1942
J. Phys. Chem. 1993,97, 1942-1951
Mechanism, Configurational Degeneracy, and Mass Action in Transition-State Rate Functions for Adsorption-Desorption Reactions A. Brad Anton School of Chemical Engineering, Cornell University, Ithaca, New York 14853-5201 Received: April 6, 1992; In Final Form: December 28, 1992
A discussion of classical transition-state theory for adsorption and desorption rates is presented, the purpose
of which is to demonstrate through several examples how the reaction mechanism and the configurational degeneracy of the adsorbed layer conspire to determine mass actions in rate functions and the relationship between preexponential factors and molecular degrees of freedom. An approximate cell model for the degeneracy of ordered layers is introduced, and the efficacy of the model is illustrated through analyses of several representative systems: e.g., CO on Ni( 110) at low coverages, where independent-molecule lattice gas behavior is identified; CO on Ni(ll0) at high coverages, where desorption converts the overlayer from (2x1) to ~ ( 4 x 2 symmetry; ) and H2on (2X 1)-Si( loo), where the rate of recombinative desorption displays first-order kinetics due to site pairing. Finally, the principles of these analyses are used to develop a model that describes the behavior of the hexagonal compression structure that forms for CO on Ru(001) at high coverages. transitions in the magnitudes of these quantities at particular coverages.l2 Rarely is a detailed mechanistic comparison made to the functional dependence of the adsorption probability on coverage, even though the adsorption and desorption rates should evidencecomplementaryinformation about the same elementary processes. If one considers the adsorption-desorption process from a fundamental point of view, one recognizes that this approach may be inadequate and can lead to misleading interpretations. In some cases a failure to give reliable information about elementary processes can result from the initial premise on which the analysis is based-the functional form of the desorption rate. As will be demonstrated in the ensuing discussion, assumptions about the form of the rate function correspondto particular models for the configurational degeneracy of the adsorbed layer. Once an errant form of the rate function is assumed, the measured Arrhenius parameters are forced to include spurious coverage dependencies. In the worst case, errors in E ( 9 ) produce almost perfectly compensating variations in v(9), since u(9) is usually obtained by linear extrapolation of rates measured at finite temperature to infinite temperature, and E ( 9 ) is the slope of the line on which the extrapolation is based.I3J4 Not surprisingly, values of v ( 9 ) obtained by this procedure are often difficult to reconcile with any reasonable microscopic picture of the elementary desorption process. For example, measurements of u(9) for ostensibly simple unimolecular desorption of CO from transition metal surfaces give values as low as 106s-l and as high as 1022 s-1.12 The aim of this paper is to demonstrate how information gleamed from independent means, particularly as regards the configurationaldegeneracyof adsorbed layers, can be incorporated into transition-state rate functions through the rigors of q u i librium statistical mechanics. In many cases, a few of which will be analyzed in detail below, this procedure yields more accurate forms for adsorption-desorption rate functions, and when data are analyzed with them, spurious coverage dependencies in effective rate parameters do not appear. In such circumstances one finds that measured preexponential factors adopt values more consistent with intuitive expectationsabout the role of molecular degrees of freedom in elementary reactions. The discussion begins with a pedagogic presentation of traditional transition-state theory for adsorption-desorption reactions, the purpose of which is to clarify how mass actions, preexponential factors, and activation energies are related to the
I. Introduction The consequences of interparticle interactions in adsorbed layers, particularly as regards the formation and phase behavior of ordered structures, has long been the subject of intense theoretical scrutiny.I4 Most emphasis has focused on adsorbate overlayerson single-crystallinesurfaces,since they provide nearly ideal models for comparisonof theoretical models to experimental observations.2J Computer simulation techniques have afforded the opportunityto investigatecomplex, mathematically intractable models that involve few restrictive assumptions?s6 advancing significantly our general understanding of these phenomena. A subject of q u a l importance is how interparticle interactions and the resulting configurational order within adsorbed layers are manifested in the kinetics of heterogeneous reactionsa7JIn principle, the kinetics of adsorption and desorption reflect the same interactions that lead to phase behavior, and rate measurements for these processes offer the potential to reveal fundamental information about the structure of adsorbed layers. Over years of study of adsorption-desorption kinetics, standard procedures and assumptions have evolved that guide our analysis and interpretation of experimental measurements. The kinetics of adsorption are usually described in terms of the adsorption probability S, the ratio of adsorption rate to impingement rate, and its variation with temperature and surface co~erage.~JO Measurementsof S are generally carried out at low temperatures, where the desorption rate is negligible, to simplify their interpretation. The overwhelming choice for investigation of desorption rates is temperature-programmed desorption (TPD), in which an adsorbed layer is prepared at low temperature, the temperature is "programmed" to increase linearly with time, and the flux of desorption products emanating from the surface is quantified by mass ~pectrometry.