New Modulated Molecular Beam Scattering Methods for Probing
Transcript Of New Modulated Molecular Beam Scattering Methods for Probing
2566
Langmuir 1991, 7, 25662573
New Modulated Molecular Beam Scattering Methods for Probing Nonlinear and CoverageDependentReaction Kinetics at Surfaces
D. F. Padowitz,+K. A. Peterlinz, and S. J. Sibener’
The James Franck Institute and The Department of Chemistry, The university of Chicago, Chicago, Illinois 60637
Received April 10, 1991.I n Final Form: July 29, 1991
fl A new three molecularbeam arrangement is introduced that expandsthe range and power of modulated
beam reactive scattering for studyin complex kinetics at surfaces. This paper presents two types of kineticmeasurementsthat utilizethis t reebeamarrangement. The firstmeasurementsusetwocontinuous,
independently adjustable molecular beams to establish steadystate reaction conditions,while a weaker modulated third beam induces small concentration perturbations around the selected steady state. This techniquepermits experimentallinearizationof nonlinearkinetics over a wide range of conditions,allowing us to explore the global behavior of reactions, determine coveragedependent rate constants, and isolate individual elementary steps from complex reaction mechanisms. We illustrate these capabilities with preliminaryresults for the oxidation of hydrogen to water and for the recombination kineticsof hydrogen on the Rh(111)surface. The secondgroup of measurementsuses timeresolvedspecularhelium scattering as a sensitive in situ probe of both adsorbate coverage and coveragedependent surface kinetics. The oxidation of CO on Rh(ll1) under pseudofirstorderconditions is examined with this new kinetic probe.
I. Introduction
Molecular beam scattering techniques have firmly established themselves among the major tools for studying the kinetics and dynamics of gassurface interactions. Modulated molecular beam relaxation spectrometry (MBRS), a descendent of Eigen’s pioneering work on chemical relaxation,’ extracts kinetic information by analyzing product evolution in response to a timevarying reactant beam flux. Relaxation techniques based on this principle are now widely used in reactive scattering to study heterogeneous chemical reactions.
The evaluation of surface kinetic data by transform analysis of modulated beam measurements was introduced in an early paper by Foxon, Boudry, and Joyce and has since been well studied.2 Transform analysis, widely used in electrical engineering and linear systems theory, determines the properties of a system by its response to a known input. Fourier or Laplace transforms are used to deconvolute a “transfer function”, characteristic of the system, from the observed output to a given periodic or transient input. Transform analysis of MBRS data is straightforward when the product response to changing reactant flux islinear.3 Accordingly,most studies reported to date have been on simplelinear reactions, suchassinglecomponent adsorptiondesorption a t low coverage, have used linear approximations for weakly nonlinear systems, orhave workedin a pseudofirstorder regimeovera limited coverage range.’+
In this paper we introduce several particularly useful extensions to conventional MBRS techniques, which expand the range of reaction types and reaction conditions
t Current address: Department of Chemistry, University of California, Berkeley, CA 94720.
(1)Eien, M. In Technique of Organic c h e m i e t ~I;nterscience: New
York, 1963; Vol. VIII, Part 11, p 901. ((23))FCohxaonn,,CH..TC.;.;BWoueidnrbye,rMg,.WR..;HJo. yJc.eC, Bhe.mA.. PShuyr8f.. S1e9i7. 71,96764,4,414766.9. (4) Olanfer, D. R.; Ullman, A. Int. J. Chem. Kinet. 1976,8,625. (5) Schwarz,J. A.; Madix, R. J. Surf. Sci. 1974, 46, 317. (6) Jones,R.He;Olander,D.R.;Siekhaus,W.J.;Schwarz,J. A. J. Vac.
Sci. Technol. 1972, 9, 1429. (7) Sawin, H. H.; Merrill, R. P. J. Vac. Sei. Technol. 1981, 19, 40. (8)DEvelyn, M. P.; Madix, R. J. Surf. Sei. Rep. 1984, 3, 413.
07437463/91/24072566$02.50/0
that can be studied with molecular beam techniques. Our ultimate goal is to study heterogeneous reactions, in the linearized limit, under arbitrarily chosen and precisely defined reactant coverage regimes. Our measurements are all based on the experimental strategy of performing kinetic measurements by observing the relaxation from small density perturbations, which we have experimentally introduced around carefully defined steadystate reaction conditions.
Most chemical systems are not linear. An elementary bimolecularstepin a chemicalreaction involvestwo species and the rate of reaction depends on the product of their concentrations. If both concentrations change with time, the rate expression for the mechanism will be nonlinear. Areaction willalsobe nonlinear if the rate constants change with concentration. This is frequently true for surface reactions, in which adsorption probabilities, preexponential factors, and activation energies often depend on coverages.
Besidesprohibiting experimentaldeconvolutionand the use of Fourier transform analysis methods that depend on linear response, nonlinearity frequently presents other severe mathematical difficulties in analyzing chemical reaction mechanisms. A model reaction mechanism yields a set of ordinary differential equations describing the experimentallyobservablerates. For nonlinear differential equations there are no general mathematical techniques providing analytical solutions. Numerical integration of nonlinear differential equationsmay be difficult since rate constants for elementary reaction steps may range over many orders of magnitude, leading to a set of “stiff“ differential equations, creating numerical instabilities.
However,it isoften possible experimentallyto constrain the reaction to a linear regime. If this is done, linear transform analysis may be applied to the experimental data,and the differential equations for a postulated mech
anism may be solved by series expansions or perturbation techniques, allowingthe sought after kinetic information to be obtained. The pioneers in the field, including Foxon, Olander, Madix, Weinberg, and their coworkers,28were aware of the problems of nonlinear kinetics and examined linearized MBRS, but the theory outstripped the available
0 1991 American Chemical Society
Modulated Molecular Beam Scattering Methods
(a) Nonlinear Kinetics
Langmuir, Vol. 7, No. 11, 1991 2567
(b) “Linearized”Kinetics
A
B
C
A
Reactant Beam
Flux
ReactantBeam Flux modulatedbeam + conlimous beam
Figure 1. Differentmodulation schemesthat canbe used duringsurfacereactivescatteringexperiments. Part a is of the conventional type, with reactant density spanninga large range of coverage duringthe modulation cycle. Part b, after Foxon et al.? which utilizes small perturbationsof reactant densityduringeach modulation cycle, portrays a schemethat is now possible with our new threebeam approach.
instrumentation. In succeeding years the numerous successes of MBRS obscured its limitations, which have onlyrecently been reconsidered asmore complexsituations are encountered.9
Let us now consider the general problem of conducting modulated beam experiments on nonlinear systems and discuss a general solution to the problem. Figure l a presents a schematizedview of a conventionalexperimental arrangement. When a small modulated beam flux is used (scheme A, Figure la), the response may be linear but is restricted to a lowcoverage regime. Depending on the shape of the order plot, there may be little signal at all. When an intense modulated beam is used (scheme B, Figure la), a wide coverage range is spanned during each modulation cycle, producing a nonlinear response that is difficult to analyze. This“strong modulation” schemecan further complicate matters by inducing timedependent behavior in other reactants and reaction intermediates. Stated simply, one of the main consequencesof thistypical method is that the measured kinetics represent a convolution over all coverages on the surface.
Ideally, one would like to avoid these problems by sampling the kinetics under isosteric conditions. The straightforward solution is to establish a steadystate reaction on the surface with arbitrarily chosen adsorbate coverages and to then superimpose a small modulation about the selected steady state. Such a scenario is shown in Figure lb. Modulation schemesAC here, i.e., for three different coverages,all involvesufficientlysmallexcursions from the preset steady state that relaxation is linear. If this linearization could be performed experimentally for different steadystate conditions, then reaction mechanisms or coverage dependencies of the rate constants may be determined over the global range of conditions for the nonlinear reaction.
This paper introduces a new three molecular beam scattering arrangement, which allows us to directly im
(9) Engstrom,J. R.;Weinberg, W.H.J . Chem. Phys. 1987,87,4211.
Molecular Beams H2
H2 (D2)
02
Mass Spectrometer
Rh(ll1) Surface
Figure 2. Multiplesource modulated molecular beam reactive scattering. Two continuousbeam sourcesestablishsteadystate surface conditions, which are perturbed by a third modulated beam in order to probe kinetics.
plement the approach described above and provides greater versatility for new methods. The scheme is depicted in Figure 2. We use two continuous, independently adjustable beams to establish a selected steadystate on the surface, while a weaker modulated third beam induces small concentration perturbations around the selected steady state. This is not the only way to linearize an MBRS experiment, but it is simple and direct, retains the advantages of molecular beam sources,and offersgreat flexibility in experimental design. In addition to linearization by means of weak modulation, we can use the continuous beams to move away from the nearzero coverage limit in which molecular beam experiments are sometimes forced to work, and to widely vary steadystate surface coverages, which is important for determining coveragedependent activation energies and preexponential factors. Most significantly, by controlling reactant and intermediate coverage regimes, sometimes using isotopic substitutions in the beams, we can isolate the individualelementary steps that composecomplexreaction mechanisms.
