This paper is published as part of Faraday
Discussions volume 140:
Electrocatalysis - Theory and Experiment at the Interface
Preface
Preface
Andrea E. Russell, Faraday Discuss., 2009
DOI: 10.1039/b814058h
Introductory Lecture
Discussion
General discussion
Faraday Discuss., 2009,
DOI: 10.1039/b814699n
Papers
Papers
Differential reactivity of Cu(111) and
Cu(100) during nitrate reduction in acid
electrolyte
Sang-Eun Bae and Andrew A. Gewirth,
Faraday Discuss., 2009
DOI: 10.1039/b803088j
The role of anions in surface
electrochemistry
D. V. Tripkovic, D. Strmcnik, D. van der Vliet,
V. Stamenkovic and N. M. Markovic, Faraday
Discuss., 2009
DOI: 10.1039/b803714k
Molecular structure at electrode/electrolyte
solution interfaces related to
electrocatalysis
Hidenori Noguchi, Tsubasa Okada and Kohei
Uosaki, Faraday Discuss., 2009
DOI: 10.1039/b803640c
From ultra-high vacuum to the
electrochemical interface: X-ray scattering
studies of model electrocatalysts
Christopher A. Lucas, Michael Cormack, Mark
E. Gallagher, Alexander Brownrigg, Paul
Thompson, Ben Fowler, Yvonne Gründer,
Jerome Roy, Vojislav Stamenković and Nenad
M. Marković, Faraday Discuss., 2009
DOI: 10.1039/b803523g
A comparative in situ195Pt electrochemicalNMR investigation of PtRu nanoparticles
supported on diverse carbon nanomaterials
Fatang Tan, Bingchen Du, Aaron L. Danberry,
In-Su Park, Yung-Eun Sung and YuYe Tong,
Faraday Discuss., 2009
DOI: 10.1039/b803073a
Electrocatalysis: theory and experiment at
the interface
Marc T. M. Koper, Faraday Discuss., 2009
DOI: 10.1039/b812859f
Surface dynamics at well-defined single
crystal microfacetted Pt(111) electrodes: in
situ optical studies
Iosif Fromondi and Daniel Scherson, Faraday
Discuss., 2009
DOI: 10.1039/b805040f
Spectroelectrochemical flow cell with
temperature control for investigation of
electrocatalytic systems with surfaceenhanced Raman spectroscopy
Bin Ren, Xiao-Bing Lian, Jian-Feng Li, PingPing Fang, Qun-Ping Lai and Zhong-Qun
Tian, Faraday Discuss., 2009
DOI: 10.1039/b803366h
Bridging the gap between nanoparticles
and single crystal surfaces
Payam Kaghazchi, Felice C. Simeone, Khaled
A. Soliman, Ludwig A. Kibler and Timo Jacob,
Faraday Discuss., 2009
DOI: 10.1039/b802919a
Mesoscopic mass transport effects in
electrocatalytic processes
Y. E. Seidel, A. Schneider, Z. Jusys, B.
Wickman, B. Kasemo and R. J. Behm,
Faraday Discuss., 2009
DOI: 10.1039/b806437g
Nanoparticle catalysts with high energy
surfaces and enhanced activity synthesized
by electrochemical method
Zhi-You Zhou, Na Tian, Zhi-Zhong Huang, DeJun Chen and Shi-Gang Sun, Faraday
Discuss., 2009
DOI: 10.1039/b803716g
Discussion
General discussion
Faraday Discuss., 2009,
DOI: 10.1039/b814700k
Papers
On the catalysis of the hydrogen oxidation
E. Santos, Kay Pötting and W. Schmickler,
Faraday Discuss., 2009
DOI: 10.1039/b802253d
Hydrogen evolution on nano-particulate
transition metal sulfides
Jacob Bonde, Poul G. Moses, Thomas F.