~ It is generally assumed that thedependence of the desorption rate on surface coverage reflects simple "mass actions" for the desorbing species, e.g., rdeD = k9 for unimolecular desorption, and rda = k92 for recombinative, bimolecular desorption, where k = Y exp(-E/R?') is an "effective" Arrhenius rateconstant with a coveragc-dependent preexponential factor u ( 9 ) and activation energy E(9).* Once this functional form of the rate is assumed, several procedures can be used to extract values for v(9) and E ( 9 ) from the data," and their dependence on coverage is used to infer mechanistic features of the desorption process. Interparticle interactions and the configurational order they induce are sometimesevidencedas abrupt 0022-3654/93/2097- 1942$04.OO/O
(B
1993 American Chemical Society
Transition-State Rate Functions statistical thermodynamic properties of adsorbed layers. This analysis yields rate functions appropriate for known states of configurational order or, conversely, allows the prevailing state of configurational order to be identified from the functional form of the rate. Several specific examples, corresponding to limiting circumstances of most general utility, are treated to demonstrate the formalism, which can be straightforwardly extended to other, more complicated situations. Next, a simple cell model for the statistical thermodynamics of adsorbed layers that involve local configurationalorder is introduced. Although the model involves simplifications that render it inadequate for describing ordered phase behavior over broad ranges of coverage and temperature, it is a practical approximation for kinetics analysis and gives a good conceptual account of rate data for some systems. Finally, data for several prototypical examples are analyzed in detail to demonstrate the efficacy of the overall approach.
II. Transition-State Formalism for Adsorption-Desorptim Reactioas The following discussionis a tutorial presentation of transitionstate theory, as applied to thekinetics of adsorptionanddwrption on crystalline surfaces. The derivations follow the classical treatment for gas-phase kinetics,IsJ6 modified as required to account for the special properties of two-dimensional adsorbed layers or condensed films. Although similar treatments have appeared elsewhere? this discussion serves to highlight the principles particularly relevant to the analyses presented later in this paper. Three simple, limiting models of the configurational degeneracyfor an adsorbed layer are considered- noninteracting lattice gas, a free two-dimensional gas, and a nondegenerate crystalline solid-to reveal their respective predictions for the form of adsorption and desorption rate functions. I will begin with the simplest, relevant case: one-step molecular adsorption for a noninteracting lattice gas that accommodates only one molecule per site. The effects of a "precursor state" that mediates adsorption will be considered in the specific examples of section IV. From a microscopic point of view, the adsorption and desorption reactions are trajectories on a multidimensional potential energy surface with well-defined wells for the gas-phase and adsorbed molecules, connected by a high-energy "col" or passageway." According to the precepts of transition-state theory, the reaction proceeds in either direction by establishment of thermodynamic equilibrium between the reactants and the "transition state", i.e., the configuration corresponding to the col of the potential energy surface.I6 With thisinmindand with Adesignating themolecular species in question, the elementary reactions for the overall process can be written as where u is a vacant site, and subscripts (a), (g), and (*) designate adsorbed, gas, and transition-state molecules, respectively. The rate in either direction is given by r = KuS*/b, where K is the transmission probability (typically '/z IK I116), u is the thermal molecular velocity over the barrier, b is the barrier length, and t9* = n*/Bisthecoverageof transition-state molecules established at equilibrium, given in terms of their number nc and the number of adsorption sites available in the surface layer E. When equilibrium is maintained within the individual phases, the relationship between 9 and gas-phase pressure or adsorbate coverageis provided by statistical mechanics. Based on statistical models of state occupation, expressions for the chemical potentials of the relevant species are derived and equated according to the stoichiometryof reaction 1 The chemical potentia1of species
The Journal of Physical Chemistry, Vol. 97, No. 9, 1993 1943 i is given by
and the "absolute activity" of species i,15a quantity of particular utility in the ensuing discussion, is given by exp(~,/kT) (3) In eq 2 Q is the canonical partition function for the system, ni is the number of molecules of type i, Vis the system volume, and A is the adsorbent area. Assume the "system* can be separated into three independent "phases": the gas, the layer of adsorbate and vacant sites, and the intermediate layer where the transition state exists, with correspondingcanonical partition functions Qg, Qa, and Q*.This separation involves the implicit assumption that the degrees of freedomof molecules in each phase are unaffected by the presence of molecules in the other phases. Consider the gas phase first, where molecules are indistinguishable and translate freely in three dimensions. In the independent particle approximation's x j
"8
Q,=;%
n,!