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This experimental approach is not without difficulties. The two most significant complications are extreme sensitivityrequirements and the need for in situ coverage assessment. Signal levels will be smaller than those
encountered in conventional MBRS measurements since
the modulation depth (Le., the surface coverage change induced by the incident modulated molecular beam) has
+ + . to be reduced in order to reach the linearized limit. This + D2 HD
limit has now been reached for the simple H2 + and the more complicated H2 D2 0 2 HDO reactions, aswillbe discussed later in this paper. Thesetwo problems have now been addressed in our group by using timeresolved helium specular scattering to assess adsorbate coverage, to determine the modulation depth, and to directly measure reaction kinetics. The basis for this technique is the extraordinarily large scattering cross section that adsorbates have for reducing specular helium reflectivity. The origin of this effect has been well documented1°12and is due to the longrange attractive part of the adsorbatehelium interaction potential. In this paper we shall demonstrate for the CO oxidation reaction that helium reflectivity measurements conducted during modulated beam reactive scattering measurements can be used as a kinetic response uamplifier”,gaining back much of the signal that was lost by using much smaller modulation depths than are commonly used in conven
tional MBRS measurements.
11. Experimental Section
For this work, a third beam source was installed in the UHV surfacescatteringinstrumentused in our previousstudiesof CO oxidationonRh(lll).13JT4heapparatusconsistsof a removable source chamber containing three supersonic molecular beam sourcesand a main chamberwith sampleand manipulator,mass spectrometerdetector,residualgas analyzer,Auger spectrometer, and sputter ion gun.
Thesourcechamber is dividedlaterallyintotwo sections,each with three differential regions of diffusion pumping before the main chamber. The left beam is in one section and the center and right beams together are in the other. The supersonic expansion in each beam originated from 100pm pinholes and was formed by a skimmer, collimator,and shutter. The center beam was modulated by a rotating chopper wheel with a fourslot50% total dutycyclepattern, or a dual pattern chopperwith two 25% slots and two 1%slits for incident beam timeofflight measurements.
On entering the main chamber,the left and right beams were defined by 80milaperturesand the center beam was defined by a 40milaperture. Thebeams passed througha finaldifferential pumpingregionbeforethe scatteringregion. Themain chamber scattering region was pumped primarily by a 400 L/s ion pump. The base pressure in the scattering region was 5 x Torr, rising to 6 x [email protected] at the highest beam fluxes.
The distance from chopper to sample was 21.21 cm. The left and right beams were 15’ off axis. The sample normal was usually at 45O to the beam axis, resulting in roughly 5 X 7 mm elliptical spots on the sample, which overlapped the center 2.5 X 3.25 mm modulated beam spot. Maximum side beam flues were on the order of 10 langmuir/s, or 106 Torr/cm2.
A rhodium singlecrystal sample was cut and polished by standard techniques. The surface was within 1/2O of the FCC (111)plane,asdeterminedby LaueXraydiffraction. Thesurface was prepared in UHV by cycles of Ar ion sputteringand heating in oxygen and was kept very clean by oxidation and reduction
(10) Poelsema, B.; Comsa, G. Scattering of Thermal Energy Atoms from Disordered Surfaces; Springer Tracta in Modern Physics 115; SpringerVerlag: Berlin, 1989.
(11) Poelsema, B.;Palmer, R. L.; Comsa, G. Surf. Sci. 1984, 136, 1. (12) Poelsema,B.;Comsa, G. Faraday Discuss. Chem. SOC1. 985,80,
247.
(13) Brown, L.S.;Sibener,S. J. J. Chem. Phys. 1988,89, 1163. (14)Brown, L. S.;Sibener,S. J. J. Chem. Phys. 1989, 90,2807.
Padowitz et a1,
duringtheexperimentalreaction. Surfacecleanlinesswasverified with Augerspectroscopyand,subsequently,with specularhelium reflectivity measurements.
The detectoris an Extranuclearquadrupolemass spectrometer, doubly differential pumped by two 50 L/s ion pumps and an additional Balzers 50 L/s turbomolecular pump on the ionizer region. Transmitted ions are detected with a Venetion blind electron multiplier, suitable for pulse counting. Angular acceptance of the detector is lo. Flight path from sample to ionizer is 14.45cm. At normal, the detector views a spot of =2.5mm diameter.
Producttime of arrivalwave formswere acquiredby a custom multichannel scalar system triggered by the photodetector on the chopperwheel assembly,collecting255 channelsof 520 ps. For the water reaction,goodqualitywave formsusuallyrequired 1060 min of signal averaging. A typical run of 20 min at 400 Hz chopper speed would be 5 x 106 chopper periods or “shots*, giving 5 X l(r counts/channel with signal to background of 2, and S / N of 200.
111. Results
A. ThreeBeam Scattering and Mechanism Linearization. We being this section by examining the oxidation of hydrogen to form water on the Rh(ll1) surface. This reaction is relatively complicated and has attracted considerable attention.15
There are several sources of nonlinearity in this mechanism. Hydrogen sticking probabilities and desorption rate parameters depend on both hydrogen and oxygen coverages.16J7 As the reaction probably proceeds by
sequential H + 0 and H + OH addition or OH dispro
portionation, the differential equations for the overall mechanism contain several terms that are quadratic in hydrogen coverage. These quadratic terms prohibit general solutions of the rate equations for this mechanism.
Experimentally, an examination of the rate of water production for different hydrogen and oxygen beam intensities (Figure 3) shows that no global “order” can be assigned to the reactants. The data for this coveragedependent order map were obtained with three beams. Continuous Hz and 0 2 beam pressures were varied to establish the desired steadystatereaction. The third molecular beam, which in this instance wasH2,wasmodulated at 100 Hz for phasesensitive detection. The rate of product formation rapidly saturates as hydrogen flux is increased for all oxygen fluxes measured. For high hydrogen beam intensities,the reaction islinear in oxygen coverage, but at low hydrogen flux, the yield is nonmonotonic, first increasing, then decreasing again as oxygen flux increases.
Time domain measurements for water production
further emphasize the nonlinear nature of MBRS wave
formstaken with arbitrarilychosen modulation strengths. With three beam sources, we are able to combine a modulated H2 beam with continuous beams of Dz and 0 2 to produce three isotopically substituted forms of water. This arrangement was shown in Figure 2. Figure 4 shows mass spectrometer signal vs time wave forms for the three products of this reaction. The HzO wave form has the usual symmetricform. The HDO product, however,shows a highly asymmetric wave form, with a rapid rise, sloping top, and long decay. As will be shown below, this appearance is the same as that for HD produced in an
(15) Norton,P. R. In The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis;King,D.A.,Woodruff,D. P.,Eds.;Fundamental
Studies of Heterogeneous Catalysis; Elaevier: New York, 1982; Vol. 4. (16) Thiel, P. A.; Yaks, J. T., Jr.; Weinberg, W. H. Surf. Sci. 1979,90,
121.
(17)Yaks,J.T.,Jr.; Thiel, P. A.; Weinberg,W. H. Surf. Sci. 1979,84,
427.
Modulated Molecular Beam Scattering Methods
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8
. e
Figure3. Steadystate water production vs reactant beam flux.' The pressure rise in the chamber calibrates the hydrogen and oxygen beam fluxes. No global "order" can be assigned to the reactants based on the relationship of reactant flux to H20
production.
I

2140
0 50 100 150 200 250
channel
Figure 5. HDO and HD wave forms. The essential features of the asymmetric HDO wave form are also seen in an HD wave form, indicating that the origin of the nonlinearity is similar in both reactions.
neither D2 nor 0 2 was modulated in this experiment. On first glance one would expect a continuous background signal of D20 to be produced from this arrangement. The negative modulation gives a direct measure of the extent to which the modulated H2 beam perturbs the steadystate coverages of 0, D, and OD.
The origin of the nonlinearity and the mathematical analysis can be best illustrated in the simpler HD recombination reaction. Figure 5 shows that the HD and HDO wave formsshare the same asymmetric shape. This can be quantitatively understood by examining the reaction mechanism and by then numerically simulating the result. The mechanism for this reaction can be modeled as
I 1 .
a
.
9  I 1.
a =i
0 50 100 150 200 250
channel
Figure 4. Mass spectrometer timeofarrivalwave forms of the
products from modulated H2 and continuous D2 and 0 2 beams reactingon Rh(111)at 665 K. Thechannel time is 10ps/channel.
The ratio of modulated to continuous beams is high. The HDO response is nonlinear and the D20 background shows negative modulation. Note that neither D2nor 0 2 beam is modulated and that the D2O modulation is solely due to perturbation of the steadystatesurface coverages of 0,D, and OD by the modulated
Hz beam.
overdrivenmodulation cycle. This nonlinearity is due to the changing coverage of D species on the surface during the H2 pulse. The most remarkable wave form is that for D20, which shows negative modulation. Remember that
H
+
D
Kd
+ HD,
(3)
Thismechanismcanbe simulatedby numericalintegration of the system:
d=[H1 2P~z(l([HI k [D]))  2kd[HI2 kd[HI[D] (4)
dt
d[D1  WD,(~([HI 4 [Dl)) 2kd[DI2 kd[HI[D] (5)
dt
d[HDl,/dt = kd[Hl [Dl
(6)
where Pincorporates the incident beam flux and the zerocoverage sticking coefficient.
