Oscillations and Bistability in the Catalytic Formation of Water on

J. Phys. Chem. C 2009, 113, 17045–17058

17045

Oscillations and Bistability in the Catalytic Formation of Water on Rhodium in High Electric Fields J.-S. McEwen,*,† P. Gaspard,† T. Visart de Bocarme´,‡ and N. Kruse‡ Centre for Nonlinear Phenomena and Complex Systems, Campus Plaine - CP 231, UniVersite´ Libre de Bruxelles, B-1050 Brussels, Belgium, and Chemical Physics of Materials, Campus Plaine - CP 243, UniVersite´ Libre de Bruxelles, B-1050 Brussels, Belgium ReceiVed: March 4, 2009; ReVised Manuscript ReceiVed: August 17, 2009

A comprehensive theory for the adsorption of H2 and O2 on a nanometric rhodium field emitter tip is developed to describe the equilibrium properties, the adsorption-desorption kinetics, as well as its observed nonlinear reaction behavior and oscillatory states. The basis is a kinetic mean-field model for hydrogen, oxygen, and subsurface oxygen which takes into account the anisotropy of the tip’s surface. The resulting model reproduces the correct anisotropy, period and form of the oscillations, as well as the bistability diagram for a varying temperature, hydrogen pressure, and external electric field as observed in a field ion microscope. 1. Introduction Despite the large progress made in recent years to provide a sound understanding of the oscillatory behavior of various catalytic surface reactions, there still are a number of questions apparently unsolved.1 One of these open questions concerns the so-called materials gap (i.e., the difference in morphology and chemical composition of the catalyst in surface science studies and that used in heterogeneous catalysis). Indeed, in surface science, oriented 2D single crystals are mainly used,2-4 whereas multifaceted 3D metal particles, usually deposited at high dispersion on an oxide support, are present in catalysis. To approximate the 3D morphology shape of a single nanometersized metal particle, in the absence of an oxide support, the apex of a field emitter tip can be considered as one of the most suitable candidates. In fact, near-atomic resolution of the tip can be achieved under the operating conditions of a field ion microscope (FIM) where an external electric field (of the order of 10 V/nm) is applied to either image the surface structure at low temperatures or reacting adsorbates at higher temperatures. The FIM can thus be used to investigate the cooperative effects and concerted behavior in a nonlinear chemical reaction since a large number of facets are simultaneously exposed on the surface of the tip.5 In our previous work,6 we investigated the oxidation of nanosized rhodium facets in the presence of a high external electric field using a FIM. Corresponding density functional theory (DFT) calculations were done on Rh(001), Rh(011), and Rh(111). We concluded that the field promotes the oxidation of the tip since a reduction of the activation barrier for oxygen incorporation into the surface was found. Such a behavior was also recently reported to occur on a Al(001) surface.7 The purpose of the present work is to present the details of a model that we have recently reported on8 concerning the catalytic formation of water from H2/O2 on rhodium as observed in a FIM.9,10 In recent years, the FIM has been used as a flow reactor to image, with nanoscale resolution, catalytic surface reactions. * To whom correspondence should be addressed. E-mail: jmcewen@ ulb.ac.be. † Centre for Nonlinear Phenomena and Complex Systems. ‡ Chemical Physics of Materials.

When the tip is exposed to a pressure of hydrogen and oxygen the catalytic formation of water is observed for sufficiently high temperatures (i.e., greater than 300 K). Moreover, a bistable behavior is observed for which a phase diagram was established between 400 and 500 K.9,10 Self-sustained oscillations are found to occur as well for which the shape, morphology, and structural changes were characterized and local chemical information was obtained with pulsed field desorption mass spectroscopy (PFDMS).10 On the theoretical side, a kinetic model for the formation of water on a rhodium field emitter tip has not been attempted before,8 although some work has been performed in the formation of water on platinum field emitter tips.11 However, in that work, a cylindrical symmetry was assumed throughout the model.11 A theoretical understanding of the interaction between the different crystal planes, which may exert a profound influence on the kinetics,5 is therefore still to be elaborated for such systems. Here we shall construct a model to fill this need, by taking into account the different properties of all of the nanofacets of the metal tip that determine the behavior of the oscillations. This will be done by taking into account the large number of studies of hydrogen on rhodium single crystal surfaces. In these studies, a number of experimental and theoretical studies have examined the adsorption12-14 and the desorption kinetics.15-19 In particular, a structure dependence can be deduced when comparing the temperature programmed desorption (TPD) spectra of hydrogen on rhodium single crystals.15-19 A clear structure dependence can also be seen when examining a number of experimental and theoretical studies on the interaction of oxygen with rhodium single crystal surfaces.20,21 In particular, the temperature programmed desorption spectrum has been determined on all low Miller-index rhodium surfaces.22-26 In the various adsorption studies, the sticking coefficient was derived from the coverage build-up curves.24,25,27 For coverages below 0.5 ML, oxygen atoms have been determined to adsorb onto hollow sites on Rh(001) and Rh(111).20 On Rh(011), they have been determined to adsorb onto long bridge and hollow sites up to coverages of 1 ML27,28 for which a series of surface reconstructions have been reported to occur.29 For coverages exceeding these values, it had been recently demonstrated that a surface oxide forms on all three surfaces.21 In these studies,

10.1021/jp901975w CCC: $40.75  2009 American Chemical Society Published on Web 09/10/2009

17046

J. Phys. Chem. C, Vol. 113, No. 39, 2009

scanning tunneling microscopy experiments (STM) along with supporting density functional theory (DFT) calculations have led to the observation of an O(ad)-Rh-O(sub) surface oxide trilayer on Rh(001),30 Rh(011),29 and Rh(111)31 surfaces which differs from the Rh2O3 stoichiometry that one encounters in the bulk oxide. Interestingly, although oxides have reported on higher index surfaces [such as on Rh(711) and Rh(012)],32-34 it is only in recent studies that a O(ad)-Rh-O(sub) surface oxide trilayer was observed on vicinal surfaces35,36 or even on rhodium nanoparticles.37 On rhodium field emitter tips, diffusion of oxygen into the bulk seems to occur likewise in positive38-41 and negative electric fields,42 and recent high resolution transmission electron microcopy (HRTEM) experiments indicate that nanosized patches of RhO2 can form in the near surface region.43 Although an atomistic picture for the formation of these oxides is still missing, it appears that their properties depend on the underlying substrate in a crucial way.6 Thus, it should come as no surprise that an anisotropy occurs with regard to the oxidation of a field emitter tip in which each nanofacet will interact with oxygen in a different manner.41 In this paper, we will show in detail that one can explain the anisotropy, as observed in the field ion microscopy experiments when a rhodium tip is exposed to H2 and O2, by taking into account the different rate constants of each nanofacet extending our short report.8 We will incorporate the various characteristics of each nanofacet with a set of activation energies for the various reactions that will occur on the tip, while taking into account the symmetry of the underlying lattice, as detailed in section 3.3. In section 3.2, we will explain how we consider the influence of these rate constants on the external electric field. The kinetic parameters will be obtained through experimental data and density functional theory calculations as will be demonstrated in section 4 and 5. They will also be obtained by a modeling of the bistability diagram and the kinetic oscillations, as detailed in sections 7.1 and 7.2. In doing so, we will arrive at a comprehensive model of the bistability and the oscillatory phenomena involved in this system. In particular, we will show that our resulting model not only gives the correct anisotropy, period, and form of the oscillations but also gives an excellent agreement with the experimentally determined bistability diagram for a varying temperature and hydrogen pressure. We will also demonstrate the predictability of our model by calculating the desorption rate of various neutral species that cannot be measured with field ion microscopy experiments. The paper ends with a summary of what has been accomplished.

McEwen et al.

Figure 1. (a) Field ion micrographs of a clean (001)-oriented Rh tip imaged by neon at PNe )10-3 Pa, T ) 55 K, F ∼ 35 V nm-1. The radius of curvature is about 10 nm. (b) Ball model of the surface structure encountered in (a).

case of rhodium, tips are prepared by electrochemically etching a thin wire (Rh: 0.127 mm diameter, 99.8% purity) in a molten mixture of NaCl and NaNO3 salts.45 The surface of the tip is usually cleaned by cycles of field evaporation and thermal annealing, followed eventually by sputtering with neon ions for deep cleaning and sharpening. High resolution FIM micrographs of the clean tip specimens were taken in neon gas (purity 99.999%) at 55 K and served as a reference to calculate values of the field strength. All gas pressures mentioned in the paper are measured with an ion gauge and corrected for ion gauge sensitivity. Traces of CO gas usually produced at the hot tungsten filament has no influence on the observed phenomena. Standard FIM images are taken with a high dynamic range CCD camera (512 × 512 pixels, 16 bits per pixel). For in situ FIM reaction studies, a high sensitivity video camera with a time resolution of 20 ms was used. Individual snapshots of the videotape are digitized by means of a frame grabber. Local brightness is then evaluated by pixel analysis of the digitized data. 3. Theory 3.1. Reaction Network. Hydrogen. The inclusion of only pure hydrogen into the gas phase allows several reaction pathways to occur. In the absence of oxygen in the gas phase, the system is completely described by the following reaction scheme: kaH

H2(gas) + 2Ø(ad) y\z 2 H(ad)