(4)
where qg = (V/A3)qw,with A = h/(2rmkT)l/*as the de Broglie wavelength for translational motion and qw as the partition function for the remaining degrees of freedom-rotational, vibrational, and electronic where appropriate. It is implicitly assumed that the zero-point potential energy of the gas-phase molecules defines the energy zero for the entire system, thus no Boltzmann factor appears in Qg. According to eq 3, the activity of molecules in the gas phase is
(5) Note that the linear dependenceof X, on P / k T = ng/V,the "mass action" of the gas phase, is a direct consequence of its configurational degeneracy, as quantified by the factor n,! in eq 4. The lattice gas partition function for the adsorbed layer is given by1J5
and the corresponding activity is (7) In these expressions 9, = n,/B is the fraction of available sites occupied by adsorbed molecules, ea is the zero-point potential energy of chemisorbed molecules relative to the gas phase, and qa includes only contributions from vibrational motions for the adsorbed molecules, since they are localized at sites. Again, the mass actions of adsorbed molecules and vacant sites, given by 9, and 1 - O,, respectively, are a direct consequence of the configurational degeneracy of the adsorbed layer, as quantified by the factor E!/n,!(B - n,)! in eq 6. Two reasonable models can be postulated for the partition function of the transition state. In the first case, transition-state molecules are assumed to translate freely in two dimensions over the vacant sites in the chemisorbed layer, in which case
by analogy to eq 4. The molecular partition function q* includes contributionsdue to rotation, vibration, and free translation within
Anton
1944 The Journal of Physical Chemistry, Vol. 97, No. 9, 1993
the area over vacant chemisorption sites, which is given by B( 19 , ) / p , where p is the surface site density (area-I). For this circumstance one finds
(9) In the second case the transition-state molecules are assumed to be localized at individual sites over the B( 1 - 9,) vacancies in the chemisorbed layer, in which case
by analogy to eq 6, and
where q* includes only vibrationalcontributions. Since formation of the transition state involves a significant increase in potential energy relative to the adsorbed states, 9* O.50 /J70K
0.01 6
/
t A
v,
0.012
-
o.ooa
-
0.004
-
365K
1 '. E
W
u) 0)
D
L
0.000 I
0.55
0.50
0.60
0.65
ac (mi) Figure 3. Rate of CO desorption from Ru(001) vs coverage at constant temperature in the regime where the compression structure exists.
the procedureoutlinedin thecontext of the first exampleof section IV,above. The relevant elementary reactions are
- -
co,,, cow
CO,,,
(60)
S i n e there are no vacancies in the adsorbed layer, the activity function for the transition state is AI = ( 9 * / q I )exp(c$/kT) by analogy to the treatment in section I1 for evaporation and condensation of a two-dimensional solid. The resulting rate expressions are
for promotion of adsorbed molecules to the transition state, and The odd functional form of X, leads to unique behavior for the compression layer. It is instructive to investigate the predictions of this for a representative model of the coverage-dependent molecular partition function. To make this analysis as transparent as possible, assume the dependence of q, on 9, is due only to an increase of the potential energy of adsorption 4 with increasing coverage due to repulsions, and let qvdesignate the remaining vibrational contribution. Then, qC(9,)= qvexp[t,(d)/kTI. Evaluating X, according to eq 58 gives
(i)+ cc
exp(
9,t; kT
)
(59)