The qualitative behavior does not depend on the rate constants, only on the ratio of the continuous and modulated fluxes. Figure 6 shows the simulated results, and correspondingFouriertransfer function plots (showing the product phase and amplitudefor allinputfrequencies), for three different ratios of the modulated D2 intensity to
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HD Waveforms Transfer Fn.
D0.5 0.4
0.3
0.2
0.1
0.0 0.4 0.8
0.5
0.4
0.3 5 5I
h
0.2
Y
0.1
~ 0.0 0.4 0.8 0.0
0.5
0.4
0.3
0.2
0.1
0.0 0.4 0.8 5 6 1 8 910
msec
Figure6. HD simulationvs modulationstrength. As the strong modulation is reduced, the wave form becomes more symmetric, the odd harmonics of the transfer function approach the singlestep arc, and the even harmonics vanishall indications that the kinetics are approaching a linear regime.
the continuous H2 intensity. The 5:l (strong modulation limit) plot shown in Figure 6 resembles the HD wave form of Figure 5. Examinationa of the simulation shows that the rapid rise is due to the increase in adsorbed H, the sagging top to decline in adsorbed D. The long tail is the recovery of D as H decreases;the product of the two drops less quickly than H alone. The transfer function, shown to the right, does not fall on the "semicircle" as a singlestep reactionwouldPV6 More importantly, significanteven harmonics appear. Since the reactant beam modulated by the chopper is a square wave, it contains only odd harmonics, and the presence of even frequencies in the response indicatesnonlinearity. In the strongmodulation limit, as the modulated D coverage increases the steadystate D coverage dropsthe overallkinetics are therefore nonlinear as the kinetic equations depend on the product of two timevarying quantities. Reducing the relative intensity of the modulated beam reduces the nonlinearity, as seen in the second and third panels of Figure 6. Here, we clearly see that as the modulation is decreased, the wave form becomes symmetric,the transfer function approaches the singlestep arc, and the even harmonics vanish. In this weak perturbation limit the reaction becomes linear.
Finally, we perform a perturbation theory analysis of the data to extract the desired kinetic information from the linearized wave forms. Note that a perturbation treatment allows us to analytically solve the differential equationsforthe HD mechanism,yieldingphysicalinsight into the elementary steps of the reaction. The reaction is second order in hydrogen concentration, and thus nonlinear:
Padowitz et a1.
8
H, i= 2H
(7)
k
The sticking coefficient is s, the desorption rate k . Let the reactant flux be P and the surface concentration of hydrogen [HI, then
d[H]/dt = 2sP  2k[HI2
(8)
+ + Now considera smalltimedependentfluxsuperimposed
on a continuous background P = po tp(t). We wish to obtain a solution in the form [HI = h~ th(t). We then have
+ + d(h0 + eh(t)) = 25030 tp(t)) 2k(h0 th(t))2 (9) dt
9+  d h w = 2sp0+ c2sp(t) 2 k h t  4kch&(t) 
dt
dt
2 k ~ ~ h ((t1)0~)
If this is to hold for all values oft over some interval, then the coefficientsof eachpower of e must equate. For zeroth order we have the steadystate component
dho/dt = 0 = 28pO  2 k h t
(11)
This yields ho = ( s p ~ / k ) l t/h~e, steadystate surface coverage. The firstorder perturbation is
dh(t)/dt = 2sp(t) 4kh&(t)
(12)
This is the desired linear response. Substituting the steadystate ho
dh(t)/dt = 2sp(t)  4 k h ( t ) d G
(13)
or
dh(t)/dt 2sp(t) k'h(t)
(14)
where the effective rate constant k' = 4(skp0)'/~.In Arrhenius form we have
A' exp(E'/kT) = 4 4 s p 4 exp(E/kT) (15)
with A' = 4(sp0A)'/~and E' = E/2. Finally, we relate the measured rates for the linearized system to the actual rates:
A = Af2/16sp0 E = 2E'
(16)
The activation energy is obtained directly, the preexpo
nential requires knowledge of the beam flux and sticking coefficient, or of the steadystate coverage ho. A similar analysis may be found in ref 5.
An Arrhenius plot of our experimental pseudofirstorder rates is shown in Figure 7. The pseudofirstorder rate constants are E', = 10.0 kcal/mol, A' = lo7s'. The true secondorder constants are Ea = 20 kcal/mol, A =
cm2/s. This is in good agreement with earlier [email protected]
We now return to the water reaction and demonstrate that the linearized limit can also be reached for this more
complicated system (before the signal becomes too small to d'etect!). Figure 8 is a plot of the even harmonics in the Fourier transform of the HDO product wave form.' Since the reactant beam modulated by the chopper is a square wave, it contains only odd harmonics, and the presence of even frequencies in the response indicates nonlinearity.
As the strength of the modulation is reduced, the per
(18)Engle,.T.;Kuipem, H.Surf.Sci. 1979,90,162. (19) Verhelj, L.K.; Hugenschmidt,M.B.;Anton, A. B.;Pole", B.; Comsa, G.Surf.Sci. 1989,210,1.
Modulated Molecular Beam Scattering Methods
lb
1
1 /T
Figure 7. HD order rates for
HAr+rheDniruescopmlobt.inTathioenlinaerearpizleodtteodr
pseudofirstfor two runs
differing slightly in continuous beam flux. The true second
order rates are derived from them as described in the text: E,
= Z'.th;e preexponentialis approximately cm2/s,using an
estimated continuousbeam flux of 10 ML/s.
% modulation
Figure 8. Linearizationof the HDO response. The amplitude of the even harmonics of the HDO wave form indicates nonlinearity. Modulation is measured as the ratio of the modulated beam to the sum of modulated H2 and continuous D2 beams. With smallermodulation,the amplitudesof the even harmonics systematically decrease and approach zero as the response becomes linear.
turbation measured by the depression of D2O production decreases and the amplitudes of the even harmonics decrease. Modulation is measured as the ratio of the modulated beam to the sum of modulated H2 and continuous D2 beams. The perturbation of steadystate surface coverage can be determined from the transient depression of the continuous D2O background. The perturbation decreases from 6096 to less than 596 as the intensity of the modulation is reduced. With smaller perturbations, the response becomes more linear. The even harmonics approach zero, clearly demonstrating the experimental linearization of the kinetics in the limit of
+ weak modulations. More extensive results for hydrogen
oxidation20and H2 D2 recombination21on Rh(ll1) are forthcoming.
B. Helium Reflectivity as a n i n S i t u Probe of
Surface Coverage and Kinetics. As shown above, the use of three beam sources provides excellent control of reactant flux, but an in situ method of assessing adsorbate surface concentration is also necessary in order to extract the desired kinetic rate constants. Furthermore, signal levels in the linearized limit can be extremely low since much smaller modulation depths are used than in conventional singlebeam MBRS experiments. These two problems have now been addressed in our group by using
(20) Padowitz, D. F.; Sibener, S. J. Surf.Sci. 1991, 254, 125. (21) Padowitz,D.F.;Sibener,S.J.J.Vuc.Sci. Technol.A 1991,9,2289.
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1
0
0.9 3 0,s
bIC 0.7
E 8 0.6
8 0.5
3 g 0.4
0.3 0.2 0.1
0
 0.1
0.5
 1 25
1.5 0
2
2.5 3
3.5 4 4.5 5 5.5
0 0.2 0.4 0.6 0.8 1
0' 0
Figure 9. Specular helium reflectivity vs relative oxygen coverage. Solid and dashed lines are the calculated fits to the data using eq 17. Parameters obtained from the fit to the data:
ZO= 44.2 A2and B = 0.826. Experimentalconditions: TS= 525 K,Ei = 63 meV, and Bi = 0, = 45O.
timeresolved helium specular scattering to assess adsorbate coverage, reactant modulation depth, and even reaction kinetics when warranted. In this section we shall demonstrate for the CO oxidation reaction that helium reflectivity measurements conducted during modulated beam reactive scattering measurements can be used as a kinetic response "amplifier",gaining back much of the signal that was lost by using much smaller modulation depths than are commonly used in conventional MBRS measurements.