2. Experimental Section The FIM used for these studies was described in detail elsewhere.44 The latter provides micrographs of the extremity of a sharp tip revealing the atomic structure of the surface. A resolution of 2-3 Å can be obtained at cryogenic temperature conditions using inert gas, helium or neon, as imaging species. Image resolution and contrast usually diminish with increasing temperature. The atom-resolving capabilities of the FIM technique are most evident in the appearance of layer edges (step atoms) or small (mainly high Miller index) layer planes. This is clearly visible in Figure 1 where we compare a field ion micrograph with a ball model in a nearly 1:1 scale. Dynamic phenomena in adsorbed layers during ongoing reaction processes at elevated temperatures can be imaged with nanoscale lateral resolution. Features like layer edges can also occasionally be resolved under reaction conditions (i.e., at temperatures of 400-600 K). Samples are prepared in the form of a nearly hemispherical shape with a radius of curvature of several nanometers. In the

(1)

kdH

kdiff

H(ad) + Ø(ad) y\z Ø(ad) + H(ad)

(2)

The adsorption of hydrogen on rhodium is dissociative and its desorption associative in eq 1. Here, kdH, kaH, and kdiff are respectively the rate constants for the associative desorption, dissociative adsorption, and diffusion of hydrogen on rhodium. The relative rates of kdH, kaH and kdiff on the various nanofacets of the tip will determine how fast these processes evolve during the reaction. For hydrogen on rhodium under our experimental conditions (temperatures ranging from 400 to 550 K and hydrogen pressures spanning from 5 × 10-4 to 3 × 10-2 Pa), diffusion of hydrogen atoms is dominant. Very low activation barriers on single crystal surfaces46-49 and on field emitter tips50,51 have been found for this process. Moreover, we suppose that hydrogen can only diffuse into neighboring empty sites so that the coadsorption of oxygen hinders its diffusion on a

Formation of Water on Rhodium

J. Phys. Chem. C, Vol. 113, No. 39, 2009 17047

rhodium field emitter tip, which is consistent with the results of Gomer’s pioneering works.51 Our model will also not take into account the precursor state of hydrogen. This can be justified by the fact that at the experimental temperatures and pressures the coverage of hydrogen will be low so that the details of the adsorption mechanism can be neglected. Oxygen. Exposing the surface of a field emitter tip to oxygen on the other hand will lead to a significant oxygen coverage at the experimental temperatures for oxygen pressures as low as 5 × 10-4 Pa. We have therefore taken into account its adsorption (kã) and desorption (k˜d) into and from an intrinsic or an extrinsic precursor state. Once in the precursor state, the oxygen molecule can dissociatively chemisorb onto the surface (ka). On the other hand, chemisorbed oxygen atoms can also desorb associatively into its precursor state (with a corresponding rate kd). Moreover, as a number of experiments mentioned in the introduction strongly indicate the formation of subsurface oxygen, we have allowed oxygen to penetrate the surface. For simplicity, we restrict subsurface diffusion to just one layer below the surface so that an oxide trilayer can be formed (kox) or reduced (kred). However, since atomic oxygen binds strongly to rhodium (with desorption energies ranging from 2.5 to 3.5 eV),27 we neglect its diffusion on the surface at the temperatures of the experiments. The pathways of oxygen interaction with rhodium can be summarized as follows: ˜k a

O2(gas) + surface y\z O2(pre) + surface

(3)

˜kd

ka

O2(pre) + 2Ø(ad) y\z 2 O(ad)

(4)

kd

kox

O(ad) + Ø(sub) y\z Ø(ad) + O(sub)

(5)

kred

where Ø(ad) and Ø(sub) denote a vacant surface adsorption site and subsurface site respectively and O2 (pre) denotes the adsorption of an oxygen molecule in its precursor state. Water Formation. Finally, exposing the rhodium crystal to both oxygen and hydrogen allows the catalytic formation of water on rhodium. For this reaction, experiments on platinum52 and on rhodium53-55 have demonstrated that the addition of the first H atom to chemisorbed oxygen is the slow step whereas the addition of a second hydrogen atom leading to H2O is fast, i.e. the lifetime of the OH species is relatively short. Indeed, a simple Arrhenius estimate of the mean lifetime of an OH species on a rhodium surface gives τ ∼ 10-10 s at 550 K since the activation barrier for the addition of a second hydrogen atom is about 0.3 eV.56 On the other hand, the energy barrier for the first step in the formation of OH on Rh(111) is significantly larger and has been experimentally determined to be around 1.0 eV.56 This also agrees with density functional calculations on the formation of OH on Rh(011) and Rh(111) in which the barrier has been determined to range from 0.8 to 1.0 eV.57-59 This allows us to simplify our reaction mechanism of water formation when modeling the FIM experiments. Accordingly, we can neglect the OH intermediate and consider the following reaction: kr

2H(ad) + O(ad) 98 3Ø(ad) + H2O(gas)

(6)

with the turnover frequency of H2O formation being simply proportional to both the oxygen and the hydrogen coverages. Since the temperature of the tip (400-550 K) is well above the desorption temperature of water [around 300 K for water on Rh(001)60], the water molecules that are formed on the surface are assumed to desorb rapidly. Moreover, as can be noticed from eq 6, we suppose it to be irreversible, since once water desorbs it has no chance to readsorb under the conditions of our experiments (vanishing water pressure, small sample size). Ionization. Exposing the surface of a field emitter tip between 400 and 550 K to hydrogen and oxygen in fields of about 10 V/nm will also yield several ionic species, such as H2O+, H3O+, and O2+ in varying proportions depending on the field strength.10 For example, the production of ionized water can proceed via several reaction pathways11

H2O(ad) f Ø(ad) + H2O+(gas) + e-(Me)

(7)

H2O(ad) + H(ad) f 2Ø(ad) + H3O+(gas) + e-(Me) (fast) (8) 2H2O(ad) f OH(ad) + Ø(ad) + H3O+(gas) + e-(Me) (slow)

(9)

where the second reaction has been found to be faster than the third one on platinum.11 The relative yield of each ionic species for this system has been measured in a number of PFDMS experiments on rhodium in which the yield of ionized water species was found to be higher than that of the oxygen ions.10 As the desorption of the neutral gas molecules, the ionization is also an activated process. However, the experimental conditions of field strengths and temperatures are such that the field ionization probability will be low with respect to the number of oxygen, subsurface oxygen, and hydrogen atoms adsorbed on the tip. Thus, the ionization should only perturb weakly the populations of neutral species and the ionic fraction of species only play the role of imaging gases. We will therefore assume that the ionization is passively driven by the neutral species. 3.2. Electric Field. The geometry of the rhodium tip is supposed to approach a paraboloid within our model

z)

R x2 + y2 2 2R

(10)

of radius of curvature R at the apex of Cartesian coordinates x ) y ) 0 and z ) R/2. This approximation can be justified by the fact that the tip keeps its shape even when it is exposed to hydrogen41 and oxygen.9 Indeed, although a rhodium field emitter tip is known to reconstruct when it is exposed to such gases,40 the shape transformations are limited to the very apex of the tip, conserving largely the overall tip-shank geometry. The electric field distribution can be expressed by61,62

F)

F0

(11)

r2 1+ 2 R

where r ) (x2 + y2)1/2 is the radial distance with respect to the symmetry axis of the paraboloid and F0 is the magnitude of the electric field at the apex of the tip. This is schematically depicted in Figure 2.

17048


McEwen et al. into account in such an expansion of the kinetic parameters into kubic harmonics, eventually by adding higher-order X(n) coefficients.

Figure 2. Schematic representation of the electric field near the apex of a positively charged metallic needle in the form of a paraboloid. Since the field emitter tip is metallic, the electric field F is perpendicular to the surface.

TABLE 1: Cartesian Components (ξ, η, ζ) of the Unit Normal Vector n and Its Spherical Angles for the Principal Surface Orientations of Miller Indices (hkl) for an Underlying fcc Crystal Such As Rhodium: n ) (ξ, η, ζ) ) (sin θ cos O, sin θ sin O, cos θ)

4. Hydrogen on Rhodium 4.1. Adsorption-Desorption Kinetics. This process is described by the network eq 1 where the adsorption is dissociative and the desorption associative. Moreover, the hydrogen coverage will be very small at our experimental temperatures and pressures and even smaller when oxygen is coadsorbed. We will therefore simply assume that the sticking of hydrogen on rhodium will follow a Langmuirian dependence on coverage. The resulting kinetic equation for the hydrogen coverage, θH, in the absence of oxygen is then given by

dθH ) 2kaHPH2(1 - θH)2 - 2kdHθH2 dt

(14)

(hkl)

ξ

η

ζ

θ

φ

The hydrogen pressure in an electric field F is related to the field-free pressure reading given by

(001) (011) (111)

0 0 1/3

0 1/2 1/3

1 1/2 1/3

0 π/4 0.95531

π/2 π/4

PH2(F) ) PH2(0)eβRH2F /2

We notice that the different properties depend on the magnitude F of the local electric field. This dependence can be expanded in powers of the electric field as

1 X(F) ) X(0) - dXF - RXF2 + · · · 2

2

(15)

where RH2 = 0.000568 eV(V/nm)-2 is the effective polarizability of H2 when subjected to an external electric field66,67 and β ) (kBT)-1. On the other hand, the adsorption rate constant is given by

(12)

kaH )

SH0as

√2πmH kBT

(16)

2

where dX can be regarded as an effective dipole moment and RX as an effective polarizability. 3.3. Anisotropy. The surface of the tip is composed of different facets, each having their own rate constants. This is because the kinetic parameters such as activation energies or sticking coefficients depend on the orientation of the facet. The latter is specified by the unit vector n ) (ξ,η,ζ) normal to the mean shape of the tip (taken to be a paraboloid). The Cartesian components of this vector are given in Table 1 for the principal orientations. The flanks at remote distance correspond to the orientations (hk0) and the angle θ ) π/2. The above description of the tip allows us to expand the kinetic parameters into the kubic harmonics.63-65 The values of each kinetic parameter on the three main orientations of Table 1 then univoquely fix the first three coefficients of this expansion as