Jaramillo, Jens K. Nørskov and Ib
Chorkendorff, Faraday Discuss., 2009
DOI: 10.1039/b803857k
Influence of water on elementary reaction
steps in electrocatalysis
Yoshihiro Gohda, Sebastian Schnur and Axel
Groß, Faraday Discuss., 2009
DOI: 10.1039/b802270d
Co-adsorbtion of Cu and Keggin type
polytungstates on polycrystalline Pt:
interplay of atomic and molecular UPD
Galina Tsirlina, Elena Mishina, Elena
Timofeeva, Nobuko Tanimura, Nataliya
Sherstyuk, Marina Borzenko, Seiichiro
Nakabayashi and Oleg Petrii, Faraday
Discuss., 2009
DOI: 10.1039/b802556h
Aqueous-based synthesis of ruthenium–
selenium catalyst for oxygen reduction
reaction
Cyril Delacôte, Arman Bonakdarpour, Christina
M. Johnston, Piotr Zelenay and Andrzej
Wieckowski, Faraday Discuss., 2009
DOI: 10.1039/b806377j
Size and composition distribution dynamics
of alloy nanoparticle electrocatalysts
probed by anomalous small angle X-ray
scattering (ASAXS)
Chengfei Yu, Shirlaine Koh, Jennifer E. Leisch,
Michael F. Toney and Peter Strasser, Faraday
Discuss., 2009
DOI: 10.1039/b801586d
Discussion
General discussion
Faraday Discuss., 2009,
DOI: 10.1039/b814701a
Papers
Efficient electrocatalytic oxygen reduction
by the blue copper oxidase, laccase,
directly attached to chemically modified
carbons
Christopher F. Blanford, Carina E. Foster,
Rachel S. Heath and Fraser A. Armstrong,
Faraday Discuss., 2009
DOI: 10.1039/b808939f
Steady state oxygen reduction and cyclic
voltammetry
Jan Rossmeisl, Gustav S. Karlberg, Thomas
Jaramillo and Jens K. Nørskov, Faraday
Discuss., 2009
DOI: 10.1039/b802129e
Intrinsic kinetic equation for oxygen
reduction reaction in acidic media: the
double Tafel slope and fuel cell applications
Jia X. Wang, Francisco A. Uribe, Thomas E.
Springer, Junliang Zhang and Radoslav R.
Adzic, Faraday Discuss., 2009
DOI: 10.1039/b802218f
A first principles comparison of the
mechanism and site requirements for the
electrocatalytic oxidation of methanol and
formic acid over Pt
Matthew Neurock, Michael Janik and Andrzej
Wieckowski, Faraday Discuss., 2009
DOI: 10.1039/b804591g
Surface structure effects on the
electrochemical oxidation of ethanol on
platinum single crystal electrodes
Flavio Colmati, Germano Tremiliosi-Filho,
Ernesto R. Gonzalez, Antonio Berná, Enrique
Herrero and Juan M. Feliu, Faraday Discuss.,
2009
DOI: 10.1039/b802160k
Electro-oxidation of ethanol and
acetaldehyde on platinum single-crystal
electrodes
Stanley C. S. Lai and Marc T. M. Koper,
Faraday Discuss., 2009
DOI: 10.1039/b803711f
Discussion
General discussion
Faraday Discuss., 2009,
DOI: 10.1039/b814702g
Concluding remarks
All dressed up, but where to go?
Concluding remarks for FD 140
David J. Schiffrin, Faraday Discuss., 2009
DOI: 10.1039/b816481a
PAPER
www.rsc.org/faraday_d | Faraday Discussions
Steady state oxygen reduction and cyclic
voltammetry
Jan Rossmeisl,*a Gustav S. Karlberg,a Thomas Jaramillob
and Jens K. Nørskova
Received 6th February 2008, Accepted 28th March 2008
First published as an Advance Article on the web 20th August 2008
DOI: 10.1039/b802129e
The catalytic activity of Pt and Pt3Ni for the oxygen reduction reaction is
investigated by applying a Sabatier model based on density functional
calculations. We investigate the role of adsorbed OH on the activity, by
comparing cyclic voltammetry obtained from theory with previously published
experimental results with and without molecular oxygen present. We find that the
simple Sabatier model predicts both the potential dependence of the OH coverage
and the measured current densities seen in experiments, and that it offers an
understanding of the oxygen reduction reaction (ORR) at the atomic level. To
investigate kinetic effects we develop a simple kinetic model for ORR. Whereas
kinetic corrections only matter close to the volcano top, an interesting outcome
of the kinetic model is a first order dependence on the oxygen pressure.