This extremely sensitive method for making in situ coverage and kinetic measurements of surface reactions utilizes specular He scattering to directly monitor adsorbate coverages. Poelsema and Comsa1012demonstrated that the large He scatteringcross sectionsfrom disordered adsorbates on low index surfaces results in a sharp attenuation of the specular He signal as adsorbate coverages increase. A relatively simple model accounts for the attentuation by assuming that the cross section for diffusely scattering the He wave can be associated with the (large) elastic scattering cross section that exists between the He atom and the adsorbate. Allowing for a random overlap, they derived a "lattice gas formula":
100is the specular He intensity from the bare surface, l e A is the specular He intensity at coverage B of species A, Z;A
is the cross section associated with that molecule for the
selected incident angle and beam energy, and q is the surface atom density. The ratio 18A/Imis defined here as the He reflectivity for that adsorbate coverage. As shown
in Figure 9, He reflectivity was calibrated against surface
coverage by utilizing thermal desorption or, in this case, titration to determine relative coverages. Here we dosed
a Rh(ll1) surface with 0 2 and monitored the He reflec
tivity for a roomtemperature, i.e., 63meV, He beam a t 45' incident angle and then we determined the relative
oxygen coverages at various dosing times by titrating the
adsorbed oxygen with CO and measuring the resultant
COZsignal. We can then plot He reflectivity against the relative 0 coverage, 0'0, where 1represents a saturated disordered 0 overlayer. Disordering, measured by the disappearance of diffraction peaks, occurs below 400 K with no new ordering at higher temperatures. Since the absolute 0 coverage (relative to the Rh atom density), Bo, was unknown, we fit our data to eq 1with do = BB'o where 0'0 is the measured coverage, and B is the absolute 0 coverage relative to the Rh atom density. Equation 17
fits our data up to 8'0 = 0.65, with 20 = 44.2 A2, q = 0.13
2572 Langmuir, Vol. 7,No. 11, 1991
0.1
& 0.08
z
%, 0.06
B
0.04 5.
0.02
0
0
0.2 f a '
0
 1 " I " " I ' " ' I ' " ' I " '
10
20
30
40
0.02
50
Time (mS)
Figure 10. COzintensityand He reflectivitywave forms for CO oxidation on 525 K Rh(ll1). The COz wave form,the solid line, was Fourier analyzed to give a rate constant,while the He wave form was analyzed to give reactantcoveragesduringthereaction. Thecombinationof informationderivedfrom thetwowaveforms allowed us to determine the coveragedependent reaction rate constant. For this case with 2's = 525 K and at the He reflec
tivity derived oxygen coverage of 0.350, we find R = 55 sl.
A+, and B = 0.826. Root et a1.22measured B = 0.83 using
XPS. A similar experiment for CO on R h ( l l 1 ) produced
Zco = 148 A2.23
The empirical calibrations, combined with the model
calculations, were used to determine the in situ, inetantaneous coverages of adsorbates in modulated molecular beam experiments. Here we present the example of CO oxidation reaction on Rh(ll1). Brown and Sibener13 determined the reaction kinetics for CO oxidation in the
low CO coverage regime by using a continuous 02 beam and a modulated CO beam. The rate expression for this
reaction is
d&o/dt = Sco(Qdco,T)~co  K(~o,~co,T)Qco (18)
Ico, = ~ ( ~ , , ~ c , ~ n ~ , ~ c , (19)
Zco, is the timedependent flux of COz leaving the surface; Bco and 00 are the timedependent CO and 0 coverages, respectively; Sco(Bo,Bco,T) is the coverage and tempera
turedependent sticking coefficient for CO; Zco is the incident CO flux;and K(Bo,Bco,T)is the coverageand tem
peraturedependent rate constant. For a sufficiently low CO intensity, 80 is constant and the kinetics follow a
pseudofirstorder behavior:
ddco/dt = ~ c o ( ~ o ,~K~'(Bcoo,neco (20)
zco, = wBo,neco
(21)
where K'(Bo,T) = K(B0,8co,T)Bo. K', the measured firstorder rate constant, was determined from the phase lag of the first Fourier component of the product COz wave
form (solid line) in Figure 10.
In order to determine the steadystate oxygen coverage as well as the coveragedependent sticking probability, a COHe mixture is used for the incident modulated beam.
lo is determined for this beam by extrapolating from hightemperature (>650 K)measurements where CO has an
insignificant residence time and thus no effect on the scattering intensity. During the modulated beam experiment, the rotatable mass spectrometer was changed to look at either the product COZintensity at normal or the
He intensity at specular. He reflectivity (dashed line) is
calculated by dividing the experimental He scattering intensities by the extrapolated intensities determined immediately prior to immediately after the modulated
(22) Root, T.W.;Schmidt,L.D.;Fisher, G.B.Surf.Sci. 1983,134,30.
Padowitz et al.
0.2
Time (mS)
Figure 11. COSintensityand 8co wave forms for CO oxidation on Rh(ll1). OCO was calculated from the in situ He reflectivity. For the experimental conditions used, we find that the CO coveragevaried from0to 1.2% duringthisscatteringexperiment. This 1.2% change is much less than the 35% oxygen coverage, clearly supporting the the assumption of pseudofirstorder kinetics. Asseeninthisfigure,desorption rateconstantsobtained from the C02 wave form and the timeresolved He reflectivity data are, within experimental uncertainty, identical.
beam kinetics measurements. Here a dual chopper wheel, with a square wave and timeofflight pattern, is used to modulate the mixed COHe beam. The He signal shown in Figure 10 therefore disappears at the midpoint of the COz wave since the chopper blocks the incident beam.
Sinceboth CO and 0are adsorbed on the surfaceduring the modulated beam experiment, there are two surface speciesto attenuate the He signal. The contribution from 0 was determined from the reflectivity a t low BCO or low Zco,, where there is an insignificant contribution from adsorbed CO and thus the He reflectivity is largest. Here Zeo/Zoo = 0.0825, which corresponds to 60 = 0.350. The exponential decay in He signal after 0.5 ms is due to the rise if Bco during the open chopper part of the modulation period. Solving explicitly for this part of the wave form
Zco can be measured from the direct beam flux, and K', the pseudofirstorder rate constant, can also be determined, as explained above. If Bco(t) can be determined from He reflectivity, as will be shown, and Zco is known,
then Sco(B0) is readily calculable from eq 22. Assuming that CO occupies random surface sites un
occupied by 0 and thus the area associated with the CO cross section overlaps randomly with 0 covered sites in the mixed overlayer, the cross section for small CO
coverages can be expressed
zeco,eo/b=o e x ~ (  ~ ~ : c o e c o /6(0l))
(23)
or
where l e o is the He signal from the 0covered surface and
Z.go,~coisthe He signal from the COand 0covered surface. Figure 11shows the comparison of the COz desorption wave form to the Bco wave form calculated from the He
wave form. The rate constant calculated from the Bco
wave form is identical, within experimental error, with the rate constant from the COz wave form, thus demonstratingthe validity of this new method. Sothe coveragedependent rate constants can be determined from the product wave forms and in situ He reflectivity measurements, or simply from the He scattering measurements by themselves, since the measurements monitor the time
Modulated Molecular Beam Scattering Methods
evolution of the surfacespecies. With the addition of flux meters to our machine to measure the incident reactant flux, we can use the He reflectivity derived reactant coverage to determine the coveragedependent sticking coefficients as well. By combining product wave form measurements with in situ He reflectivity measurements, we can measure and characterize the coveragedependent behavior of the kinetic parameters, the rate constants, coverages, and sticking coefficients for a wide range of incident reactant pressures and surfacetemperatures. This approach is general and will work well for any surface chemical system that has at most two wellcharacterized adsorbates on a low index surface.
IV. Conclusion
In this paper we have described several particularly usefulextensionsto conventionalMBRS techniques, which expand the range of reaction times and reaction conditions that can be studied with molecular beam techniques. The methods discussed employ independently adjustable, continuous sources of each reactant to establish steadystate conditions on the surface, which are then perturbed by a modulated flux of one reactant. In situ specular helium scatteringfrom the same modulated beam can then be used to assessor confirm coverages,modulation depths, sticking coefficients, and reaction rates.23 This approach allows us to explore nonlinear reaction kinetics using a very general combination of experimental and analytical strategies based on linearity in the limit of small perturbations around a steady state.
(23)Peterlinz, K.A.;Curtiss, T.J.; Sibener, S.J. J. Chem. Phys., in
prees.
Langmuir, Vol. 7, No. 11, 1991 2573
 + THhDis alinmditthheasmnoorwe bceoemnprleicaactheeddHfozr thDe2ei+mp02lePH2HtDDO2
+  reactions, adding to the success of the CO 02 COZ
system. When linearized, the coupled differential equations that described reaction kinetics can be solved by perturbation theory, yieldingphysicalinsight intoreaction mechanismsforarbitrarily chosen andprecisely measured surface coverage regimes. Both HD and HDO reactions have been moved from nonlinear to linear regimes by reducing the intensity of the modulated flux relative to continuous reactant flux, and the reactant coveragesused in the COz reaction have been calibrated and measured precisely with in situ He scattering.
To datewe have been concerned with controlling surface reactant coverages in order to examine reaction rates. At a more fundamental level, chemisorption energies have been observed to change with coverage.16 In the future our focus will move from kinetics to dynamics, by probing the disposal of reaction exoergicity into product degrees of freedom as a function of coverage. This will allow us to examine in greater detail howadsorbate density changes the potential energy surfaces that govern heterogeneous reactions.
Acknowledgment. This work was supported in part
by the Officeof Naval Research and by the NSF Materials
Research Laboratory program a tthe University of Chicago (NSFDMR8819860).
Registry No. Hz,1333740;H,12385136;CO,630080; Rh, 7440166.