X ) X(0) + X(4)(ξ4 + η4 + ζ4) + X(6)(ξ2η2ζ2) + · · · (13) Moreover, since the surface adsorption areas vary from one nanofacet to another, we have redefined the coverages so that the same definition of coverage applies to all the facets, as detailed in the Appendix. We remark that such an approach is justified because the atomic scale (Rh lattice constant of 0.38 nm) is significantly smaller than the scale of the nanopatterns and the tip’s radius of curvature (∼10 nm). Furthermore, we notice that the surfaces with more densily covered step edges or kinks can be taken

with mH2 ) 2 × 1.008 amu and where SH0 is the initial sticking coefficient of hydrogen at zero coverage. As explained in Appendix A, the reference unit area is taken to be the same on all the orientations of the tip, with as ) 10 Å2. Turning to the desorption rate, one can show that it is given by68

kdH )

H2 SH0askBTZint

hλH2 2(qH3 )2

0 -β(EdH-FddH) e-β(EdH-FddH) ) kdH e

(17) where q3H is the partition function for the center-of-mass vibrations of an adsorbed hydrogen atom. We have also H2 for, respectively, the introduced λH2 ) h/(2πmH2kBT)1/2 and Zint thermal wavelength of H2 and the partition function of the internal degrees of freedom of an H2 molecule in the gas phase. As for EdH, it is the associative desorption energy of hydrogen, which can be expressed in terms of the binding energy of H on rhodium (V0,H) as well as the electronic dissociation energy of an H2 molecule (De,H) such that EdH ) 2V0,H - De,H.19 Moreover, 0 is the pre-exponential factor for desorption and ddH is an kdH effective dipole moment which reflects the dependence of the desorption energy on the external electric field. Here we assume that the pre-exponential factor for desorption 0 ) is a constant which is fixed by an Arrhenius analysis of (kdH the temperature programmed desorption (TPD) curves at low coverage on Rh(111).19 In addition, we will assume that the 0 , and ddH will be independent of the surface values of SH0, kdH



TABLE 2: Various Constants Used in the Kinetic Mean Field Model for Hydrogen on a Rhodium Field Emitter Tip SH0 0 kdH EdH(001) EdH(011) EdH(111) ddH

0.3 3 × 1010 s-1 0.75 eV 0.64 eV 0.70 eV 0.005 eV nm V-1

orientation, the values of which are given in Table 2. The values of these constants have been adjusted so as to model the experimental field ion microscopy experiments while remaining consistent with various other experiments for hydrogen adsorption on single crystal surfaces. In particular, the value of the experimental sticking coefficient for dissociative adsorption at zero coverage on Rh(113) is actually quite close to the one deduced here since it has been measured with a value of 0.3 roughly up to a coverage of 0.5 ML, above which it decreases rapidly.19 As for desorption energies, we incorporated an anisotropy in the model (as can be seen in Table 2) so that they are consistent with TPD experiments for hydrogen on Rh(001), Rh(011), and Rh(111). The details in the modeling of the TPD experiments will be presented elsewhere.69 To get the desorption energies on the other facets of the tip, we then expanded these desorption energies with cubic harmonics, using eq 13. The validity of such an approach can be verified. Here we compare the desorption energy of another facet of the tip using such an approach with the corresponding value as obtained when examining the TPD spectra. On the Rh(113) surface, the resulting desorption energies are 0.70 eV in both cases. Thus, such an expansion seems suitable here for the adsorption of hydrogen on single crystal rhodium surfaces. Obviously, we do not expect such a good agreement for all surface orientations, but do stress that only low Miller index surfaces were considered in eq 13 and other surface orientations could be added to such an expansion if a more accurate modeling of the potential energy surface of the field emitter tip were required. [We further remark here that the three surfaces chosen in our expansion are the ones with the most symmetry planes associated with them. Thus, the next planes in our expansion could be potentially chosen with symmetry considerations in mind, such as the {113} and the {012} surfaces.70] As for the value of ddH, it was chosen so as to be in line with a number of density functional theory calculations in which an applied positive external field resulted in a decrease of the hydrogen’s binding energy.71 4.2. Diffusion Kinetics. As mentioned above, the rapidity of hydrogen diffusion on rhodium allows us to assume that a quasi-equilibrium situation is established so that the diffusive current remains close to zero on the time scales of the kinetic processes other than diffusion. The vanishing of the current, JH ) 0, can be shown to be equivalent to requiring69 that

θH(r, t) )

1 - θO(r, t)

(18)

1 + eβ[UH(r)-µH(t)]

Equation 18 can also be argued to hold true since, in the absence of oxygen, θH(r,t) will eventually attain its equilibrium coverage value if one sets:

1 1 UH(r) ) - EdH(r) - ddH(r)F(r) + RH2F(r)2 + cst 2 2 (19)

[

]

where r denotes the position on the tip’s surface (see eq 10). Moreover, when the system attains its equilibrium state, the quantity µH(t) is no longer time dependent and is equal to the equilibrium chemical potential of hydrogen. In Figure 3, we show the equilibrium distribution of hydrogen at 500 K in the absence of oxygen. As can be seen in the Figure, the hydrogen coverage is smallest on the {011} planes since the desorption energy of hydrogen (as given in Table 2) is at its lowest on these facets. In the case of hydrogen and oxygen coadsorption, the system will be away from equilibrium as long as the reaction of water proceeds. Differentiating eq 18 with respect to time and summing over all of the facets of the tip one can show that the time dependence of the effective chemical potential, µH(t), is governed by

∑ w[(1 - θO)∂tθH + θH∂tθO]

dµH(t) facets ) kBT dt

∑ wθH(1 - θH - θO)

(20)

facets

where the weight w takes into account the various equivalent facets on the tip. For example, on a (001)-oriented tip, w ) 4 since there are four {011} planes. [When referencing a particular plane of the field emitter tip, we will henceforth denote it with curly brackets, since each facet has other equivalent ones on the tip. However, when referencing the (001) plane we will use ordinary brackets since there is only one (001) facet on a (001)oriented tip.] 5. Oxygen on Rhodium 5.1. Adsorption-Desorption Kinetics. We will assume that O2 adsorbs on rhodium in a molecular precursor state which can then dissociate and chemisorb onto the surface. Moreover, if the activation barriers separating the precursor O2 (pre) from the gas molecule and the chemisorbed state 2O(ad) are low, O2(pre) will rapidly desorb or dissociate into 2O(ad) and thus rapidly equilibrate with O(ad). In such a case, the precursor is in quasi-equilibrium and dθO2(pre)/dt ) 0 which allows us to simplify our kinetic equations. On the other hand, various kinetic models have been constructed to describe an oxide state.2,72-76 In our case, the oxide state consists of an O(ad)-Rh-O(sub) surface trilayer oxide. Various theoretical and experimental works have shown6,77 that subsurface oxygen by itself is unstable and that oxygen only goes subsurface when the outer layer is completely saturated. In itself, this process is not identical to the presence of subsurface oxygen as for other metals such as Pd(011).2,74,76 Nevertheless, a surface trilayer oxide is observed on Pd(011)21 as well as on Rh(001), Rh(011) and Rh(111) which can be described in both cases with the presence of oxygen both on and below the surface. The trilayer on rhodium should thus correspond to a state in which both the oxygen coverage, θO, and subsurface site occupation, θs, are equal to 1. The formation of the trilayer or the oxidation of the metal surface in our model can therefore be envisaged as the passage of oxygen below the surface while the reverse process is its reduction. Thus, with these approximations, the following kinetic equations result for the oxygen and the subsurface oxygen site occupations, which completely determine the evolution of the system in the absence of hydrogen

17050


dθO 2 (kãKPO2θØ2 - kdθO2) ) dt 1 + KθØ2 -koxθO(1 - θs) + kredθsθØ

McEwen et al.

(21)

TABLE 3: Various Constants Used in the Kinetic Mean-Field Model for Oxygen on a Rhodium Field Emitter Tipa (hkl)

dθs ) koxθO(1 - θs) - kredθsθØ dt

(22)

where θØ ) 1 - θO and K ) ka/k˜d. The pressure dependence of oxygen on the electric field is similar to that of hydrogen βRO2F2/2

PO2(F) ) PO2(0)e

(23)

where RO2 = 0.0011 eV(V/nm)-2 is the effective polarizability of O2 when subjected to an external electric field.66,67 On the other hand, the adsorption rate of the precursor is given by

kã )

SO0as

(24)

√2πmO kBT

SO0

AKs (eV)

(001) 0.95 0.070 (011) 0.95 0.075 (111) 0.6 0.082

Eox dox (eV Ered dred (eV Ed (eV) nm V-1) (eV) nm V-1) (eV)

Ad (eV)

1.52 1.63 1.68

-0.6 -0.8 -0.5 -0.7 -0.4 -0.5

0.0350 0.0250 0.0200

1.47 1.57 1.59

0.0200 0.0175 0.0150

3.50 3.20 2.85

Bd (eV)

a The desorption energies of oxygen on rhodium together with the constants Ad and Bd give the dependence on the oxygen coverage due to the lateral interactions, as determined from the experimental TPD spectra.