Importantly, the conclusion obtained from the simple Sabatier model still
persists: an intermediate binding of OH corresponds to the highest catalytic
activity, i.e. Pt is limited by a too strong OH binding and Pt3Ni is limited
by a too weak OH binding.
Introduction
The largest challenge in proton-exchange membrane fuel cell (PEMFC) catalysis
concerns the high overpotentials required to drive the oxygen reduction reaction
(ORR).1 Sluggish ORR kinetics account for the majority of the voltage drop in
PEMFCs, limiting state-of-the-art systems to operating voltages of only 0.7 V,
far from the equilibrium potential of 1.2 V. The highest-performance PEMFCs
all rely upon Pt to catalyze this reaction despite the high cost of this material,
a problem which will be further exacerbated by its scarcity. In recent years, much
effort has been devoted to improving the ORR activity of Pt, with secondary goals
of improving catalyst stability and reducing catalyst cost. A number of different
approaches have been taken in this regard, however three in particular have been
shown to be effective: (1) alloying Pt with other transition metals, e.g. Co, Ni or
Fe,2–4 (2) modifying Pt’s lattice constant via Pt (or Pt-alloy) overlayer structures,5–8
and (3) overlaying other metals, such as Au, onto Pt supports.9 To date, the most
active ORR catalyst was recently reported by Stamenkovic et al.,5 whose experiments revealed an order of magnitude improvement of Pt3Ni(111) over Pt (111).
The relationship between OH adsorption and the ORR is a critical issue in
PEMFC catalysis. In the following work, we investigate this by applying a previously
a
Center for Atomic-scale Materials Design, Department of Physics, Technical University of
Denmark, Lyngby, DK-2800, Denmark
b
Center for Nano-particle Functionality, Department of Physics, Technical University of
Denmark, Lyngby, DK-2800, Denmark
This journal is ª The Royal Society of Chemistry 2008
Faraday Discuss., 2008, 140, 337–346 | 337
published theoretical model based on first-principles.2,10 Our general aim is to elucidate the role of adsorbed OH in the ORR catalytic activity. The only inputs to our
model are density functional theory (DFT) calculations and standard molecular
tables.
The paper is divided into two parts. In the first part we compare the output
from our theoretical Sabatier model to recent experimental results for Pt(111) and
Pt3Ni(111),5 as these are well-known ‘‘benchmarks’’ for ORR catalysis. We find
that the key features observed in experimental cyclic voltammetry in an O2-free
environment are predicted by the simple Sabatier model. In the second part of the
paper, we develop a more detailed kinetic model which allows us to compare the
calculated and measured steady state current densities in the presence of O2. We
find that the ORR current can indeed be expressed as a function of the OH-coverage,
noting however that it is the coverage of OH under operating conditions (in the
presence of dissolved O2) that is used and not the coverage measured in normal
CV experiments.
Cyclic voltammetry
We begin by examining the potential dependence of the OH coverage resulting
from water dissociation during cyclic voltammetry in an O2-free environment. The
reaction to consider is:
H2O(l) 4 OH* + H+ + e
(1)
By introducing a theoretical counterpart to the standard hydrogen electrode,10
we re-write reaction (1) at zero potential as:
H2O(l) 4 OH* + 1/2H2(g)
(2)
The free energy of this reaction is defined as DGOH ¼ DE + DZPE TDS, where
DZPE and TDS account for changes in zero point energy and entropy. DE is the
reaction energy as calculated using DFT calculations. DGOH corresponds to the
change in free energy of reaction (1) at zero potential. At finite potentials (U s 0)
the chemical potential of electron in reaction (1) is changed by eU. According
to a previously developed method for calculating cyclic voltammograms based
on density functional theory,11 the coverage of OH as a function of potential can
be derived from the coverage dependence of DGOH.