Langmuir 1991, 7, 25662573
New Modulated Molecular Beam Scattering Methods for Probing Nonlinear and CoverageDependentReaction Kinetics at Surfaces
D. F. Padowitz,+K. A. Peterlinz, and S. J. Sibener’
The James Franck Institute and The Department of Chemistry, The university of Chicago, Chicago, Illinois 60637
Received April 10, 1991.I n Final Form: July 29, 1991
fl A new three molecularbeam arrangement is introduced that expandsthe range and power of modulated
beam reactive scattering for studyin complex kinetics at surfaces. This paper presents two types of kineticmeasurementsthat utilizethis t reebeamarrangement. The firstmeasurementsusetwocontinuous,
independently adjustable molecular beams to establish steadystate reaction conditions,while a weaker modulated third beam induces small concentration perturbations around the selected steady state. This techniquepermits experimentallinearizationof nonlinearkinetics over a wide range of conditions,allowing us to explore the global behavior of reactions, determine coveragedependent rate constants, and isolate individual elementary steps from complex reaction mechanisms. We illustrate these capabilities with preliminaryresults for the oxidation of hydrogen to water and for the recombination kineticsof hydrogen on the Rh(111)surface. The secondgroup of measurementsuses timeresolvedspecularhelium scattering as a sensitive in situ probe of both adsorbate coverage and coveragedependent surface kinetics. The oxidation of CO on Rh(ll1) under pseudofirstorderconditions is examined with this new kinetic probe.
I. Introduction
Molecular beam scattering techniques have firmly established themselves among the major tools for studying the kinetics and dynamics of gassurface interactions. Modulated molecular beam relaxation spectrometry (MBRS), a descendent of Eigen’s pioneering work on chemical relaxation,’ extracts kinetic information by analyzing product evolution in response to a timevarying reactant beam flux. Relaxation techniques based on this principle are now widely used in reactive scattering to study heterogeneous chemical reactions.
The evaluation of surface kinetic data by transform analysis of modulated beam measurements was introduced in an early paper by Foxon, Boudry, and Joyce and has since been well studied.2 Transform analysis, widely used in electrical engineering and linear systems theory, determines the properties of a system by its response to a known input. Fourier or Laplace transforms are used to deconvolute a “transfer function”, characteristic of the system, from the observed output to a given periodic or transient input. Transform analysis of MBRS data is straightforward when the product response to changing reactant flux islinear.3 Accordingly,most studies reported to date have been on simplelinear reactions, suchassinglecomponent adsorptiondesorption a t low coverage, have used linear approximations for weakly nonlinear systems, orhave workedin a pseudofirstorder regimeovera limited coverage range.’+
In this paper we introduce several particularly useful extensions to conventional MBRS techniques, which expand the range of reaction types and reaction conditions
t Current address: Department of Chemistry, University of California, Berkeley, CA 94720.
(1)Eien, M. In Technique of Organic c h e m i e t ~I;nterscience: New
York, 1963; Vol. VIII, Part 11, p 901. ((23))FCohxaonn,,CH..TC.;.;BWoueidnrbye,rMg,.WR..;HJo. yJc.eC, Bhe.mA.. PShuyr8f.. S1e9i7. 71,96764,4,414766.9. (4) Olanfer, D. R.; Ullman, A. Int. J. Chem. Kinet. 1976,8,625. (5) Schwarz,J. A.; Madix, R. J. Surf. Sci. 1974, 46, 317. (6) Jones,R.He;Olander,D.R.;Siekhaus,W.J.;Schwarz,J. A. J. Vac.
Sci. Technol. 1972, 9, 1429. (7) Sawin, H. H.; Merrill, R. P. J. Vac. Sei. Technol. 1981, 19, 40. (8)DEvelyn, M. P.; Madix, R. J. Surf. Sei. Rep. 1984, 3, 413.
07437463/91/24072566$02.50/0
that can be studied with molecular beam techniques. Our ultimate goal is to study heterogeneous reactions, in the linearized limit, under arbitrarily chosen and precisely defined reactant coverage regimes. Our measurements are all based on the experimental strategy of performing kinetic measurements by observing the relaxation from small density perturbations, which we have experimentally introduced around carefully defined steadystate reaction conditions.
Most chemical systems are not linear. An elementary bimolecularstepin a chemicalreaction involvestwo species and the rate of reaction depends on the product of their concentrations. If both concentrations change with time, the rate expression for the mechanism will be nonlinear. Areaction willalsobe nonlinear if the rate constants change with concentration. This is frequently true for surface reactions, in which adsorption probabilities, preexponential factors, and activation energies often depend on coverages.
Besidesprohibiting experimentaldeconvolutionand the use of Fourier transform analysis methods that depend on linear response, nonlinearity frequently presents other severe mathematical difficulties in analyzing chemical reaction mechanisms. A model reaction mechanism yields a set of ordinary differential equations describing the experimentallyobservablerates. For nonlinear differential equations there are no general mathematical techniques providing analytical solutions. Numerical integration of nonlinear differential equationsmay be difficult since rate constants for elementary reaction steps may range over many orders of magnitude, leading to a set of “stiff“ differential equations, creating numerical instabilities.
However,it isoften possible experimentallyto constrain the reaction to a linear regime. If this is done, linear transform analysis may be applied to the experimental data,and the differential equations for a postulated mech
anism may be solved by series expansions or perturbation techniques, allowingthe sought after kinetic information to be obtained. The pioneers in the field, including Foxon, Olander, Madix, Weinberg, and their coworkers,28were aware of the problems of nonlinear kinetics and examined linearized MBRS, but the theory outstripped the available
0 1991 American Chemical Society
Modulated Molecular Beam Scattering Methods
(a) Nonlinear Kinetics
Langmuir, Vol. 7, No. 11, 1991 2567
(b) “Linearized”Kinetics
A
B
C
A
Reactant Beam
Flux
ReactantBeam Flux modulatedbeam + conlimous beam
Figure 1. Differentmodulation schemesthat canbe used duringsurfacereactivescatteringexperiments. Part a is of the conventional type, with reactant density spanninga large range of coverage duringthe modulation cycle. Part b, after Foxon et al.? which utilizes small perturbationsof reactant densityduringeach modulation cycle, portrays a schemethat is now possible with our new threebeam approach.
instrumentation. In succeeding years the numerous successes of MBRS obscured its limitations, which have onlyrecently been reconsidered asmore complexsituations are encountered.9
Let us now consider the general problem of conducting modulated beam experiments on nonlinear systems and discuss a general solution to the problem. Figure l a presents a schematizedview of a conventionalexperimental arrangement. When a small modulated beam flux is used (scheme A, Figure la), the response may be linear but is restricted to a lowcoverage regime. Depending on the shape of the order plot, there may be little signal at all. When an intense modulated beam is used (scheme B, Figure la), a wide coverage range is spanned during each modulation cycle, producing a nonlinear response that is difficult to analyze. This“strong modulation” schemecan further complicate matters by inducing timedependent behavior in other reactants and reaction intermediates. Stated simply, one of the main consequencesof thistypical method is that the measured kinetics represent a convolution over all coverages on the surface.
Ideally, one would like to avoid these problems by sampling the kinetics under isosteric conditions. The straightforward solution is to establish a steadystate reaction on the surface with arbitrarily chosen adsorbate coverages and to then superimpose a small modulation about the selected steady state. Such a scenario is shown in Figure lb. Modulation schemesAC here, i.e., for three different coverages,all involvesufficientlysmallexcursions from the preset steady state that relaxation is linear. If this linearization could be performed experimentally for different steadystate conditions, then reaction mechanisms or coverage dependencies of the rate constants may be determined over the global range of conditions for the nonlinear reaction.
This paper introduces a new three molecular beam scattering arrangement, which allows us to directly im
(9) Engstrom,J. R.;Weinberg, W.H.J . Chem. Phys. 1987,87,4211.
Molecular Beams H2
H2 (D2)
02
Mass Spectrometer
Rh(ll1) Surface
Figure 2. Multiplesource modulated molecular beam reactive scattering. Two continuousbeam sourcesestablishsteadystate surface conditions, which are perturbed by a third modulated beam in order to probe kinetics.
plement the approach described above and provides greater versatility for new methods. The scheme is depicted in Figure 2. We use two continuous, independently adjustable beams to establish a selected steadystate on the surface, while a weaker modulated third beam induces small concentration perturbations around the selected steady state. This is not the only way to linearize an MBRS experiment, but it is simple and direct, retains the advantages of molecular beam sources,and offersgreat flexibility in experimental design. In addition to linearization by means of weak modulation, we can use the continuous beams to move away from the nearzero coverage limit in which molecular beam experiments are sometimes forced to work, and to widely vary steadystate surface coverages, which is important for determining coveragedependent activation energies and preexponential factors. Most significantly, by controlling reactant and intermediate coverage regimes, sometimes using isotopic substitutions in the beams, we can isolate the individualelementary steps that composecomplexreaction mechanisms.