TABLE 4: Various Constants Used in the Kinetic Mean-Field Model for Oxygen on All of the Facets of a Rhodium Field Emitter Tip K0 EK AKO k0ox 0 kred s Ared k0d

0.2525 -0.178 eV 0.158 eV 5 × 1011 s-1 1.85 × 1013 s-1 0.3 eV 6 × 1013 s-1

2

with mO2 ) 2 × 16.0 amu and reference area as ) 10 Å2, taken independently of the orientation. Here, SO0 is the initial sticking coefficient of oxygen. In our modeling, a structure-dependence for S0O was assumed (the values at low Miller index planes being given in Table 3). As for K, an Arrhenius dependence on temperature was assumed in which the incorporation of an activation barrier (EK) and a mean field oxygen coverage dependence (AKO) was found necessary in the model. Indeed, these parameters are needed to explain the dependence of the total sticking coefficient on temperature23,25 as well as the TPD spectra of oxygen on Rh(001), Rh(011) and Rh(111).22-24,26 The resulting dependence of K on θO, θs, and temperature is then given by

K ) K0e-β(EK+AKθO+AKθs) O

s

(25)

where the values of K0, EK, and AKO are given in Table 4. As can be seen from eq 25, there is no dependence on the electric field which is here assumed for a matter of simplification and for lack of data. On the other hand, AKs in eq 25 governs a

feedback dependence of subsurface oxygen on oxygen adsorption. This feedback is known to be essential to explain the oscillatory behavior in this context.2,74,76 A structure-dependence for AKs was found to be necessary as well and its values are given in Table 3. This structure-dependence was adjusted in order to reproduce the experimentally observed oscillations, along with their corresponding nanoscale patterns that will be presented in section 7.2. 5.2. Subsurface Oxygen and Desorption Kinetics. We now come back to the determination of the constants that govern the diffusion of oxygen atoms into the bulk. These reactions are crucial for both the bistability of the O2-H2 reaction and the oscillations so that their reaction rates were chosen in consequence. We have here also used DFT6 to estimate the oxidation and reduction barriers as well as their dependence on the electric field. In particular, for the hopping barrier of an oxygen atom to go from the inner to the outer layer of the surface trilayer oxide on an underlying Rh(001), Rh(011) and a Rh(111) substrate [denoted here as Eox(hkl)], an effective dipolar dependence on the external electric field was incorporated into our model, dox(hkl), such that

Figure 3. Equilibrium state of the tip at temperature T ) 500 K, hydrogen pressure PH2 ) 10-3 Pa, and electric field F ) 10 V/nm: (a) as a plot of the partial coverage θH in the disk of radial coordinate θ ) arctg(r/R) and polar angle φ. The apex is at θ ) 0 and the flanks at θ ) π/2, corresponding to r ) ∞; (b) as a density plot in the same disk.



(26)

Rh(011):

µO ) -1.25 eV

(32)

0 where kox is its pre-exponential factor. A similar dependence was also found for the inverse process, with an effective dipole of dred(hkl) and a barrier of Ered(hkl) in the absence of an external field. In our recent DFT calculations, it was found that both barriers decreased on all three surfaces with the application of an external electric field.6 Moreover, it was found that this decrease depends on the underlying substrate, with the largest decrease being found on Rh(001). We have incorporated this information in our kinetic model in which both the barriers and the effective dipoles were made anisotropic. The values of these constants are given in Tables 3 and 4. However, for the reduction of the oxide, a mean-field dependence on subsurface oxygen is required in order to reproduce the equilibrium phase transition between the metallic and oxidized surfaces

Rh(111):

µO ) -1.16 eV

(33)

kox ) k0oxe-β(Eox-doxF)

kred ) k0rede-β(Ered-dredF+Aredθs) s

(27)

0 s (the pre-exponential factor) and Ared govern its where kred dependence on subsurface oxygen. More precisely, the equilibrium state of the model is given by the detailed balance conditions which imply:

1

θO,eq ) 1+

θs,eq )

(28) kd

kãKPO2

1 kred 1+ kox

(29) kd kãKPO2

However, these equations are implicit since the rate constants depend on the partial coverages due to the lateral interactions. They can nevertheless be solved numerically. The resulting subsurface oxygen coverage dependence on pressure is a sigmoidal curve, featuring an equilibrium phase s given in Table 4 is transition. The value of the constant Ared directly responsible of this phase coexistence between a metallic surface at low pressure and an oxide surface covered with an O(ad)-Rh-O(sub) trilayer at high pressure. Indeed, the surface remains metallic as long as θs,eq = 0 and becomes a surface oxide if θO,eq = θs,eq = 1. To determine the transition to the surface oxide we may suppose that, approximately, it happens for θs,eq = 1/2, i.e., at the pressure

PO2 =

( )

kred 2 kd kox kãK

µO ) -1.32 eV

PO2 )

O2 kBTZint

λO23

e2βµO

(34)

which is a further verification. Figure 4 shows that the model agrees with the DFT calculations.21,78 Turning now to the desorption energy at low coverage in the absence of an external electric field, we can write Ed ) 2V0 De where V0 denotes the binding energy of an oxygen atom with the rhodium surface and De its electronic dissociation energy,19 as it was similarly done in the hydrogen case (see above). We have found that Ed decreases when a positive external electric field, F, is applied on a number of single crystal surfaces using DFT. Varying the oxygen coverage caused the effective dipole, dd, to range from -0.005 eV/(V/nm) to -0.035 eV/(V/nm).71 Consequently, the desorption energy decreased with positive values of the external field which is in line with what was found previously for gold tip surfaces79 and for a variety of rhodium tip surface oxides.6 However, the desorption rate is so low in our experimental temperature range (400-550 K) that a small decrease in its binding energy at the experimental field values will not have a significant effect. We will therefore ignore its dependence on the external electric field in our model. It has also been found through a modeling of various TPD experiments that the desorption energy decreases with increasing oxygen coverages due to the repulsive interactions between the adsorbed oxygen atoms. We have approximated this decrease in a mean-field manner in which the desorption rate now depends on the oxygen coverage

kd ) k0de-β(Ed-dOF+AdθO+BdθO ) 2

(35)

(30)

This is an approximation to the condition of equality of the chemical potentials of atomic oxygen in both the adsorbate and the surface oxide. On the other hand, DFT calculations by Mittendorfer et al.21 have shown that this transition occurs at the following chemical potentials for the different surface orientations:

Rh(001):

while the bulk oxide occurs at µO ) -1.25 eV. Finally, if one denotes λO2 ) h/(2πmO2kBT)1/2 as the thermal O2 as the partition function for the wavelength of oxygen and Zint internal degrees of freedom of an O2 molecule in the gas phase then the oxygen pressure, as given in eq 30, should coincide with

(31)

Figure 4. Equilibrium phase diagram of oxygen on rhodium in the model and according to the DFT calculations showing the line of transition between the metallic phase and the phase with the surface oxide.

17052


McEwen et al.

This gives values of Ad and Bd as given in Table 3 when modeling the TPD spectra [i.e., using eqs 21 and 22 in the absence of an electric field and with PO2 ) 0].22-26 It also fixes the value of the prefactor, kd0, which is given in Table 4. The resulting TPD spectra was found to range from 600 to 1400 K depending on the underlying substrate. Moreover, it has also been found that the coverage dependence of the resulting desorption energies as obtained with DFT calculations are also consistent with the aforementioned values69 of the coefficients Ad and Bd, which serves as a further check. However, despite this decrease of the desorption energy with increasing oxygen coverage, it still does not increase the desorption rate significantly in the temperature range of interest (400-550 K). Accordingly, the desorption of oxygen turns out to be negligible in the bistability regime and under oscillatory conditions. 6. Complete Model As mentioned in section 3.1, we will assume that the rate of water production is governed by the reaction given in eq 6, so that the complete model is therefore given by the following equations:

∂θH ) 2kaHPH2θØ2 - 2kdHθH2 - 2krθHθO ∂t ∂θO ∂t

)

(36)

2 (kãKPO2θØ2 - kdθO2) - koxθO(1 - θs) + 1 + KθØ2 kredθsθØ - krθHθO (37)

∂θs ) koxθO(1 - θs) - kredθsθØ ∂t

(38)

∑ w[(1 - θS)∂tθH + θH∂tθO]

dµH facets ) kBT dt

∑ wθH(1 - θH - θO)

(39)

facets

where θØ ) 1 - θH - θO and θH is given by eq 18. The reaction constant, kr, that appears above is given by

kr ) k0r e-β(Er-drF+Ar θH+Ar θO) H

O

(40)

where k0r is the pre-exponential factor in the formation of water, Er is its energy barrier, and dr is its effective dipole which will govern the dependence of the reaction constant on an externally applied electric field. Equation 40 also shows that the reaction constant will depend in a mean-field manner on the hydrogen and oxygen coverages through ArH and ArO, respectively. This was found necessary in order to reproduce the observed bistability and oscillations, as will be detailed in the next sections. One can also argue that these mean field parameters take into account various other reaction pathways that lead to the observed nonlinear behavior on rhodium. The resulting parameter values that we deduced in our modeling of the bistability and the oscillations are given in Table 5. The differential equations given above for the coverages hold for each facet of the tip, whereas the differential equation for the effective (nonequilibrium) chemical potential of hydrogen globally couples all of the facets. This is due to the rapidity of hydrogen diffusion and the corresponding immobility of oxygen

TABLE 5: Various Constants Used in the Kinetic Model of the Formation of Water on a Rhodium Tip kr0 Er (001) Er (011) Er (111) ArH ArO dr