We will consider the case of adsorbed OH in contact with liquid water and will,
therefore, for simplicity assume an ice-like water–hydroxyl layer. Such model
systems have been intensively studied in surface science due to their close resemblance to realistic metal-water interfaces.12 Choosing this model system for the
metal–water interface means that the total coverage of OH and H2O is always 2/3
of a monolayer (due to the supply of H2O molecules from the ‘‘real’’ liquid phase).13
As can be seen from a previously published detailed investigation,14 there are two
different coverage regimes for the OH–H2O interaction for such layers on
Pt(111)—up to 1/3 of a monolayer and above 1/3 of a monolayer of OH. The origin
of these different regimes is two-fold; first, the OH–H2O interaction is stronger than
both the OH–OH and the H2O–H2O interaction, and second, the hydrogen scrambling in these overlayers is very facile,15 meaning that the relaxation time for finding
the most stable overlayer is fast.
For OH coverages less than 1/3 of a monolayer, every OH molecule will have three
H2O molecules as nearest neighbors. As the coverage of OH exceeds 1/3 of a monolayer, however, ‘‘defects’’ in the OH/H2O adlayer appear where OH molecules
are nearest neighbors. Since the hydrogen bond formed between OH molecules is
just half as strong as the hydrogen bond formed between OH and H2O,14 the latter
situation will result in OH molecules less strongly bound to the surface. Looking
338 | Faraday Discuss., 2008, 140, 337–346
This journal is ª The Royal Society of Chemistry 2008
Fig. 1 Top and side view of two configurations with total coverage of OH and H2O of 2/3 of
a monolayer. To the left, 1/3 of a monolayer of OH and H2O, and to the right, 4/9 of a monolayer of OH and 2/9 of a monolayer of H2O. The differential adsorption energy for OH is
DE ¼ 0.45 eV for 1/3 of monolayer of OH and DE ¼ 0.86 eV when the OH coverage exceeds
1/3 of a monolayer. For details about the density functional theory calculations see ref 2.
at the reaction energy DE of reaction (2) versus OH coverage, this corresponds to DE
being constant for OH coverages less than or equal to 1/3 of a monolayer. When the
OH coverage exceeds 1/3, DE would become more positive. This jump in DE is
verified with explicit density functional theory calculations in Fig. 1.
The jump in DE is important since it shows that under the potentials of interest in
this study the coverage of OH cannot exceed 1/3 of a monolayer. We also note that
this value of the coverage compares well to recent experiments (0.4 ML).5
With a weak interaction between hydroxyls in the interesting coverage interval
(<1/3 ML), the coverage dependence in DGOH will only arise from configurational
entropy. Using the configurational entropy of non-interacting particles, DS ¼
kln((1 qOH)/qOH), for 0 < qOH < 1/3, we can therefore write the potential and
coverage dependence of the reaction free energy of reaction (1) as
DG(qOH,U) ¼ DGOH kln((1 qOH)/qOH) eU
(3)
Here DGOH is calculated for the standard condition of 1/3 of a monolayer
coverage OH and 1/3 of a monolayer H2O, this structure is depicted to the left in
Fig. 1. Assuming that reaction (1) is in equilibrium for all potentials (DG(U,qOH)
¼ 0), this leads to the following expression for the coverage
QOH(U) ¼ (1/3) 1/{1 + exp[(DGOH eU)/kT]}
(4)
In the following we will use DGOH equal to 0.80 eV and 0.93 eV for Pt and P3Ni
respectively, as was reported in ref. 2. The analytical expression for coverage shown
in eqn (4) is determined solely by the equilibrium with water and protons and is the
theoretical counterpart to the OH coverage obtained from cyclic voltamograms in
This journal is ª The Royal Society of Chemistry 2008
Faraday Discuss., 2008, 140, 337–346 | 339
Fig. 2 (a) Experimental (dashed lines with circles) coverage of OH versus potential for Pt3Ni
(black) and pure Pt (red) obtained from integration of cyclic voltammograms.5 The solid lines
show the coverage as calculated using eqn (4) with input from density functional theory and
standard tables. (b) The same as Fig. 4a but with the experimental coverage scaled so that
the max. coverage is 1/3.
an O2-free environment. Computational and experimental5 results are co-plotted in
Fig. 2, revealing a good agreement despite the relative simplicity of the theoretical
model.