2568 Langmuir, Vol, 7, No. 11, 1991
This experimental approach is not without difficulties. The two most significant complications are extreme sensitivityrequirements and the need for in situ coverage assessment. Signal levels will be smaller than those
encountered in conventional MBRS measurements since
the modulation depth (Le., the surface coverage change induced by the incident modulated molecular beam) has
+ + . to be reduced in order to reach the linearized limit. This + D2 HD
limit has now been reached for the simple H2 + and the more complicated H2 D2 0 2 HDO reactions, aswillbe discussed later in this paper. Thesetwo problems have now been addressed in our group by using timeresolved helium specular scattering to assess adsorbate coverage, to determine the modulation depth, and to directly measure reaction kinetics. The basis for this technique is the extraordinarily large scattering cross section that adsorbates have for reducing specular helium reflectivity. The origin of this effect has been well documented1°12and is due to the longrange attractive part of the adsorbatehelium interaction potential. In this paper we shall demonstrate for the CO oxidation reaction that helium reflectivity measurements conducted during modulated beam reactive scattering measurements can be used as a kinetic response uamplifier”,gaining back much of the signal that was lost by using much smaller modulation depths than are commonly used in conven
tional MBRS measurements.
11. Experimental Section
For this work, a third beam source was installed in the UHV surfacescatteringinstrumentused in our previousstudiesof CO oxidationonRh(lll).13JT4heapparatusconsistsof a removable source chamber containing three supersonic molecular beam sourcesand a main chamberwith sampleand manipulator,mass spectrometerdetector,residualgas analyzer,Auger spectrometer, and sputter ion gun.
Thesourcechamber is dividedlaterallyintotwo sections,each with three differential regions of diffusion pumping before the main chamber. The left beam is in one section and the center and right beams together are in the other. The supersonic expansion in each beam originated from 100pm pinholes and was formed by a skimmer, collimator,and shutter. The center beam was modulated by a rotating chopper wheel with a fourslot50% total dutycyclepattern, or a dual pattern chopperwith two 25% slots and two 1%slits for incident beam timeofflight measurements.
On entering the main chamber,the left and right beams were defined by 80milaperturesand the center beam was defined by a 40milaperture. Thebeams passed througha finaldifferential pumpingregionbeforethe scatteringregion. Themain chamber scattering region was pumped primarily by a 400 L/s ion pump. The base pressure in the scattering region was 5 x Torr, rising to 6 x [email protected] at the highest beam fluxes.
The distance from chopper to sample was 21.21 cm. The left and right beams were 15’ off axis. The sample normal was usually at 45O to the beam axis, resulting in roughly 5 X 7 mm elliptical spots on the sample, which overlapped the center 2.5 X 3.25 mm modulated beam spot. Maximum side beam flues were on the order of 10 langmuir/s, or 106 Torr/cm2.
A rhodium singlecrystal sample was cut and polished by standard techniques. The surface was within 1/2O of the FCC (111)plane,asdeterminedby LaueXraydiffraction. Thesurface was prepared in UHV by cycles of Ar ion sputteringand heating in oxygen and was kept very clean by oxidation and reduction
(10) Poelsema, B.; Comsa, G. Scattering of Thermal Energy Atoms from Disordered Surfaces; Springer Tracta in Modern Physics 115; SpringerVerlag: Berlin, 1989.
(11) Poelsema, B.;Palmer, R. L.; Comsa, G. Surf. Sci. 1984, 136, 1. (12) Poelsema,B.;Comsa, G. Faraday Discuss. Chem. SOC1. 985,80,
247.
(13) Brown, L.S.;Sibener,S. J. J. Chem. Phys. 1988,89, 1163. (14)Brown, L. S.;Sibener,S. J. J. Chem. Phys. 1989, 90,2807.
Padowitz et a1,
duringtheexperimentalreaction. Surfacecleanlinesswasverified with Augerspectroscopyand,subsequently,with specularhelium reflectivity measurements.
The detectoris an Extranuclearquadrupolemass spectrometer, doubly differential pumped by two 50 L/s ion pumps and an additional Balzers 50 L/s turbomolecular pump on the ionizer region. Transmitted ions are detected with a Venetion blind electron multiplier, suitable for pulse counting. Angular acceptance of the detector is lo. Flight path from sample to ionizer is 14.45cm. At normal, the detector views a spot of =2.5mm diameter.
Producttime of arrivalwave formswere acquiredby a custom multichannel scalar system triggered by the photodetector on the chopperwheel assembly,collecting255 channelsof 520 ps. For the water reaction,goodqualitywave formsusuallyrequired 1060 min of signal averaging. A typical run of 20 min at 400 Hz chopper speed would be 5 x 106 chopper periods or “shots*, giving 5 X l(r counts/channel with signal to background of 2, and S / N of 200.
111. Results
A. ThreeBeam Scattering and Mechanism Linearization. We being this section by examining the oxidation of hydrogen to form water on the Rh(ll1) surface. This reaction is relatively complicated and has attracted considerable attention.15
There are several sources of nonlinearity in this mechanism. Hydrogen sticking probabilities and desorption rate parameters depend on both hydrogen and oxygen coverages.16J7 As the reaction probably proceeds by
sequential H + 0 and H + OH addition or OH dispro
portionation, the differential equations for the overall mechanism contain several terms that are quadratic in hydrogen coverage. These quadratic terms prohibit general solutions of the rate equations for this mechanism.
Experimentally, an examination of the rate of water production for different hydrogen and oxygen beam intensities (Figure 3) shows that no global “order” can be assigned to the reactants. The data for this coveragedependent order map were obtained with three beams. Continuous Hz and 0 2 beam pressures were varied to establish the desired steadystatereaction. The third molecular beam, which in this instance wasH2,wasmodulated at 100 Hz for phasesensitive detection. The rate of product formation rapidly saturates as hydrogen flux is increased for all oxygen fluxes measured. For high hydrogen beam intensities,the reaction islinear in oxygen coverage, but at low hydrogen flux, the yield is nonmonotonic, first increasing, then decreasing again as oxygen flux increases.
Time domain measurements for water production
further emphasize the nonlinear nature of MBRS wave
formstaken with arbitrarilychosen modulation strengths. With three beam sources, we are able to combine a modulated H2 beam with continuous beams of Dz and 0 2 to produce three isotopically substituted forms of water. This arrangement was shown in Figure 2. Figure 4 shows mass spectrometer signal vs time wave forms for the three products of this reaction. The HzO wave form has the usual symmetricform. The HDO product, however,shows a highly asymmetric wave form, with a rapid rise, sloping top, and long decay. As will be shown below, this appearance is the same as that for HD produced in an
(15) Norton,P. R. In The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis;King,D.A.,Woodruff,D. P.,Eds.;Fundamental
Studies of Heterogeneous Catalysis; Elaevier: New York, 1982; Vol. 4. (16) Thiel, P. A.; Yaks, J. T., Jr.; Weinberg, W. H. Surf. Sci. 1979,90,
121.
(17)Yaks,J.T.,Jr.; Thiel, P. A.; Weinberg,W. H. Surf. Sci. 1979,84,
427.
Modulated Molecular Beam Scattering Methods
Langmuir, Vol. 7, No. 11, 1991 2569
8
. e
Figure3. Steadystate water production vs reactant beam flux.' The pressure rise in the chamber calibrates the hydrogen and oxygen beam fluxes. No global "order" can be assigned to the reactants based on the relationship of reactant flux to H20
production.
I

2140
0 50 100 150 200 250
channel
Figure 5. HDO and HD wave forms. The essential features of the asymmetric HDO wave form are also seen in an HD wave form, indicating that the origin of the nonlinearity is similar in both reactions.
neither D2 nor 0 2 was modulated in this experiment. On first glance one would expect a continuous background signal of D20 to be produced from this arrangement. The negative modulation gives a direct measure of the extent to which the modulated H2 beam perturbs the steadystate coverages of 0, D, and OD.
The origin of the nonlinearity and the mathematical analysis can be best illustrated in the simpler HD recombination reaction. Figure 5 shows that the HD and HDO wave formsshare the same asymmetric shape. This can be quantitatively understood by examining the reaction mechanism and by then numerically simulating the result. The mechanism for this reaction can be modeled as
I 1 .
a
.
9  I 1.
a =i
0 50 100 150 200 250
channel
Figure 4. Mass spectrometer timeofarrivalwave forms of the
products from modulated H2 and continuous D2 and 0 2 beams reactingon Rh(111)at 665 K. Thechannel time is 10ps/channel.
The ratio of modulated to continuous beams is high. The HDO response is nonlinear and the D20 background shows negative modulation. Note that neither D2nor 0 2 beam is modulated and that the D2O modulation is solely due to perturbation of the steadystatesurface coverages of 0,D, and OD by the modulated
Hz beam.
overdrivenmodulation cycle. This nonlinearity is due to the changing coverage of D species on the surface during the H2 pulse. The most remarkable wave form is that for D20, which shows negative modulation. Remember that
H
+
D
Kd
+ HD,
(3)
Thismechanismcanbe simulatedby numericalintegration of the system:
d=[H1 2P~z(l([HI k [D]))  2kd[HI2 kd[HI[D] (4)
dt
d[D1  WD,(~([HI 4 [Dl)) 2kd[DI2 kd[HI[D] (5)
dt
d[HDl,/dt = kd[Hl [Dl
(6)
where Pincorporates the incident beam flux and the zerocoverage sticking coefficient.