for the activity of molecules in the compression layer, where t i = dc,/db,. Note that X, does not include mass action in the traditional sense, but that the dependence of 4 on 9, introduces an effective mass action that ultimately controls the variation of adsorption and desorption rates with coverage. Note also the significanceof the quantity 9 , ~ ' :Each time a molecule is added toot removed from thecompressionlayer, the zero-point potential energy of every other molecule in the layer changes. In effect, t i is the change in potential energy per molecule, and multiplication by 9, sums this contribution for all molecules in the layer. Their product repredents the energy required to "spread" molecules laterally and make a "hole" for an additional molecule at any coverage or, alternatively,the energy recovered when one molecule is removed and all others "relax" to a new hexagonal structure that covers the available vacancy. It follows that 9,4' is the difference between the integral and differential energies of adsorption.' CO adsorption on Ru(001) is mediated by a precursor state; thus the development of transition-state rate expression follows
for the reverse process. Finally, the expression for the overall desorption rate is
and the adsorption probability is given by
s-
rk L T L -Pg &pc
In these expressions kcp= r,,, k, is the rate constant in eq 62, T is the trapping probability and k , is the rate constant for desorption from the precursor state to the gas phase, as described by q 37. A set of isothermal desorption rate measurements for this system, spanning a temperature range of 330 to 370 K, is shown in Figure 3. The adsorption probability approaches zero at temperatures where the desorption rate is appreciable;3IJ* thus k , >> kpcand rdm= kcp. The preexponential contributions to the transition-state rate constant can be grouped into a single coefficient vcp and the rate function can be arranged into the following useful form for comparison to the data: kT[ln (rdm) - In (v,)] - t, - Qct,' (65) Since the right side is a function of coverage only, the left should be a constant if rates at various temperatures are compared at fixed values of the coverage. A plot of kT In (rda) vs kT at 9, = 0.55 mL gives a straight line (not shown), and from its slope uCp= 1 X 10I4f0.3 ML/s is obtained. The magnitude of ucp can
The Journul of Physical Chemistry, Vol. 97, No. 9, 1993 1951
Transition-State Rate Functions
CO/Ru(001)
4
I thank Bob Merrill for stimulating discussions of the subtleties of transition-state theory and Mike DeAngelis for his skillful collection, analysis, and presentation of the desorption data for the CO/Ni( 110) and CO/Ru(001) systems. Refer-
aod Notea
(1) Clark,A. TheoryofAdsorpfionandCara1ysis;AcademicPrw: New
0.50
0.55
0.60
0.65
oc (mi> Figure 4. Data of Figure 3, rearranged according to eq 65 with vcp = 1 X 10’‘ ML/s for the prccxponential factor. be reconciled well with conversion of vibrational to frustrated rotational motion in the formation of the transition state.32 Once ucp is known, q 65 indicates that all of the data for desorption from the compression layer should collapse to a common line when displayed as kT[ln ( r d g ) - In (41. The data of Figure 3 are plotted according to this prescription in Figure 4, and the agreement with the preceding hypotheses is excellent. A detailed report that includes measurements of the adsorption probability as a function of coverage and temperature for this interesting system is in preparation.32
v.
summary I have demonstrated a rational procedure for incorporating information about the configurational degeneracy of adsorbed layers into transition-state rate expressions for adsorption and desorption. A simple cellmodel for the configurationaldegeneracy of ordered layers is introduced, the basic tenets of which are substantiated by the ability of the model to account successfully for the behavior of several systems known to involve configurational order, e.g., CO on Ni(ll0) and Hz on Si(l00). Finally, the principles of these analyses are extended to describe the anomalous behavior of the hexagonal compression structure that CO forms on Ru(001) at high coverages. Taken in its entirety, the discussion serves to emphasize the extreme importance of accurately characterizing the configurational state of adsorbed layers when attempting to extract fundamental information about them from their kinetics of adsorption and desorption. Acknowledgment. I acknowledgegratefully the support of the National Science Foundation on Contracts No. CTS-8808655 andNo.DMR-9121654 (ComellMaterialsScienceCenter).Also,