Steady state current in the presence of molecular oxygen
We consider the following reaction steps for ORR:
O2(g) + 4H+ + 4e 4 HOO* + 3H+ + 3e 4 H2O(l) + O* +
2H+ + 2e 4 H2O(l) + HO* + H+ + e 4 2H2O(l)
(5)
An alternative to the first reduction step is dissociation of molecular oxygen (O2(g)
4 2O*). However, it has previously been shown that the dissociative mechanism is
less important for the metals and potentials of interest in this study.16
Based on a detailed theoretical model which includes the effect of the local field in
the double layer,17 we have calculated the potential free energy diagram for the ORR
pathway on Pt(111) at 0.9 V vs. NHE, see Fig. 3. From this graph there are only two
candidates for the rate-limiting step—formation of OOH and removal of OH. We
note that no barriers are included in the model.
Since the adsorption energy of all the intermediates of the ORR can be linearly
related to the adsorption energy of oxygen, DEO,18,19 this energy is a suitable
descriptor. The observation that the binding energy of hydroxyl, OH*, scales
Fig. 3 The free energy diagram for the ORR steps 1–4 at 0.9 V on Pt (111). Notice that there
are only two only steps which are uphill: the formation of OOH (step 1) and the formation of
H2O (step 4). This allows us to simplify the kinetic model to only consider these two reaction
steps.
340 | Faraday Discuss., 2008, 140, 337–346
This journal is ª The Royal Society of Chemistry 2008
linearly with the binding energy of oxygen indicates that the binding energy of
hydroxyl could just as well have been chosen. In the Sabatier analysis we determine
the reaction step related with the most positive change in free energy, and since the
rate of a step is proportional to exp(DG), this step will be the most difficult along
the ORR reaction. The Sabatier volcano for ORR is defined by:
DG(U) ¼ Max[DG1(DEO,U), DG2(DEO,U), DG3(DEO,U), DG4(DEO,U)]
(6)
Where DG1–4 are the reaction free energies for the four elementary reduction steps of
reaction (5) and DG(U) is the most positive change in free energy encountered along
the ORR pathway. Fig. 4a shows the relative activities of Pt and Pt3Ni as a function
of their DFT-calculated DEO according to the Sabatier model. As shown in Fig. 3, it
is either DG1(DEO,U) or DG4(DEO,U) that defines DG(U).
In order to predict ORR current density computationally, we begin with the Tafel
equation
jc ¼ j0exp(ahe/kT)
(7)
where jc is the cathode current density, j0 the exchange current, h is the overpotential
and a is the transfer coefficient. We will only consider a ¼ 1, but the model could be
made for other values of a. We use the definition of h h U0 U in order to rewrite
the Tafel equation:
jc ¼ j0exp[a(eU0 eU)/kT ] ¼ jlimitexp[a(DG0 eU)/kT]
(8)
where jlimit can be written as:
jlimt ¼ j0exp[a(eU0 DG0)/kT ]
(9)
The physical meaning of the term jlimit is the current density achieved if all surface
reactions are exothermic, i.e. the highest possible turn-over-frequency per site in an
electrochemical cell with minimal diffusion limitations. Clearly, this term is dependent on a number of factors, including the electrode structure and its interface
with the liquid electrolyte. In previous work for hydrogen evolution, we found
that jlimit 200 sites1 s1 (or in terms of surface area, 96 mA cm2) for Pt(111) fit
experimental data well, and we will utilize this value in this work as well.20 As the
Fig. 4 (a) The volcano at U ¼ 0, which defines DG0 as a function of DEO, Either DG1 or DG4 is
the limiting step. The vertical dashed line represents zero overpotential. The distance from the
solid volcano to the horizontal dashed line is the overpotential at j ¼ jlimit. (b) Current density
(jc) from experiment from ref. 5 (dashed lines). and from the model (solid lines). The current
density is calculated using eqn (8) with the DG0 for Pt (red lines) and Pt3Ni (black lines)
from Fig. 4a as input.