The qualitative behavior does not depend on the rate constants, only on the ratio of the continuous and modulated fluxes. Figure 6 shows the simulated results, and correspondingFouriertransfer function plots (showing the product phase and amplitudefor allinputfrequencies), for three different ratios of the modulated D2 intensity to
2570 Langmuir, Vol. 7, No. 11,1991
HD Waveforms Transfer Fn.
D0.5 0.4
0.3
0.2
0.1
0.0 0.4 0.8
0.5
0.4
0.3 5 5I
h
0.2
Y
0.1
~ 0.0 0.4 0.8 0.0
0.5
0.4
0.3
0.2
0.1
0.0 0.4 0.8 5 6 1 8 910
msec
Figure6. HD simulationvs modulationstrength. As the strong modulation is reduced, the wave form becomes more symmetric, the odd harmonics of the transfer function approach the singlestep arc, and the even harmonics vanishall indications that the kinetics are approaching a linear regime.
the continuous H2 intensity. The 5:l (strong modulation limit) plot shown in Figure 6 resembles the HD wave form of Figure 5. Examinationa of the simulation shows that the rapid rise is due to the increase in adsorbed H, the sagging top to decline in adsorbed D. The long tail is the recovery of D as H decreases;the product of the two drops less quickly than H alone. The transfer function, shown to the right, does not fall on the "semicircle" as a singlestep reactionwouldPV6 More importantly, significanteven harmonics appear. Since the reactant beam modulated by the chopper is a square wave, it contains only odd harmonics, and the presence of even frequencies in the response indicatesnonlinearity. In the strongmodulation limit, as the modulated D coverage increases the steadystate D coverage dropsthe overallkinetics are therefore nonlinear as the kinetic equations depend on the product of two timevarying quantities. Reducing the relative intensity of the modulated beam reduces the nonlinearity, as seen in the second and third panels of Figure 6. Here, we clearly see that as the modulation is decreased, the wave form becomes symmetric,the transfer function approaches the singlestep arc, and the even harmonics vanish. In this weak perturbation limit the reaction becomes linear.
Finally, we perform a perturbation theory analysis of the data to extract the desired kinetic information from the linearized wave forms. Note that a perturbation treatment allows us to analytically solve the differential equationsforthe HD mechanism,yieldingphysicalinsight into the elementary steps of the reaction. The reaction is second order in hydrogen concentration, and thus nonlinear:
Padowitz et a1.
8
H, i= 2H
(7)
k
The sticking coefficient is s, the desorption rate k . Let the reactant flux be P and the surface concentration of hydrogen [HI, then
d[H]/dt = 2sP  2k[HI2
(8)
+ + Now considera smalltimedependentfluxsuperimposed
on a continuous background P = po tp(t). We wish to obtain a solution in the form [HI = h~ th(t). We then have
+ + d(h0 + eh(t)) = 25030 tp(t)) 2k(h0 th(t))2 (9) dt
9+  d h w = 2sp0+ c2sp(t) 2 k h t  4kch&(t) 
dt
dt
2 k ~ ~ h ((t1)0~)
If this is to hold for all values oft over some interval, then the coefficientsof eachpower of e must equate. For zeroth order we have the steadystate component
dho/dt = 0 = 28pO  2 k h t
(11)
This yields ho = ( s p ~ / k ) l t/h~e, steadystate surface coverage. The firstorder perturbation is
dh(t)/dt = 2sp(t) 4kh&(t)
(12)
This is the desired linear response. Substituting the steadystate ho
dh(t)/dt = 2sp(t)  4 k h ( t ) d G
(13)
or
dh(t)/dt 2sp(t) k'h(t)
(14)
where the effective rate constant k' = 4(skp0)'/~.In Arrhenius form we have
A' exp(E'/kT) = 4 4 s p 4 exp(E/kT) (15)
with A' = 4(sp0A)'/~and E' = E/2. Finally, we relate the measured rates for the linearized system to the actual rates:
A = Af2/16sp0 E = 2E'
(16)
The activation energy is obtained directly, the preexpo
nential requires knowledge of the beam flux and sticking coefficient, or of the steadystate coverage ho. A similar analysis may be found in ref 5.
An Arrhenius plot of our experimental pseudofirstorder rates is shown in Figure 7. The pseudofirstorder rate constants are E', = 10.0 kcal/mol, A' = lo7s'. The true secondorder constants are Ea = 20 kcal/mol, A =
cm2/s. This is in good agreement with earlier [email protected]
We now return to the water reaction and demonstrate that the linearized limit can also be reached for this more
complicated system (before the signal becomes too small to d'etect!). Figure 8 is a plot of the even harmonics in the Fourier transform of the HDO product wave form.' Since the reactant beam modulated by the chopper is a square wave, it contains only odd harmonics, and the presence of even frequencies in the response indicates nonlinearity.
As the strength of the modulation is reduced, the per
(18)Engle,.T.;Kuipem, H.Surf.Sci. 1979,90,162. (19) Verhelj, L.K.; Hugenschmidt,M.B.;Anton, A. B.;Pole", B.; Comsa, G.Surf.Sci. 1989,210,1.
Modulated Molecular Beam Scattering Methods
lb
1
1 /T
Figure 7. HD order rates for
HAr+rheDniruescopmlobt.inTathioenlinaerearpizleodtteodr
pseudofirstfor two runs
differing slightly in continuous beam flux. The true second
order rates are derived from them as described in the text: E,
= Z'.th;e preexponentialis approximately cm2/s,using an
estimated continuousbeam flux of 10 ML/s.
% modulation
Figure 8. Linearizationof the HDO response. The amplitude of the even harmonics of the HDO wave form indicates nonlinearity. Modulation is measured as the ratio of the modulated beam to the sum of modulated H2 and continuous D2 beams. With smallermodulation,the amplitudesof the even harmonics systematically decrease and approach zero as the response becomes linear.
turbation measured by the depression of D2O production decreases and the amplitudes of the even harmonics decrease. Modulation is measured as the ratio of the modulated beam to the sum of modulated H2 and continuous D2 beams. The perturbation of steadystate surface coverage can be determined from the transient depression of the continuous D2O background. The perturbation decreases from 6096 to less than 596 as the intensity of the modulation is reduced. With smaller perturbations, the response becomes more linear. The even harmonics approach zero, clearly demonstrating the experimental linearization of the kinetics in the limit of
+ weak modulations. More extensive results for hydrogen
oxidation20and H2 D2 recombination21on Rh(ll1) are forthcoming.
B. Helium Reflectivity as a n i n S i t u Probe of
Surface Coverage and Kinetics. As shown above, the use of three beam sources provides excellent control of reactant flux, but an in situ method of assessing adsorbate surface concentration is also necessary in order to extract the desired kinetic rate constants. Furthermore, signal levels in the linearized limit can be extremely low since much smaller modulation depths are used than in conventional singlebeam MBRS experiments. These two problems have now been addressed in our group by using
(20) Padowitz, D. F.; Sibener, S. J. Surf.Sci. 1991, 254, 125. (21) Padowitz,D.F.;Sibener,S.J.J.Vuc.Sci. Technol.A 1991,9,2289.
Langmuir, Vol. 7, No. 11, 1991 2571
1
0
0.9 3 0,s
bIC 0.7
E 8 0.6
8 0.5
3 g 0.4
0.3 0.2 0.1
0
 0.1
0.5
 1 25
1.5 0
2
2.5 3
3.5 4 4.5 5 5.5
0 0.2 0.4 0.6 0.8 1
0' 0
Figure 9. Specular helium reflectivity vs relative oxygen coverage. Solid and dashed lines are the calculated fits to the data using eq 17. Parameters obtained from the fit to the data:
ZO= 44.2 A2and B = 0.826. Experimentalconditions: TS= 525 K,Ei = 63 meV, and Bi = 0, = 45O.
timeresolved helium specular scattering to assess adsorbate coverage, reactant modulation depth, and even reaction kinetics when warranted. In this section we shall demonstrate for the CO oxidation reaction that helium reflectivity measurements conducted during modulated beam reactive scattering measurements can be used as a kinetic response "amplifier",gaining back much of the signal that was lost by using much smaller modulation depths than are commonly used in conventional MBRS measurements.