7 × 1012 s-1 0.79 eV 0.79 eV 0.75 eV -0.27 eV -0.145 eV -0.0075 eV nm V-1

on rhodium, as explained in section 3.1. The facets are defined as the points of a grid. We can take advantage of the symmetry of the [001]-oriented tip under the group C4V, expressing the 4-fold symmetry including the reflections by the orthogonal axes as well as the median lines. The grid can thus be defined on an octant between the horizontal axis and the first median line. We have taken the grid formed by the 21 points which is shown in Figure 5 along with the corresponding coordinates of φ and θ. 7. Results 7.1. Bistability. As mentioned in the Introduction, a bistable behavior was previously observed on rhodium field emitted tips when it was exposed to a pressure of oxygen and hydrogen.9,10 More precisely, the tip was experimentally imaged at about 10 V/nm, where the PH2/PO2 ratio was varied from a hydrogenrich (9:1, PO2 ) 1.5 × 10-3) to a hydrogen-poor environment. In these experiments, it was concluded that for small PH2/PO2 the surface is predominantly covered by oxygen, whereas for large PH2/PO2 it is mainly covered by hydrogen. In an intermediate region the system becomes bistable, with hysteresis extending over a certain range of PH2/PO2 which narrows considerably toward higher temperatures. This behavior is similar to what was observed in other systems, such as those examined by Gorodetskii et al.5 involving the catalytic formation of water on platinum. However, the role of the external electric field for the observed bistability has remained unknown since the bistability phase diagram was always only measured at one external electric field value. In this work, we have explored the bistability diagram’s dependence on the external electric field. Field ion imaging of surface reactions was performed as follows. First, the tip temperature was raised to values of interest (∼400-550 K). Then oxygen or hydrogen was introduced into the microscope chamber at pressures ranging from 10-4 to 10-2 Pa. After pressure equilibration, an electric field of 8-16 V/nm was

Figure 5. Coordinates of the 21 points of the grid used to model a FIM tip.

Formation of Water on Rhodium applied in order to provoke field ionization and image formation. Under well repeatable H2/O2 ratios, structural transformations of the imaged surface were clearly visible. In a hydrogen rich H2/O2 mixture, the main features of a metallic surface such as those observed under low-temperature conditions were still visible; that is, (001) and (111) planes were discernible along with their respective symmetries. When decreasing the H2 partial pressure while keeping the O2 pressure constant, local transformations were observed where a “granular” structure was observed without any indication of the symmetry of the underlying bulk material. Local chemical analysis of these regions by means of short field pulses has previously shown the presence of RhxOy in the mass spectra indicating the presence of a surface oxide,9 whereas the surface composition in the presence of a H2 rich gas mixture was metallic. Depending on temperature (400 K < T < 500 K), local and erratic fluctuations of the surface composition were observed within restricted values of H2/O2 ratios. This defines a region of bistability of the system. These results were treated theoretically with our kinetic model. Using AUTO, a published program package which contains continuation algorithms for constructing bifurcation diagrams,80 the bistability phase diagram, at a constant experimental value of PO2 ) 5 × 10-4 Pa, was determined in the PH2-T plane using our model eqs 36-39. We first examine the dependence in the formation of water on the exerted hydrogen pressure and compare it to what we see in experiment. In order to do this, let us further make the assumption that the catalytic activity of the tip is correlated with its brightness, as was deduced in a previous experimental study.10 In this case very good agreement between experiment and theory is obtained. More precisely, at high hydrogen pressures (in which the ratio of the hydrogen to oxygen pressures, H2/O2, is 20.0), the amount of water produced is significant and agrees with experiment since a very bright tip is observed, as can be seen when comparing panels a and g in Figure 6. Correspondingly, the subsurface oxygen coverage is very small at this pressure, as depicted in Figure 6d. However, experimentally, one can notice that the very apex part of the tip is relatively dark. Previous FIM experimental studies with a platinum field emitter tip have claimed that this darkness is associated with a significant hydrogen coverage.5 This correlation agrees well with our model, since the hydrogen coverage is at its largest at the (001) facet under these conditions. The coverage distribution is in fact similar to what we obtained in Figure 3. This is most likely due to the fact that hydrogen binds more strongly to rhodium at the (001) facet than on the other facets of the tip, as can be seen from Table 2. In the bistable regime, the tip is divided into four quadrants, with the high water production areas being concentrated around the {111} facets. This is again in very good agreement with experiment as seen when examining panels b and h of Figure 6. As for the subsurface oxygen coverage, it forms a nanometric cross-like structure [Figure 6e] which correlates well with the granular cross-like structure as seen in the experiment. At low hydrogen pressures [panels c, f, and i in Figure 6 in which the ratio of hydrogen to oxygen pressures is 0.2], one can again see a nice agreement with experiment since the amount of water produced is small and, in our model, is mostly concentrated at the shank of the tip, i.e. where the hydrogen coverage is at its largest. On the other hand, subsurface oxygen completely covers the tip’s visible surface [Figure 6f] which should correspond to the granular structure as seen in experiment [Figure 6c].


Figure 6. Series of micrographs as a part of a video sequence showing the structural transformation of the surface at 450 K, 12.3 V/nm, and PO2 ) 5.0 × 10-4 Pa starting from a hydrogen-rich gas mixture (pressure ratio of H2/O2 ) 20.0) (a), to the bistability regime (H2/O2 ) 10.0) (b), and finally to a hydrogen poor mixture (H2/O2 ) 0.2) (c). The images for the subsurface oxygen site occupation within the theoretical model are shown on a logarithmic scale in (d) (H2/O2 ) 20.0), (e) (H2/O2 ) 2.0), and (f) (H2/O2 ) 0.2). The bright areas in the theoretical model correspond to a high site occupation value while the dark areas have a vanishing site occupation. The corresponding turnover frequency in the theoretical model is shown on a logarithmic scale in (g) through (i) where the bright areas correspond to a turnover frequency of about 16 s-1 while the dark areas have a vanishing turnover frequency.

Figure 7. Bistability diagram at two values of the external field and for PO2 ) 5 × 10-4 Pa. The circles and stars indicate the experimental pressures (with corresponding error bars) for which the structural transformation occurred when decreasing and increasing the hydrogen pressure, respectively. The area in between marks the region of bistability. The full lines mark the theoretical delimitation of the bistability region with the kinetic model as presented in the text.

Finally we depict the bistability diagram in Figure 7, for which our theoretical model shows a remarkable agreement with experiment. Here, we have measured and modeled the bistability diagram at two external field values, 11 V/nm and 12.3 V/nm. For PH2 less than 5 × 10-3 Pa and at 400 K, we find that the tip is covered with oxygen. We shall define this region as the oxygen-covered surface phase. On the other hand, the water formation rate will increase with temperature. Eventually, this

17054


will lead to the reduction of the oxide so that hydrogen can adsorb onto the surface of the tip. This situation occurs around 500 K for relatively low hydrogen pressures (at values of PH2 ) 3 × 10-3 Pa and higher). We will define this region as the Oad/Had-covered surface phase. In between these two regions is a bistable regime which narrows considerably toward higher temperatures. As one can see in Figure 7, a larger bistability region is obtained when the applied external electric field is increased. The main features of the bistability diagram can be understood by considering the effects of (i) the initial sticking coefficient of H2 and O2 on rhodium, (ii) the energy barrier Er in the formation of water, (iii) the desorption energy of hydrogen, and (iv) the hopping barriers involved in the formation and the reduction of the oxide. The influence of the sticking coefficient on the bistability diagram can be simply understood by examining the adsorption terms for hydrogen and oxygen as given in eq 14. Indeed, since the adsorption rate of hydrogen is linearly proportional to both its pressure and its initial sticking coefficient, a change of its sticking coefficient value will effectively shift the bistability diagram to lower or higher hydrogen pressure values. A similar argument also applies to the oxygen adsorption rate. To understand the fundamental dependence of the bistability diagram on the energy barrier in the formation of water, Er, one should first remark that if its value was too high with respect to the temperature of the tip, oxygen would always cover the tip in our experimental pressure and temperature range. Indeed, this is due to the high desorption rate of hydrogen between 400 and 500 K with respect to desorption rates of oxygen as deduced from various TPD spectra.69 In addition to Er, the transition from the oxygen-covered region to the Oad/Had covered surface region will also depend on its increase with the external field and the significant mean-field repulsion between hydrogen atoms in the production rate of water (ArH). As for the effective dipolar coefficient, dr, its value controls the increase of the bistability region with the external electric field. On the other hand, a meanfield repulsion between hydrogen atoms in the catalytic formation of water is necessary if one wants to explain the temperature dependence of the bistability region in a quantitative fashion: in its absence one finds that the dependence of the bistability region in the PH2-T plane is too steep. This mean-field repulsion can also be argued when considering the temperature programmed desorption spectrum from mixed oxygen and hydrogen ad-layers.60 Indeed, these desorption spectra show a shift in the peaks toward lower temperatures with increasing amounts of hydrogen at low oxygen coverages: an indication of an repulsion between the hydrogen atoms, as demonstrated in the modeling of a number of adsorbate systems.81,82 Although in principle the desorption energies should be fixed by our modeling of the TPD spectra of hydrogen on various single crystal surfaces,69 we have found a small adjustment of their values necessary so as to explain the bistability phase diagram. In fact, the values of the desorption energies as deduced by Kreuzer et al.19 are slightly too low in order to explain the experimentally determined bistability region. This does not come as a surprise since the properties of the various nanofacets of the tip are expected to be somewhat modified with respect to those of extended single crystal surfaces. Finally, the dependence of the bistability diagram on the hopping barriers in the formation and the reduction of the oxide is more subtile. This dependence arises through the feedback term in the adsorption of oxygen. We have found that the width of the bistability region depends on the energy difference Eox

McEwen et al.