This journal is ª The Royal Society of Chemistry 2008
Faraday Discuss., 2008, 140, 337–346 | 341
lattice parameters of Pt-alloys change only by a few percent relative to pure Pt, the
number of sites cm2 is fairly constant and jlimit can be effectively considered material-independent, unlike exchange current density. By incorporating jlimit into the
analysis, we can thus rewrite the Tafel equation, eqn (8) in such a manner that all
material dependence is concentrated within only one computable parameter, DG0.
Revisiting eqn (8), DG0 is the largest negative change in free energy along the
ORR pathway at U ¼ 0, and in Fig. 4a, we show the dependence of DG0 on DEO,
simply the Sabatier volcano at U ¼ 0. Using the DG0 from Fig. 4a together with
eqn (8) we get the polarization curves shown in Fig. 4b co-plotted with the experimentally polarization curves. Since eqn (8) does not include the diffusion limitations
specific to the experiments of ref 5, the model and experiments should only be
compared in the upper half of Fig. 4b.
Relating ORR activity to OH coverage from water dissociation
In the case of Pt-alloy ORR electrocatalysts, it has become commonplace to explain
changes in activity by relating measurements of ORR reaction rates in O2-saturated
solutions to the coverage of OH, measured in O2-free solutions. Oftentimes in this
analysis, it has been rationalized that OH is a ‘‘poison’’ that blocks surface sites
from the O2 reactant, leading to the logical deduction that less OH adsorption on
the surface translates directly to improved ORR catalysis.5,8,21,22
Consider the measured ORR current densities for Pt (111) and Pt3Ni (111) in
Fig. 4b and the OH coverage from water dissociation during cyclic voltammetry
in an O2-free environment shown in Fig. 2. Careful inspection of the experimental
data shows that there is no straightforward correlation between ORR current
density and the OH coverage determined by the O2-free CV. In the case of
Pt(111), for example, the OH coverage is fairly constant at 0.4 ML in the potential
region of 0.82 < V < 1.00. Still, within this same region, its ORR current density
increases by several orders of magnitude from nearly zero to approximately 5 mA
cm2, where the reaction encounters significant transport limitations. A similar
phenomenon is observed on the Pt3Ni(111) in the potential region of 0.87 < V <
1.00, where OH coverage decreases moderately from 0.4 ML to 0.2 ML, and yet
ORR current density increases several orders of magnitude from nearly zero to
approximately 6 mA cm2 (its transport-controlled limit).
Another noteworthy point concerns different shifts in potential between the
Pt(111) and Pt3Ni(111) data in the case of OH coverage (Fig. 2) versus that of
ORR current density (Fig. 4b). In terms of OH coverage, the data for the two materials are shifted by 100 mV, while in the case of ORR current density, the half-wave
potential is shifted by only 60 mV. The lack of a one-to-one correspondence
between these two shifts also raises doubts as to whether OH is only playing
a role as a site-blocker. The theoretical model presented above offers a different
interpretation.
Fig. 2 and 4b show that our theoretical model reproduces the key features
observed in the experiments. For instance, the calculations reproduce a smaller
potential shift between the ORR polarization curves (Fig. 4b) than the potential shift
for the OH coverage curves (Fig. 2). Based on the calculations, the physical reason
for this observation is that the nature of the rate determining step changes as we go
from Pt (111) to Pt3Ni(111). While Pt is limited by too strong of a bond to adsorbed
OH, (DG4 > DG1), this is not true of Pt3Ni(111), which instead is limited by the OOH
formation step (DG1 > DG4). This fundamental difference between the two materials
is graphically represented in Fig. 4a, where the two materials are situated on the
opposite sides of the volcano. Had both materials been located on the same side
of the volcano, one would have expected the ORR current density to shift by
approximately the same amount as the coverage 130 mV, which is indicated
with the dotted lines in Fig. 4a.
