of surface tension. The studies of Barker and Young revealed a decrease in µf with the adsorption of H or O. Worthy of note, electrocapillary studies on polycrystalline Pt in acidic electrolyte show a similar dependence of surface tension on potential.56 Thus, these findings, as well as the model results, suggest that adsorbed species have a pronounced influence on surface tension. Thus, the putative effect of potential on surface tension is presumably primarily an effect of the adsorbate structure of the catalyst. Comparison of effective rate constants (rows 3 and 4 in Table 1) obtained from the model fits to the in situ data of Ferreira et al. at 0.75 and 0.95 V indicate that the value used at 0.75 V is approximately 2 orders of magnitude lower than that at 0.95 V. Our model leaves one plausible explanation for this. At the low potential of 0.75 V, the oxide coverage on the Pt surface is estimated to be low, i.e., in the range of 0.2-0.4.57 Correspondingly, the rate of Pt-O dissolution will be significantly reduced. Low Pt-O coverage favors the electrochemical route of dissolution of bare Pt, which remains, however, insignificant due to the unfavorable overpotential. As discussed previously, the Pt-O interaction energy is likely to influence the dissolution rate. We can empirically incorporate this effect into the kinetic equation of Pt-O dissolution,

( )

k∞3 ) kω3 ) λθPtO exp

ωθPtO RT

(21)

where Figure 10. Comparison of initial particle radius distributions obtained from TEM micrographs in Ferreira et al.10 and final PRD model prediction. Input parameters and experimental conditions are given in Table 1. An initial log-normal distribution with r0 ) 1.45 nm and σSD ) 0.3 was used.

(

λ ) a∞3 exp -

)( )(

∆G∞3 cH+ cO2 RT cref+ cref H O2

)

y3

(22)

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is a parameter collecting the potential independent factors. Equation 21 assumes that interactions between Pt-O species destabilize particles, causing an increase in the dissolution rate with oxide coverage. To estimate the applicability of eq 21, Ferreira et al.’s data at 0.75 and 0.95 V can be used to correlate the effective rate constant with fractional oxide coverage. At 0.95 V, we assume a complete monolayer layer of oxide is formed. The interaction energy of nearest neighbor oxide species is estimated to be 24-35 kJ mol-1 for polycrystalline Pt.58 With values for the oxide coverage, interaction energy (ω ) 24 kJ mol-1), and the effective rate constants in Table 1, eq 21 can be used to estimate the coverage at 0.75 V. For Ferreira et al.’s study, we predict an oxide coverage of 0.4 at 0.75 V; this value lies in the range reported by Mathias et al., who experimentally determined the surface oxide coverage in a CCL of a PEFC at 0.75 V to be between 0.2 and 0.4.56 This analysis suggests that the interaction energy of surface oxide species may play a role in Pt oxide dissolution kinetics. It moreover implies that with detailed knowledge of surface coverage, surface tension, and the initial PRD, the model can be used to predict ECSA loss. However, as discussed before, the type and coverage of oxygenated surface species on bulk Pt, let alone Pt nanoparticles, is still controversial. Additionally, the interaction energy used corresponds to experiments on bulk Pt; deviations in oxide interaction parameters are to be expected for Pt nanoparticles. Last, the rate of dissolution has been experimentally determined to be 7.2 × 10-13 mol m-2 s-1 for 10 wt % Pt/C at 0.9 V vs SHE in 0.57 M perchloric acid.20 This value is in good agreement with the effective rate constants that we present in Table 1 for the fits of experimental data at potentials above Ec. All in all, parameters used to model experimental data can be related to, and lie within acceptable ranges of, available independent experimental values. 5. Discussion In the effort to develop rational strategies in degradation modeling, a consistent potentiostatic model establishes the fulcrum for any potential cycling or variable potential model. A remarkable feature of the Pt nanoparticle dissolution model presented here is its simplicity, afforded by the analytical parametric solution. In comparison to known ECSA loss models, with numerous parameters, the Pt nanoparticle dissolution model has essentially two independent parameters, the rate constant and surface tension, both of which were correlated to available experimental data. The model links PRD analysis on the basis of TEM micrographs with ECSA loss data. From our model results and literature analysis, it seems instructive to distinguish a high potential region, in which the ECSA loss mechanism and dissolution rate are only weakly dependent on potential, a transition region, and a region of low potential. Our current approach ignores the low potential region; in this region, the total rates of Pt dissolution are rather small, and other neglected phenomena, such as redeposition, should be accounted for. In this region there is a complex interplay of processes, making the precise determination of relevant parameters from experimental studies difficult. Comparatively, the high constant potential region is most suitable for discriminating mechanisms and for systematically evaluating effects of parameters. In the high potential region, in which ECSA loss is most severe, oxides are the most abundant surface species. It is the species most affected by degradation. This insight could be