Figure 8. Series of FIM micrographs covering the complete oscillatory cycle. The temperature, electric field and partial pressures of oxygen and hydrogen are fixed to T ) 550 K, F0) 12 V/nm, and PO2)PH2 ) 2 × 10-3 Pa.

- Ered at the various facets of the tip. A smaller difference in the energy barriers resulting in a larger bistability region. 7.2. Oscillations. We now turn to the determination of the oscillations within our model. At temperatures near 500 K and higher, oscillations are observed which can be described as follows. Starting from a metallic surface, a Rh oxide is first formed at the topmost (001) layer of the tip. The oxide layer anisotropically expands toward the peripheral regions of the tip with a preference for {011} zone lines. It is noteworthy to realize that the density of kink sites in facets lying along these directions is high and that the distance between kinks in layer bridge positions remains unchanged (0.380 nm). Surface oxidation proceeds until the complete invasion of the visible surface area, including {111} planes. This is associated with a decrease of the overall brightness. The oscillation cycle is closed by a sudden reduction of the surface oxide from the outskirts toward the top, with a considerable increase of the brightness. Structural features (such as layer edges) of the metallic surface are recovered at this stage of the cycle. We show these oscillations in Figure 8 determined here at T ) 550 K, F0 ) 12 V/nm, and PO2 ) PH2 )2 × 10-3 Pa. Within our model, we obtain self-sustained oscillations at F0) 12 V/nm, PO2 ) 2 × 10-3 Pa for which the hydrogen pressure, PH2, is set to 4 × 10-3 Pa. One can examine how the corresponding oxygen and subsurface oxygen site occupations evolve as a function of time during the oscillatory cycle on the



Figure 11. Experimental time dependence in the global brightness of the field emitter tip in the oscillatory regime. Experimental conditions are the same as in Figure 8. Figure 9. Time evolution of the (a) oxygen and (b) subsurface oxygen site occupations at the (001) plane (solid lines), the {011} planes (dotted lines), and at the {111} planes (dash-dotted lines) in the oscillatory regime. The temperature, electric field, and partial pressures of oxygen and hydrogen are T ) 550 K, F0) 12 V/nm, PO2 ) 2 × 10-3 Pa, and PH2 ) 4 × 10-3 Pa, respectively.

Figure 10. Time evolution of the (a) hydrogen coverage and the (b) water turnover frequency in the oscillatory regime. The line types as well as the temperature, electric field and partial pressures of oxygen and hydrogen are the same as in Figure 9.

(001), {011}, and {111} facets. This evolution is given in Figure 9. The corresponding hydrogen coverages and turnover frequencies are given in Figure 10, in which an oscillatory period of 41 s is obtained. We have found that this period may vary with 0 0 and kred values without changing the structural features the kox 0 0 in a noticeable way. Indeed, if one increases kox and kred while 0 0 /kred fixed, the oscillatory period increases while if keeping kox we decrease their values the oscillatory period decreases in a corresponding fashion. It is instructive to compare such results to the recent experimental observations on low Miller-indexed surfaces regarding the reaction of hydrogen with the O(ad)-RhO(sub) surface oxide55,83 in which the reactivity of H2 with the surface oxide is similar to CO.55 In these experiments, it was concluded that hydrogen does not adsorb onto this surface oxide. Moreover, the bulk Rh2O3 oxide is found catalytically inactive.84-87 Examining Figure 9, we can see that our model is in agreement with such experimental results. Indeed, when the surface oxide is formed (between t ) 15 and 40 s) θH ) 10-5 whereas when it is reduced it increases by 3 orders of magnitude. A similar behavior can also be found with regards to the catalytic activity: In our model the enhancement of the catalytic activity of Rh seems correlated to the transition from the surface oxide to the

metallic phase, supporting the idea that the oxide is a source oxygen for the formation of water (a conclusion which was previouly obtained for the CO-oxidation reaction).84 The corresponding experimental time dependence in the global brightness of the field emitter tip is given in Figure 11, where about the same oscillation period is obtained. As mentioned in the last section, previous atom-probe studies have directly correlated the catalytic activity of the tip with its brightness.10 This conclusion was drawn by analyzing the intensities at 18, 19, and 32 amu (corresponding to H2O+, H3O+, and O2+, respectively) in the spectra at 450 K. Accordingly, the H2O+/H3O+ peak was always larger than the O2+ peak, which leads to the conclusion that the O2+ contribution to the brightness is of minor importance. However, if one compares the time evolution of the global brightness with that of the water yield within our model, it is difficult to see such a correlation between the catalytic activity of the tip with its brightness. To understand this phenomenon further, we performed some additional experiments to examine the H2O+/H3O+ to O2+ peak ratio in the temperature and pressure regime where the oscillations are observed. Shortly after the observed spike in the brightness (see Figure 11), we find the peak ratio to be large since all of the oxygen is reacted off. Afterward, as the tip’s brightness decreases, the ratio gets smaller because of oxygen accumulation on the surface. This leads us to conclude that the O2+ signal should not be completely ignored when analyzing the tip’s brightness. Our theoretical model supports such a conclusion. Indeed, the evolution of the oxygen coverage as given in Figure 9 displays the same features as the evolution of the tip’s brightness since first a sharp increase occurs followed by a slow decay, in both situations. This leads us to conclude that its brightness is in fact a convolution of H2O+/H3O+ and O2+ ions. This conclusion is supported even further if one reexamines the water turnover frequency as well as the oxygen and subsurface oxygen site occupations during one oscillatory cycle, as given in Figures 10 and 9, respectively. Between t ) 0 and 10 s, the water production rate is large but drops rapidly afterward. On the other hand, around 15 s, the oxygen and subsurface oxygen site occupations are now significant but slowly decay afterward. Thus, one arrives exactly at the same conclusion as the one obtained with our atom-probe experiments: the image brightness at the beginning of an oscillatory cycle is due to the production of water ions but, as the oxide invades the tip, the O2+ ions start to contribute significantly to the image brightness. Regarding the catalytic production and subsequent desorption of neutral water, we can summarize our arguments as follows.

17056


McEwen et al.

Figure 12. Series of FIM micrographs covering the complete oscillatory cycle as well as the corresponding time evolution of the subsurface oxygen distribution on a logarithmic scale as obtained within our kinetic model. Starting from a surface in the quasi-metallic state (a) and (d), an oxide layer invades the topmost plane and grows along the {011} facets forming a nanometric cross-like structure (b) and (e). The oxide front spreads to finally the whole visible surface area (c) and (f). The temperature, electric field and partial pressure of oxygen are as in Figure 8. On the other hand, the hydrogen pressure is PH2 )2 × 10-3 Pa in the FIM experiments and 4 × 10-3 Pa in our kinetic model. The subsurface oxygen site occupation scale is the same as Figure 6.

Since (i) the temperature of our rhodium tip is well beyond the TPD peak of water on Rh(001) [a fact that allowed us to simplify our model by assuming that water desorbed immediately after it was formed, as detailed around eq 6] and (ii) the imaging fields are relatively low, the largest amounts of water will desorb in neutral form. Indeed, several 104 ions/s of H2O+/H3O+ species were expected to form by the reaction of hydrogen with rhodium oxide surfaces in atom-probe experiments;88 however, the actual maximum count rates were measured to be smaller by more than a factor of 10. Thus, even if we take into account an increased water ionization probability at protruding step and kink sites, the desorption of neutral water remains the dominating process. The correlation found with the tip’s brightness and the amount of subsurface oxygen in the t ) 10-40 s time frame of the oscillatory cycle leads us to compare the structural features found in the FIM experiments with those found for subsurface oxygen in the theoretical model. This is done in Figure 12, which shows excellent agreement between experiment and theory. Indeed, experimentally the beginning of the oscillatory period is marked by a bright area that extends no further than the {012} planes (Figure 12a). In our model, the subsurface oxygen coverage extends slightly beyond the {012} facets (Figure 12d). At 13 s, the cross-like nanometric structure is clearly formed (Figure 12b). As for our model, the nanometric cross-like shape is also formed and extends out to the shank of the tip (Figure 12e). Finally, at t ) 36 s, one can notice that the brightness of the tip has dramatically reduced. The oxide now has invaded nearly all of the visible area of the tip, except in areas near the {111} facets (Figure 12c). Remarkably, this is exactly what we obtain in our model (Figure 12f). 8. Conclusions and Outlook In this paper we formulated a general theory to model the nanoclock behavior as observed in a field ion microscope when a rhodium field emitter tip is exposed to hydrogen and oxygen. To arrive at such a model, the cooperative effects and concerted behavior for the observed nonlinear phenomena was taken into

account. The kinetic parameters were obtained through experimental data and density functional theory calculations of O and H adsorbed on low Miller indexed rhodium surfaces as a function of coverage and the external electric field. The theory is consistent with the temperature dependence of the total sticking coefficient and the temperature programmed desorption rates on Rh(001), Rh(011), and Rh(111). Here, the anisotropy of the tip was taken into account and the diffusion of hydrogen allows the various nanofacets to interact. The formation of subsurface oxygen is also incorporated into the model which is determined to be crucial in order to explain the experimental behavior. Our resulting model was set up to give not only the correct anisotropy, period and form of the oscillations but also to provide excellent agreement with the experimentally determined bistability diagram for a varying temperature, hydrogen pressure and external electric field. The remarkable occurrence of subsurface oxygen at such low oxygen pressures was facilitated by the presence of a positive external field, since it decreased the barrier for subsurface incorporation. Thus, the electric field, in the present study, turned out to bridge the gap between low-pressure single-crystal surface studies and highpressure heterogeneous catalysis. The present study thus allowed us to understand elemental processes considered relevant for catalysis, on a fundamental level. On the one hand, the role of the external electric field for our chemical clock’s behavior was essential, since in its absence there was no subsurface oxygen and thus no oscillations present. On the other hand, the instabilities as encountered here on the nanoscale could not be explained by a reaction-diffusion model since the diffusion of hydrogen was too fast. Indeed, in order to be able to explain the nanosized pattern formation occurring during the oscillations we had to take into account the intrinsic reaction behavior of all the facets that are simultaneously exposed at the tip’s surface. In this way, much progress has been made since the work of Gorodetskii et al.5 An important step beyond the comprehensive model presented in this paper would be to refine the discretization approach that we adopted in our paper. In particular, the assumption that the