This extremely sensitive method for making in situ coverage and kinetic measurements of surface reactions utilizes specular He scattering to directly monitor adsorbate coverages. Poelsema and Comsa1012demonstrated that the large He scatteringcross sectionsfrom disordered adsorbates on low index surfaces results in a sharp attenuation of the specular He signal as adsorbate coverages increase. A relatively simple model accounts for the attentuation by assuming that the cross section for diffusely scattering the He wave can be associated with the (large) elastic scattering cross section that exists between the He atom and the adsorbate. Allowing for a random overlap, they derived a "lattice gas formula":
100is the specular He intensity from the bare surface, l e A is the specular He intensity at coverage B of species A, Z;A
is the cross section associated with that molecule for the
selected incident angle and beam energy, and q is the surface atom density. The ratio 18A/Imis defined here as the He reflectivity for that adsorbate coverage. As shown
in Figure 9, He reflectivity was calibrated against surface
coverage by utilizing thermal desorption or, in this case, titration to determine relative coverages. Here we dosed
a Rh(ll1) surface with 0 2 and monitored the He reflec
tivity for a roomtemperature, i.e., 63meV, He beam a t 45' incident angle and then we determined the relative
oxygen coverages at various dosing times by titrating the
adsorbed oxygen with CO and measuring the resultant
COZsignal. We can then plot He reflectivity against the relative 0 coverage, 0'0, where 1represents a saturated disordered 0 overlayer. Disordering, measured by the disappearance of diffraction peaks, occurs below 400 K with no new ordering at higher temperatures. Since the absolute 0 coverage (relative to the Rh atom density), Bo, was unknown, we fit our data to eq 1with do = BB'o where 0'0 is the measured coverage, and B is the absolute 0 coverage relative to the Rh atom density. Equation 17
fits our data up to 8'0 = 0.65, with 20 = 44.2 A2, q = 0.13
2572 Langmuir, Vol. 7,No. 11, 1991
0.1
& 0.08
z
%, 0.06
B
0.04 5.
0.02
0
0
0.2 f a '
0
 1 " I " " I ' " ' I ' " ' I " '
10
20
30
40
0.02
50
Time (mS)
Figure 10. COzintensityand He reflectivitywave forms for CO oxidation on 525 K Rh(ll1). The COz wave form,the solid line, was Fourier analyzed to give a rate constant,while the He wave form was analyzed to give reactantcoveragesduringthereaction. Thecombinationof informationderivedfrom thetwowaveforms allowed us to determine the coveragedependent reaction rate constant. For this case with 2's = 525 K and at the He reflec
tivity derived oxygen coverage of 0.350, we find R = 55 sl.
A+, and B = 0.826. Root et a1.22measured B = 0.83 using
XPS. A similar experiment for CO on R h ( l l 1 ) produced
Zco = 148 A2.23
The empirical calibrations, combined with the model
calculations, were used to determine the in situ, inetantaneous coverages of adsorbates in modulated molecular beam experiments. Here we present the example of CO oxidation reaction on Rh(ll1). Brown and Sibener13 determined the reaction kinetics for CO oxidation in the
low CO coverage regime by using a continuous 02 beam and a modulated CO beam. The rate expression for this
reaction is
d&o/dt = Sco(Qdco,T)~co  K(~o,~co,T)Qco (18)
Ico, = ~ ( ~ , , ~ c , ~ n ~ , ~ c , (19)
Zco, is the timedependent flux of COz leaving the surface; Bco and 00 are the timedependent CO and 0 coverages, respectively; Sco(Bo,Bco,T) is the coverage and tempera
turedependent sticking coefficient for CO; Zco is the incident CO flux;and K(Bo,Bco,T)is the coverageand tem
peraturedependent rate constant. For a sufficiently low CO intensity, 80 is constant and the kinetics follow a
pseudofirstorder behavior:
ddco/dt = ~ c o ( ~ o ,~K~'(Bcoo,neco (20)
zco, = wBo,neco
(21)
where K'(Bo,T) = K(B0,8co,T)Bo. K', the measured firstorder rate constant, was determined from the phase lag of the first Fourier component of the product COz wave
form (solid line) in Figure 10.
In order to determine the steadystate oxygen coverage as well as the coveragedependent sticking probability, a COHe mixture is used for the incident modulated beam.
lo is determined for this beam by extrapolating from hightemperature (>650 K)measurements where CO has an
insignificant residence time and thus no effect on the scattering intensity. During the modulated beam experiment, the rotatable mass spectrometer was changed to look at either the product COZintensity at normal or the
He intensity at specular. He reflectivity (dashed line) is
calculated by dividing the experimental He scattering intensities by the extrapolated intensities determined immediately prior to immediately after the modulated
(22) Root, T.W.;Schmidt,L.D.;Fisher, G.B.Surf.Sci. 1983,134,30.
Padowitz et al.
0.2
Time (mS)
Figure 11. COSintensityand 8co wave forms for CO oxidation on Rh(ll1). OCO was calculated from the in situ He reflectivity. For the experimental conditions used, we find that the CO coveragevaried from0to 1.2% duringthisscatteringexperiment. This 1.2% change is much less than the 35% oxygen coverage, clearly supporting the the assumption of pseudofirstorder kinetics. Asseeninthisfigure,desorption rateconstantsobtained from the C02 wave form and the timeresolved He reflectivity data are, within experimental uncertainty, identical.
beam kinetics measurements. Here a dual chopper wheel, with a square wave and timeofflight pattern, is used to modulate the mixed COHe beam. The He signal shown in Figure 10 therefore disappears at the midpoint of the COz wave since the chopper blocks the incident beam.
Sinceboth CO and 0are adsorbed on the surfaceduring the modulated beam experiment, there are two surface speciesto attenuate the He signal. The contribution from 0 was determined from the reflectivity a t low BCO or low Zco,, where there is an insignificant contribution from adsorbed CO and thus the He reflectivity is largest. Here Zeo/Zoo = 0.0825, which corresponds to 60 = 0.350. The exponential decay in He signal after 0.5 ms is due to the rise if Bco during the open chopper part of the modulation period. Solving explicitly for this part of the wave form
Zco can be measured from the direct beam flux, and K', the pseudofirstorder rate constant, can also be determined, as explained above. If Bco(t) can be determined from He reflectivity, as will be shown, and Zco is known,
then Sco(B0) is readily calculable from eq 22. Assuming that CO occupies random surface sites un
occupied by 0 and thus the area associated with the CO cross section overlaps randomly with 0 covered sites in the mixed overlayer, the cross section for small CO
coverages can be expressed
zeco,eo/b=o e x ~ (  ~ ~ : c o e c o /6(0l))
(23)
or
where l e o is the He signal from the 0covered surface and
Z.go,~coisthe He signal from the COand 0covered surface. Figure 11shows the comparison of the COz desorption wave form to the Bco wave form calculated from the He
wave form. The rate constant calculated from the Bco
wave form is identical, within experimental error, with the rate constant from the COz wave form, thus demonstratingthe validity of this new method. Sothe coveragedependent rate constants can be determined from the product wave forms and in situ He reflectivity measurements, or simply from the He scattering measurements by themselves, since the measurements monitor the time
Modulated Molecular Beam Scattering Methods
evolution of the surfacespecies. With the addition of flux meters to our machine to measure the incident reactant flux, we can use the He reflectivity derived reactant coverage to determine the coveragedependent sticking coefficients as well. By combining product wave form measurements with in situ He reflectivity measurements, we can measure and characterize the coveragedependent behavior of the kinetic parameters, the rate constants, coverages, and sticking coefficients for a wide range of incident reactant pressures and surfacetemperatures. This approach is general and will work well for any surface chemical system that has at most two wellcharacterized adsorbates on a low index surface.
IV. Conclusion
In this paper we have described several particularly usefulextensionsto conventionalMBRS techniques, which expand the range of reaction times and reaction conditions that can be studied with molecular beam techniques. The methods discussed employ independently adjustable, continuous sources of each reactant to establish steadystate conditions on the surface, which are then perturbed by a modulated flux of one reactant. In situ specular helium scatteringfrom the same modulated beam can then be used to assessor confirm coverages,modulation depths, sticking coefficients, and reaction rates.23 This approach allows us to explore nonlinear reaction kinetics using a very general combination of experimental and analytical strategies based on linearity in the limit of small perturbations around a steady state.
(23)Peterlinz, K.A.;Curtiss, T.J.; Sibener, S.J. J. Chem. Phys., in
prees.
Langmuir, Vol. 7, No. 11, 1991 2573
 + THhDis alinmditthheasmnoorwe bceoemnprleicaactheeddHfozr thDe2ei+mp02lePH2HtDDO2
+  reactions, adding to the success of the CO 02 COZ
system. When linearized, the coupled differential equations that described reaction kinetics can be solved by perturbation theory, yieldingphysicalinsight intoreaction mechanismsforarbitrarily chosen andprecisely measured surface coverage regimes. Both HD and HDO reactions have been moved from nonlinear to linear regimes by reducing the intensity of the modulated flux relative to continuous reactant flux, and the reactant coveragesused in the COz reaction have been calibrated and measured precisely with in situ He scattering.
To datewe have been concerned with controlling surface reactant coverages in order to examine reaction rates. At a more fundamental level, chemisorption energies have been observed to change with coverage.16 In the future our focus will move from kinetics to dynamics, by probing the disposal of reaction exoergicity into product degrees of freedom as a function of coverage. This will allow us to examine in greater detail howadsorbate density changes the potential energy surfaces that govern heterogeneous reactions.
Acknowledgment. This work was supported in part
by the Officeof Naval Research and by the NSF Materials
Research Laboratory program a tthe University of Chicago (NSFDMR8819860).
Registry No. Hz,1333740;H,12385136;CO,630080; Rh, 7440166.