York, 1970. (2) Weinbcrg, W. H. Annu. Rev. Phys. Chem. 1983,34, 217. (3) Bauer, E. Srrucrure urd Dynamics of Surfaces II; Schommers, W., von Blanckenhagen, P., Eds.; Springer-Verlag: Berlin, 1987;p 1 15. (4) Penson, B. N. J. Surf. Sci. Rep. 1992, 15, 1. (5) Nicholson. D.; Parsonage, N. G. Computer Simulation and the Srarisrical Mechanics of Adsorprion; Academic Press: London, 1982. (6) Roclofs, L. D. Chemis?ryandPhysicsofSolidSurfacesIV; Vaneslow, R., Howe, R., Eds.; Springer-Verlag: Berlin, 1982;p 219. (7) Lombardo, S.T.; Bell. A. T. Surf.Sci. Rep. 1991, 13, 1. (8) Zhdanov, V. P. Surf.Sci. Rep. 1991, 12, 183. (9) Weinbcrg, W. H. Dynamics of Gas-Surfoce Collisions; Rettner, C. T., Ashfold, M. N. R., Eds.; R. Sw. Chem.: Cambridge, 1991;p 171. (IO) Weinberg, W. H. Kinetics of Merface Reactions; Kreuzer, H. J., Grunze, M., Eds.; Springer-Verlag: Heidelberg, 1987;p 94. (1 1) de Jong, A. M.; Niemantsverdriet, J. W. Surf.Sci. 1990,223,355, (12) Seebauer,E.G.;Kong,A.C.F.;Schmidt,L.D.Surf.Sci.1988,193, 417. (13) Kang, H. C.; Jachimowski, T. A,; Weinbcrg, W. H. J . Chem. Phys. 1990, 93, 1418. (14) Carter, R. N.; Anton, A. B. J. Vac. Sci. Technol. A 1992, IO, 344. Scheidle, P. H.; Anton, A. B.; Lyons, K.J. Manuscript in preparation. (1 5) Hill, T. L. An In?roduc?ion ?o S?a?is?ical Thermodynamics; Addison-Wesley: Reading, MA, 1960. (16) Laidler, K.J. Chemicul Kinetics. 3rd 4.;Harper and Row: New York, 1987. (17) Doren, D. J.; Tully, J. C. hngmuir 1988, 4, 256; J . Chem. Phys. 1991,94, 8428. (18) DeAngelis, M. A,; Anton, A. B. J . VUC.Sci. Technol. A 1992,10, 3507. (19) Eyring, H.; Re,T.; Hirai, H. Proc. Narl. Acad. Sci. U S A . 1948, 44, 683. (20) McAlpin, J. J.; Pierotti, R. A. J . Chem. Phys. 1964, 41,68. (21) The configurational entropy gained by relaxing the constraint that cells maintain registry with another could lead to interesting temperaturedependent phase behavior. See,for example: Frenkel, D.; Louis, A. A. Phys. Rev. Let?. 1992, 68, 3363. (22) DeAngelis, M. A.; Glines, A. M.; Anton, A. B.J . Chem. Phys. 1992, 96, 8582. (23) Behm, R. J.; Ertl, G.; Penka, V. Surf.Sci. 1985, 160, 387. (24) Norton, P. R.; Bindner, P. E.; Jackman, T. E. Surf.Sci. 1986,175, 313. (25) Campbell, J. H.; Vajo, J. J.; Becker, C. H. J . Phys. Chem. 1992,96, 1826. (26) Sinniah, K.; Sherman, M. G.;Lewis, L. B.; Weinbcig, W. H.; Yates, Jr., J. T.; Janda, K. C. Phys. Rev. Le??.1989, 62,567. (27) Ibach, H.; Rowe, J. E. Surf.Sci. 1974,43,481. (28) Boland, J. J. Phys. Rev. Le??.1991, 67, 1539. (29) A recent paper presents a similar but more complete treatment of the same problem: D’Evelyn, M. P.; Yang, Y. L.; Sutcu, L. F. J . Chem. Phys. 1992,96,852. These authors account for half-filled dimers in the partition function and usc the observed kinetics of hydrogen desorption to delimit the energetic penalty associated with half-dimer formation. Their results are consistent with the assumption used to develop eq 52; Le., that filled and empty dimers make the dominant contributions to the partition function. (30) Williams, E. D.; Weinbcrg, W. H. Surf.Sci. 1979, 82, 93. (31) Pfniir, H.; Feulner, P.; Engelhardt, H. A.; Menzel. D. Chrm. Phys. Le??.1978,59,481. Pfniir, H.; Menzel, D. J. Chem. Phys. 1983, 79,2400. Pfniir, H.; Feulner, P.; Menzel, D. J . Chem. Phys. 1983, 79,4613. (32) DeAngelis, M. A. Ph.D.Thesis,CornellUniversity, 1992. DeAngelis, M. A.;Anton, A. B. Manuscript in preparation.