Formation of Water on Rhodium hydrogen coverage distribution remains in quasi-equilibrium can be improved by referring directly to the diffusion equation as it should be the case for a slowly diffusing adspecies in another system. Moreover, such a discretization scheme would be necessary to include noise and stochastic effects. Regarding the stochastic effects, although they were shown not to influence the oscillations,89,90 they would have to be considered when modeling the erratic fluctuations of the surface composition which are associated with the bistability phenomenon described in section 7.2. Each quadrant was here observed to change its brightness and, while doing so, revealed structural features that were not correlated with the surface structure of the original state. Moreover, in a certain range of pressures in the bistable regime each of the four quadrants was observed to react independently so that the sequence of structural transformations in the quadrants usually changed in a random manner. We notice that fluctuations have previously been studied on platinum field emitter tips.91 However, in the corresponding theoretical model, only a single crystal surface was considered and the presence of an external electric field was not taken into account.91 In our system, each quadrant of the field emitter tip changed its brightness randomly and independently of the others in the aforementioned circumstances. Thus, in order to be able to model such phenomena a refined stochastic description with independent variables on the eight octants of the tip should be carried out. This could be done starting from our model and we intend to do so in our future work. Another interesting extension of our model would be to investigate how nonlinear dependencies on the external electric field would influence the behavior of the system. In particular, it would be interesting to see their influence on the oxidation and the reduction rate of the oxide. In order to determine if these nonlinear terms are important one could try to estimate them through density functional calculations, as we did previously to estimate the effective dipolar dependence on the external field.6 Another important generalization of our model would be to allow oxygen to penetrate deeper into the bulk by allowing more than one subsurface layer to be formed and to see how it influences the bistability and the oscillations. The theoretical framework in this paper could also be applied to explain other field ion microscopy experiments where an oscillatory behavior is observed on a platinum tip when it is exposed to H2 and NO292 and efforts are currently underway to accomplish this. This is of fundamental importance, since it will eventually help understand and predict the behavior of such materials at the nanometer length scale. Acknowledgment. Dedicated to Prof. Hans Ju¨rgen Kreuzer on the occasion of his 65th birthday. T.V. and J.-S.M. (postdoctoral researchers) gratefully thank the Fonds de la Recherche Scientifique (F.R.S.-FNRS) for financial support. JSM would also like to thank Pierre de Buyl and Florian Mittendorfer for useful discussions. This research is financially supported by the “Communaute´ francaise de Belgique” (contract “Actions de Recherche Concerteés” No. 04/09-312). Appendix Common definition of coverage. We notice that the coverages θ must have a common definition if eq 13 is to be used on a surface composed of several crystal facets. We should thus distinguish the coverage Θ associated with the geometry of a given surface orientation with respect to the coverage defined by θ ) asσ in terms of the surface density σ of adsorbed atoms with respect to some common reference area as. Here, we take

J. Phys. Chem. C, Vol. 113, No. 39, 2009 17057 TABLE A1: Comparison of Different Surface Orientations for Their Coverage at Saturation, beyond Which the Formation of an Oxide Is Observeda surface

A(hkl)

Rh(100) Rh(110) Rh(111)

a /2 ) 7.22 Å a2/2 ) 10.2 Å2 (31/2)a2/4 ) 6.25 Å2 2

2

ΘO

θO

as/A(hkl)

0.5 0.95 0.5

0.69 0.93 0.8

1.385 0.980 1.6

a a ) 3.8 Å is the lattice constant of Rh. The values of ΘO corresponds to saturation with surface oxygen adatoms.

as ) 10 Å2 by convention. The relationship between both coverages is thus given by

θ Θ ) as A(hkl)

(A1)

where A(hkl) is the area of a unit cell on a surface of orientation (hkl). Table A1 compares the situation on different surface orientations. References and Notes (1) Imbihl, R. Catal. Today 2005, 105, 206. (2) Imbihl, R. Prog. Surf. Sci. 1993, 44, 185. (3) Imbihl, R.; Ertl, G. Chem. ReV. 1995, 95, 697. (4) Mikhailov, A. S.; Ertl, G. ChemPhysChem 2009, 10, 86. (5) Gorodetskii, V.; Lauterbach, J.; Rotermund, H.-H.; Block, J. H.; Ertl, G. Nature 1994, 370, 276. (6) McEwen, J.-S.; Gaspard, P.; Mittendorfer, F.; Visart de Bocarme´, T.; Kruse, N. Chem. Phys. Lett. 2008, 452, 133. (7) Sankaranarayanan, S. K. R. S.; Kaxiras, E.; Ramanathan, S. Phys. ReV. Lett. 2009, 102, 095504. (8) McEwen, J.-S.; Gaspard, P.; Visart de Bocarme´, T.; Kruse, N. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 3006. (9) Visart de Bocarme´, T.; Ba¨r, T.; Kruse, N. Ultramicroscopy 2001, 89, 75. (10) Visart de Bocarme´, T.; Beketov, G.; Kruse, N. Surf. Interface Anal. 2004, 36, 522. (11) Ernst, N.; Bozdech, G.; Gorodetskii, V.; Kreuzer, H. J.; Wang, R. C. L.; Block, J. Surf. Sci. 1994, 318, L1221. (12) Colonell, J. I.; Curtiss, T. J.; Siberner, S. J. Surf. Sci. 1996, 366, 19. (13) Johansson, M.; Lytken, O.; Chorkendorff, I. J. Chem. Phys. 2008, 128, 034706. (14) Eichler, A.; Kresse, G.; Hafner, J. Phys. ReV. Lett. 1996, 77, 1119. (15) Yates, J. T.; Thiel, P. A.; Weinberg, W. H. Surf. Sci. 1979, 84, 427. (16) Richter, L. J.; Ho, W. J. Vac. Sci. Technol. A 1987, 5, 453. (17) Nichtl-Pecher, W. Ph.D. thesis; University of Erlangen: Nu¨rnberg, 1990. (18) Kreuzer, H. J.; Jun, Z.; Payne, S. H.; Nichtl-Pecher, W.; Hammer, L.; Mu¨ller, K. Surf. Sci. 1994, 303, 1. (19) Payne, S. H.; Kreuzer, H. J.; Frie, W.; Hammer, L.; Heinz, K. Surf. Sci. 1999, 421, 279. (20) Comelli, G.; Dhanak, V. R.; Kiskinova, M.; Prince, K. C.; Rosei, R. Surf. Sci Rep. 1998, 32, 165. (21) Lundgren, E.; Mikkelsen, A.; Andersen, J.; Kresse, G.; Schmid, M.; Varga, P. J. Phys.: Condens. Matter 2006, 18, R481. (22) Fisher, G. B.; Schmieg, S. J. J. Vac. Sci. Technol. A 1983, 1, 1064. (23) Schwarz, E.; Lenz, J.; Wohlgemuth, H.; Christmann, K. Vacuum 1990, 41, 167. (24) Salanov, A. N.; Savchenko, V. I. React. Kinet. Catal. Lett. 1993, 49, 29. (25) Salanov, A. N.; Savchenko, V. I. Surf. Sci. 1993, 296, 393. (26) Peterlinz, K. A.; Sibener, S. J. J. Phys. Chem. 1995, 99, 2817. (27) Comelli, G.; Baraldi, A.; Lizzit, S.; Cocco, D.; Paolucci, G.; Rosei, R.; Kiskinova, M. Chem. Phys. Lett. 1996, 261, 253. (28) Alfe`, D.; Rudolf, P.; Kiskinova, M.; Rosei, R. Chem. Phys. Lett. 1993, 211, 220. (29) Dri, C.; Africh, C.; Esch, F.; Comelli, G.; Dubay, O.; Ko¨hler, L.; Mittendorfer, F.; Kresse, G. J. Chem. Phys. 2006, 125, 094701. (30) Gustafson, J.; Mikkelsen, A.; Borg, M.; Andersen, J. N.; Lundgren, E.; Klein, C.; Hofer, W.; Schmid, M.; Varga, P.; Ko¨hler, L.; Kresse, G.; Kasper, N.; Stierle, A.; Dosch, H. Phys. ReV. B 2005, 71, 115442.