342 | Faraday Discuss., 2008, 140, 337–346
This journal is ª The Royal Society of Chemistry 2008
This analysis shows that the simple Sabatier model accounts for the key features
observed in experiments. The higher activity of Pt3Ni(111) is a consequence of
a better compromise in bond strength to the OOH and OH intermediates. In other
words Pt3Ni is closer to the top of the volcano.
Kinetic volcano
The simple model yields a good agreement with experiments already without
a detailed description of the kinetics, indicating that kinetics will play a minor
role for the conclusions drawn. In the following we will try to prove this by developing a more detailed kinetic model taking steady-state coverages into account,
the scenario encountered under standard fuel cell operating conditions. We start
by considering the two steps needed in the kinetic model.
O2(g) + * + H+ + e / OOH* Step 1
(10)
OH* + H+ + e 4 H2O(l) Step 4
(11)
From Fig. 3 and Fig. 4a it is seen that, these two steps are the only possible ratelimiting steps. The situation becomes even more simple if we assume Step 1 to be
irreversible. This is likely since the OOH* formed rapidly proceeds to OH*
(Fig. 3), and implies that the coverage of OOH* always is low.
First we write the rate constants:
k1 ¼ k10 Min(1,exp[a1(DG1 eU)/kT )])
(12)
where k10 is the prefactor and DG1 is the change in free energy for Step 1. We introduce an upper limit to all the rate constants, if there is no barrier for a reaction step
the rate constant is ki0. The expression for k4 is analogous to k1, while k4 is given by:
k4 ¼ k40 Min(1,exp[(1 a4)(eU DG4)/kT ])
(13)
First, at steady state:
dQOH/dt ¼ (1 QOH)k1 QOHk4 + (1 QOH)k4 ¼ 0
(14)
This means that the OH-coverage can be written as:
QOH ¼ (k1 + k4)/(k1 + k4 + k4)
(15)
Here QOH denotes the OH-coverage in the presence of molecular oxygen. This is
not necessarily the same as the coverage obtained in eqn (4), which is the coverage in
the absence of molecular oxygen. The steady state current is given by
j ¼ jlimit(1 QOH)k1 ¼ jlimit[QOHk4 (1 QOH)k4] ¼
jlimit[1 (k1 + k4)/(k1 + k4 + k4)]k1 ¼ jlimitk1k4/(k1 + k4 + k4)
(16)
This is the steady state version of eqn (8). Note that the first term is similar to the
expression which many authors use to relate the oxygen reduction current to the
coverage of OH as given by cyclic voltammetry.21 However, it is important to
note that QOH and k1 are not independent, but related through eqn (15)
In the Sabatier analysis, the change in activity is seen as changes in k1 and k4.
Looking at the limit where either k1 or k4 is large, the expression above eqn (16)
approaches:
This journal is ª The Royal Society of Chemistry 2008
Faraday Discuss., 2008, 140, 337–346 | 343
j / ¼ jlimitk4
(17)
k4 [ k1,k4 j / ¼ jlimitk1.
(18)
k1 [ k4,k4
or
This is exactly the result of the Sabatier analysis. This means that in the limit of
just one rate limiting step, the Sabatier volcano is obtained. However, when k1, k4
and k4 are all of the same size, which is the case at the top of the volcano, the
Sabatier analysis is no longer exact. Instead a ‘‘kinetic volcano’’ is obtained. In
the following we will estimate this kinetic volcano.
Due to computational limitations we have no means at present to calculate exact
values for all the parameters that go into this kinetic modeling based on density
functional theory. We will, however, show that for a realistic set of parameters,
our kinetic model will give reasonable results. Concerning pre-factors, we assume
that k40 and k40 are identical. For the sake of simplicity, for this analysis we will
assume k40, k40 and jlimit to be 1 in what follows. From the kinetics of the O2 adsorption step it follows that the pre-factor of Step 1 will contain the partial pressure of
oxygen. Hence, varying k10 essentially corresponds to a variation of the oxygen
pressure. However, it is reasonable to expect k10 to be less than or equal to 1 (i.e.
less than or equal to k40 and k40) since Step 1 includes both adsorption of O2
and proton transfer to O2*, whereas Step 4 only includes the proton transfer between
OH* and H2O*. Assuming that the proton transfer in both cases have the same
pre-factor, k10 will be smaller than k40 and k40 no matter how big the oxygen partial
pressure.