Rinaldo et al. obtained from the analysis of independent experimental studies, and it is the main hypothesis in our model, which agrees well with ECSA loss data. However, an issue with the model, in its current form, is that the effective rate constant and the effective surface tension cannot be measured independently under conditions relevant for PEFC operation. To the best of the authors’ knowledge, reported experimental values for the surface tension are obtained from measurements at very high temperatures; thus, parameter extraction and application to PEFC model simulations is not warranted. Moreover, the transfer coefficient β is a largely inaccessible parameter. First principles modeling of elementary Pt-O dissolution reactions, based on ab initio calculations of the lowest free energy reaction paths of the processes in eq 5, would be the requisite approach to determine β. Results of such calculations are currently not available. Nevertheless, measured ratios of surface tensions at varying potential and oxide surface coverage can be used to evaluate model results and thus parametrize ECSA loss as a function of potential via oxide coverage and surface tension, as shown in our analysis. The effective rate constant is also amenable to experimental determination from ex situ solubility studies.20 Systematic experimental studies could thus be conducted to further validate our model. In our model, the potential dependence of Pt dissolution and concomitant ECSA loss is indirect, mediated by the dependence of Pt-O coverage and surface tension on potential. However, regardless of the dissolution mechanism, ECSA loss can be experimentally related to PRD evolution and to the accumulation of Pt ions in solution. We expect to find qualitative differences in PRD evolution and Pt ion accumulation depending on the mechanism of ECSA loss. An ECSA loss mechanism dominated by dissolution would have experimental fingerprints different from that of a ripening or agglomeration dominated mechanism. A possible experimental fingerprint for these mechanisms could be the relation between mean particle radius and ECSA loss, shown in Figure 4. Such an analysis requires data on the evolution of the PRD. Independent location TEM experiments (IL-TEM) can examine, in situ, the same catalyst region during an accelerated stress test (AST);59 thereby this novel method allows tracking of individual particles during degradation and thus direct observation of how the PRD changes with time. Another fingerprint method could be electrochemical quartz crystal microbalance (EQCM) analysis on Pt/C.9,23,60 Assuming the study is performed in a potential region where carbon corrosion is negligible or with a noncorroding support, an ECSA loss mechanism, dominated by ripening or agglomeration, would be subject to Pt mass conservation. However, if it were dominated by dissolution, the mass loss of Pt from the electrode would be significant. The current model is easily amendable to Pt mass loss analysis. Pending clarification on the issues of agglomeration/ripening and chemical versus electrochemical dissolution, we will extend our degradation model. Mobilities and concentrations of Pt ions and protons may have an influence on ECSA loss in PEFCs. Furthermore, ionic Pt in the membrane or catalyst pores may have undesirable effects that have not been considered. A full numerical solution for ionic transport is under development to gain better insight into these effects. This requires coupling the Pt dissolution model to a performance model to account for migration terms in the transport equations. Regardless of the results from the numerical investigation, it is clear from current results that increasing the mean particle radius or lowering the effective surface tension would have an influence on ECSA



decay rates. The effective surface tension may be influenced by catalyst alloying or the catalyst support material used, whereas increments in PRD mean particle size have to be weighed against losses in mass specific activity of the catalyst.

Appendix B. Following the method of characteristics, the characteristics of eq 16 are given by

6. Conclusions In the present study, a model and analytical solution for Pt nanoparticle dissolution in PEFC was presented and compared to literature data. The main results are summarized as follows. The predominant mechanism in the high potential region, corroborated by comparison with available degradation studies and independent data on surface tension, is the dissolution of Pt oxide. The presented analytical solution for Pt nanoparticle dissolution, driven by surface energy; it could be easily adapted to mechanisms other than Pt oxide dissolution. The model relates the evolution of the particle radius (PRD) evolution to the loss of electrochemically active surface area (ECSA) with time. The main parameter controlling these time-dependent functions is the surface tension. We have discussed predictive diagnostic capabilities of the model, including dependencies of dissolution phenomena on the initial particle radius distribution, surface tension, and dissolution rate. Systematic studies exploring the presented model could be used to relate effects of surface tension and dissolution rate further to particle composition, adsorbate structure (type and surface coverage of adsorbed species), and properties of the catalyst support. Our study demonstrates that the increase of the mean radius of the PRD, observed in Pt/C degradation studies, cannot be unambiguously ascribed to redeposition and/ or agglomeration processes. Future studies will have to compare specific signatures of the Pt oxide dissolution mechanism with those of other Pt mechanisms, relating effects among PRD evolution, ECSA loss, Pt ion accumulation in the cathode catalyst layer, and Pt ion effluence into fuel cell components flanking the cathode catalyst layer. Acknowledgment. We gratefully acknowledge the financial support of this work by the NSERC Collaborative Research and Development Grants program as well as by the collaborating companies, Automotive Fuel Cell Cooperation Corp. (AFCC) and Ballard Power Systems Inc. We would also like to thank David Harvey for insightful discussions.