17058


(31) Gustafson, J.; Mikkelsen, A.; Borg, M.; Ko¨hler, E. L. L.; Kresse, G.; Schmid, M.; Varga, P.; Yuhara, J.; Torrelles, X.; Quiro´s, C.; Andersen, J. N. Phys. ReV. Lett. 2004, 92, 126102. (32) Tucker, C. W. Acta Metall. 1967, 15, 1465. (33) Belton, D. N.; Fisher, G. B.; DiMaggio, C. L. Surf. Sci. 1990, 233, 12. (34) Rebholz, M.; Prins, R.; Kruse, N. Surf. Sci. 1992, 269-270, 293. (35) Gustafson, J.; Resta, A.; Mikkelsen, A.; Westerstro¨m, R.; Anderson, J. N.; Lundgren, E.; Weissenrieder, J.; Schmid, M.; Varga, P.; Kaspaer, N.; Torrelles, X.; Ferrer, S.; Mittendorfer, F.; Kresse, G. Phys. ReV. B 2006, 74, 035401. (36) Klikovits, J.; Schmid, M.; Merte, L. R.; Varga, P.; Westerstro¨m, R.; Resta, A.; Andersen, J. N.; Gustafson, J.; Mikkelsen, A.; Lundgren, E.; Mittendorfer, F.; Kresse, G. Phys. ReV. Lett. 2008, 101, 266104. (37) Nolte, P.; Stierle, A.; Jin-Phillip, N. Y.; Krasper, N.; Schulli, T. U.; Dosch, H. Science 2008, 321, 1654. (38) Kellogg, G. L. Phys. ReV. Lett. 1985, 54, 82. (39) Kellogg, G. L. Surf. Sci. 1986, 171, 359. (40) Voss, C.; Kruse, N. Surf. Sci. 1998, 409, 252. (41) Medvedev, V. K.; Suchorski, Y.; Voss, C.; Visart de Bocarme´, T.; Ba¨r, T.; Kruse, N. Langmuir 1998, 14, 6151. (42) Jansen, N. M. H.; van Tol, M. F. H.; Nieuwenhuys, B. E. Appl. Surf. Sci. 1994, 74, 1. (43) Willinger, M.; Schlo¨gl, R.; Visart de Bocarme´, T.; Kruse, N. to be published. (44) Gaussmann, A.; Kruse, N. Catal. Lett. 1991, 10, 305. (45) Mu¨ller, E. W.; Tsong, T. T. Field Ion Microscopy, Principles and Applications; Elsevier: New York, 1969. (46) Makeev, A.; Imbihl, R. Chem. Phys. Lett. 1995, 242, 221. (47) Makeev, A.; Imbihl, R. J. Chem. Phys. 2000, 113, 3854. (48) Schaak, A.; Imbihl, R. J. Chem. Phys. 2000, 113, 9822. (49) Monine, M. I.; Schaak, A.; Rubinstein, B. Y.; Imbihl, R.; Pismen, L. M. Catal. Today 2001, 70, 321. (50) DiFoggio, R.; Gomer, R. Phys. ReV. B 1982, 25, 3490. (51) Wang, S. C.; Gomer, R. J. Chem. Phys. 1985, 83, 4193. (52) Vo¨lkening, S.; Bedu¨ftig, K.; Jacobi, K.; Wintterlin, J.; Ertl, G. Phys. ReV. Lett. 1999, 83, 2672. (53) Gurney, B. A.; Ho, W. J. Chem. Phys. 1987, 87, 5562. (54) Wagner, M. L.; Schmidt, L. D. J. Chem. Phys. 1995, 99, 805. (55) Klikovits, J.; Schmid, M.; Gustafson, J.; Mikklsen, A.; Resta, A.; Lundgren, E.; Andersen, J. N.; Varga, P. J. Phys. Chem. B 2006, 110, 9966. (56) Wike, S.; Natoli, V.; Cohen, M. H. J. Chem. Phys. 2000, 112, 9986. (57) Africh, C.; Esch, F.; Li, W. X.; Corso, M.; Hammer, B.; Rosei, R.; Comelli, G. Phys. ReV. Lett. 2004, 93, 126104. (58) Michaelides, A.; Hu, P. J. Chem. Phys. 2001, 114, 513. (59) Michaelides, A.; Hu, P. J. Am. Chem. Soc. 2001, 123, 4235. (60) Gregoratti, L.; Baraldi, A.; Dhanak, V. R.; Comelli, G.; Kiskinova, M.; Rosei, R. Surf. Sci. 1995, 340, 205. (61) Miller, M. K.; Cerezo, A.; Hetherington, M. G.; Smith, G. D. W. Atom Probe Field Ion Microscopy; Monographs on the Physics and Chemistry of Materials; Clarendon Press: Oxford, 1996. (62) Visart de Bocarme´, T.; Kruse, N.; Gaspard, P.; Kreuzer, H. J. J. Chem. Phys. 2006, 125, 054704. (63) der Lage, F. C. V.; Bethe, H. A. Phys. ReV. 1947, 71, 612. (64) Watson, J. H.; Conradi, M. S. Phys. ReV. B 1990, 41, 6234.

McEwen et al. (65) Anderson, F. G.; Leung, C.-H.; Ham, F. S. J. Phys. A: Math. Gen. 2000, 33, 6889. (66) Bridge, N. D.; Buckingham, A. D. Proc. R. Soc. Lon. Ser. A 1966, 295, 334. (67) Guggenheim, E. A. In Encyclopedia of Physics; Flu¨gge, S., Ed.; Springer-Verlag: Berlin, 1959; Vol. III/2, pp 1-118. (68) Kreuzer, H. J.; Payne, S. H.; Drozdowski, A.; Menzel, D. J. Chem. Phys. 1999, 110, 6982. (69) McEwen, J.-S.; Gaspard, P.; Visart de Bocarme´, T.; Kruse, N. 2008, in preparation. (70) Jenkins, S. J.; Pratt, S. J. Surf. Sci. Rep. 2007, 62, 373. (71) McEwen, J.-S.; Gaspard, P.; Mittendorfer, F.; Visart de Bocarme´, T.; Kruse, N. 2008, in preparation. (72) Slinko, M. M.; Slinko, M. G. Catal. ReV. 1978, 17, 119. (73) Sales, B. C.; Turner, J. E.; Maple, M. B. Surf. Sci. 1982, 114, 381. (74) Bassett, M. R.; Imbihl, R. J. Chem. Phys. 1990, 93, 811. (75) Zhdanov, V. P. Surf. Sci. 1993, 296, 261. (76) Hartmann, N.; Krischer, K.; Imbihl, R. J. Chem. Phys. 1994, 101, 6717. (77) Ganduglia, M. V.; Reuter, K.; Scheffler, M. Phys. ReV. B 2002, 65, 245426. (78) Reuter, K.; Scheffler, M. Phys. ReV. B 2001, 65, 35406. (79) McEwen, J.-S.; Gaspard, P. J. Chem. Phys. 2006, 125, 214707. (80) Doedel, E. J.; Paffenroth, R. C.; Champneys, A. R.; Fairgrieve, T. F.; Kuznetsov, Y. A.; Oldeman, B. E.; Sandstede, B.; Wang, X. J. AUTO2000: Continuation and bifurcation software for ordinary differential equations; Applied and Computational Mathematics, California Institue of Technology, 2000; 2001: https//sourceforge.net/projects/auto2000/. (81) Kreuzer, H. J.; Payne, S. H. Theories of the Adsorption-Desorption Kinetics on Homogeneous Surfaces. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W. S., Zgrablich, G., Eds.; Elsevier Science: New York, 1997; Vol. 104. (82) McEwen, J.-S.; Payne, S. H.; Kreuzer, H. J.; Kinne, M.; Denecke, R.; Steinru¨ck, H.-P. Surf. Sci. 2003, 545, 47. (83) Dudin, P.; Barinov, A.; Gregoratti, L.; Kiskinova, M.; Esch, F.; Dri, C.; Africh, C.; Comelli, C. J. Phys. Chem. B 2005, 109, 13649. (84) Lundgren, E.; Gustafson, G.; Resta, A.; Weissenrieder, J.; Mikkelsen, A.; Andersen, J. N.; Ko¨hler, L.; Kresse, G.; Klikovits, J.; Biederman, A.; Schmid, M.; Varga, P. J. Electron Spectrosc. Relat. Phenom. 2005, 144-147, 367. (85) Westerstro¨m, R.; Wang, J. G.; Ackermann, M. D.; Gustafson, J.; Resta, A.; Mikkelsen, A.; Andersen, J. N.; Lundgren, E.; Balmes, O.; Torrelles, X.; Frenken, J. W. M.; Hammer, B. J. Phys.: Condens. Matter 2008, 20, 184018. (86) Gustafson, J.; Westerstro¨m, R.; Mikkelsen, A.; Torrelles, X.; Balmes, O.; Bovet, N.; Andersen, J. N.; Baddeley, C. J.; Lundgren, E. Phys. ReV. B 2008, 78, 045423. (87) Flege, J. I.; Sutter, P. Phys. ReV. B 2008, 78, 153402. (88) Kruse, N.; Abend, G.; Block, J. H. J. Chem. Phys. 1988, 88, 1307. (89) Gaspard, P. J. Chem. Phys. 2002, 117, 8905. (90) Andrieux, D.; Gaspard, P. J. Chem. Phys. 2008, 128, 154506. (91) Suchorski, Y.; Beben, J.; James, E. W.; Evans, J. W.; Imbihl, R. Phys. ReV. Lett. 1999, 82, 1907. (92) Voss, C.; Kruse, N. Appl. Surf. Sci. 1996, 94/95, 186.

JP901975W

Oscillations and Bistability in the Catalytic Formation of Water on

Recommend Documents