Other unknown parameters are the transfer coefficients a1 and a2. They are
related to potential dependent barriers, which are very difficult to address with
present theoretical methods. Since it is not possible to rigidly include the barriers
we will assume that there are no additional barriers for the proton transfer reaction.
This is included in our model by the upper limits to the rate constants. We have
checked that other models give similar qualitative results, and the conclusions drawn
in this paper are robust if we include small potential-dependent barriers.
The steady state current density for oxygen reduction is calculated based on the
full kinetics, see Fig. 5. The major effect introduced by the kinetics is observed
near the top of the volcano. This can be expected, as in this particular region of
the volcano we have relaxed the assumption of a single rate-limiting step.
Fig. 5 a. The result of a detailed kinetic analysis. (with k10 ¼ k40 ¼ 1). The volcanoes are
obtained from the kinetic analysis, the y axis is kT ln(j), j is obtained from eqn (16) with
k40, k40 ¼ 1 for various values of k10. The most realistic scenario is the case where both k40
and k40 are larger than k10, since k10 includes adsorption of O2 and proton transfer to O2*.
k40 and k40 are the prefactors related to the forward and backwards reactions of the last
proton transfer step OH* + H+ + e 4H2O*, which we consider to be fast. (b) The same as
Fig. 5a with only the volcano for k01 ¼ 0.1 and with the respective Sabatier volcano indicated
by the dashed lines.
344 | Faraday Discuss., 2008, 140, 337–346
This journal is ª The Royal Society of Chemistry 2008
Fig. 6 A log–log plot of the oxygen reduction current density versus the pre-factor k10, or
equivalently the oxygen partial pressure. The reaction order goes from 1 to zero.
Nevertheless, the activity predictions for Pt and Pt3Ni based on the kinetic model
and the reasonable assumption of k10 < k40 and k40 are essentially unchanged as
compared to the results found by the simple Sabatier model, see Fig. 5b. This
confirms the validity of the assumption of a single reaction step being rate determining is valid for both Pt and Pt3Ni, and is the reason why the simple Sabatier
approach works.
Another interesting feature to note in Fig. 5 regards the change in current as
a function of k10. Obviously, for k10 in the interesting interval in Fig. 5, the change
in current is directly proportional to k10. Bearing the connection between k10 and the
partial pressure of oxygen in mind, this indicates that the oxygen reduction current is
first order with respect to the oxygen partial pressure. We investigate this further in
Fig. 6, where the current density is plotted versus k10. A first order dependence is
clearly seen. For higher values of k10 the reaction order goes to zero, first for Pt
and later for Pt3Ni. We don’t expect that it is possible to probe the zero-order-regime
in experiments since it would require k10 to be larger than one.
Conclusion
We have found that the experimental results of the ORR on Pt3Ni were correctly
predicted by the simple Sabatier model. The predictions hold true both for the
current density and the OH coverage as functions of the potential. The Sabatier
model can be used to predict ORR activity, and is therefore a useful tool for
designing new ORR catalysts. Furthermore, we have developed a simple kinetic
model of the ORR. The kinetic model reduces to the Sabatier model in the limits
where only one reaction step is rate-determining. This kinetic model predicts that
the reaction order in oxygen pressure is one in the physically relevant region.
We find that the current density can indeed be written as (1 QOH)k1, as has often
been suggested. However, it is very important to note that QOH and k1 are connected
by eqn (15), which means that this way of writing the current does not offer insights
as to which parameters to change in order to optimize the catalytic activity. The top
of the activity volcano is defined by the Sabatier analysis.
Acknowledgements
CAMD is funded by the Lundbeck foundation. This work was supported by the
Danish Center for Scientific Computing through grant no. HDW-1103-06. TFJ
This journal is ª The Royal Society of Chemistry 2008
Faraday Discuss., 2008, 140, 337–346 | 345
acknowledges the H. C. Ørsted Postdoctoral Fellowship from the Technical University of Denmark.
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