dξ(s) ) ξ(s)2 exp(ξ(s)) ds dτ(s) )1 ds df(s) ) -ξ(s)2 exp(ξ(s))f(s) ds

(B1)

where s is a parametric variable. With initial conditions

τ(0) ) 0, ξ(0) ) ξ0, and f(0) ) f0(ξ0)

(B2)

the second differential equation in eq B1 can be solved to yield, τ ) s. Next, rearranging the third equation in eq B1 gives

df(s) dξ(s) ) -f(s) ds ds

(B3)

which can be solved resulting in

f(s) ) C1 exp(-ξ(s))

(B4)

where C1 is a constant of integration. Substituting initial conditions gives the solution as given in eq 19. In the last step, the first differential can be solved, yielding

s+

exp(-ξ(s)) - Ei(1, ξ(s)) + C2 ) 0 ξ(s)

(B5)

Next, employing initial conditions and τ ) s to eq B5, the solution as presented in eq 20 is found. Appendix C.

Appendix A. The activation Gibbs energies of electrochemical dissolution and redeposition can be written as

∆G1(r) )

equilibrium potential and rate constant of Pt dissolution and redeposition, resulting in the expressions given in eq 7.

∆G∞1

For Figures 1, 3-10, an initial distribution function of the form

j Pt j Pt βγPtV βγPtV - R1F(E(z, t) - Eeq 1 ) r r (A1)

f0(r) )

1

√2πσSDr

( ) [ ( )] ln

exp -

r r0

2

(C1)

2σSD2

was used, and for Figure 2,

(

and

∞ ∆G-1(r) ) ∆G-1

j Pt βγPtV + r

(1 - R1)F(E(z, t) - Eeq 1 ) +

(r - r0)2 1 f0(r) ) exp -2 σSD2 σSD√0.5π j Pt βγPtV r

)

(C2)

List of Symbols

(A2)

Applying the equilibrium conditions, ∆G1(r) ) ∆G-1(r) and E ) E1eq(r), eqs A1 and A2 can be solved for the r-dependent

Dependent Variables ci(z,t) E(z,t)

concentration of species i at position z and time t, mol m-3 potential, V

5784


Eeq j (r)

radius-dependent equilibrium potential of reaction j, V f(z,r,t) number of particles per unit electrode volume at time t in a size class r to r + dr, m-4 f0(r) initial PRD, m-4 ∆G(j(r) Gibbs free energy of activation for reaction j, J mol-1 J(z,r,t) source term for eq 11, m-4 s-1 k(j(r) radius-dependent rate constant for reaction j, mol m-2 s-1 SECSA(z,t) electrochemical platinum surface area per unit electrode volume, m2 m-3 Vj(z,r,t) rate of reaction j, mol m-2 s-1 θPtO(z,r,t) fractional oxide coverage on a particle sized r at coordinate z µ(r) chemical potential per metal atom as a function of particle radius, J mol-1 µ(∞) chemical potential of the bulk metal phase, J mol-1 ΦM(z,t) metal phase potential, V solution phase potential, V ΦS(z,t) Independent Variables r t z

radius of a platinum nanoparticle, m time, s coordinate, m

Input Parameters a∞j cref i Di Ec Ecoh Eeq j ∆G∞(j k3ω k∞j nj R0 r0 T0 T Rj β Γmax γPt λ σSD ω

prefactor for bulk rate constant of reaction j, mol m-2 s-1 reference concentration of species i, mol m-3 diffusion coefficient of species i, m2 s-1 critical potential, V cohesive energy, eV bulk equilibrium potential of reaction j, V bulk Gibbs free energy of activation for reaction j, J mol-1 interaction energy modified rate constant for Pt-O dissolution, mol m-2 s-1 effective bulk rate constant for reaction j, mol m-2 s-1 number of electrons transferred in reaction j characteristic radius, m mean radius of PRD, m characteristic time, s temperature, K electron transfer coefficient for reaction j transfer coefficient moles of adsorption sites per unit of platinum area, mol m-2 platinum surface tension, J m-2 parametrized constant for Pt-O dissolution, mol m-2 s-1 standard deviation of initial PRD Pt-O/Pt-O interaction parameter, J mol-1

Constants F R j Pt V

Faraday’s constant, 96 485 C mol-1 gas constant, 8.3144 J mol-1 K-1 molar volume of Pt, 9.1 × 10-6 m3 mol-1

Dimensionless Solution Variables Ci s ξ

constant of integration characteristic parametric variable dimensionless radius

Rinaldo et al. ξ0 τ

characteristic parametric variable dimensionless time

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