Analysis of Thermal Desorption Spectra of Heterogeneous Surfaces

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Langmuir 1999, 15, 6070-6082

Analysis of Thermal Desorption Spectra of Heterogeneous Surfaces† Yu. K. Tovbin* and E. V. Votyakov Department of Matter Structure, Karpov Institute of Physical Chemistry, ul. Vorontsovo Pole 10, Moscow 103064, Russia Received October 22, 1998. In Final Form: March 1, 1999 A proper analysis of thermal desorption spectra (TDS) can give important information about molecular properties of the adsorption system studied (bond energies of heterogeneous adsorption sites and their composition and structure, strengths of adspecies lateral interactions, role of adspecies mobility, etc.); however, this is complicated because all these factors are distributed over the total spectrum. To solve this problem, it is convenient to use a lattice model since this enables us to extend an approach of traditional chemical kinetics for nonideal reactionary systems in such a way that it jointly takes care of surface heterogeneity and adspecies lateral interactions. In this paper, this model is taken as a base to consider fundamental issues of TDS interpretation dealing with a heterogeneous distribution function of adsorption centers over bond energy, on one hand, and the number of peaks and shape of TDS, on the other hand. Moreover, we discuss the validity of phenomenological approaches to interpret TDS; among them are a phenomenological equation employing an effective activation desorption energy and pre-exponential factor and two recently proposed treatments invoking a so-called entropic factor into adsorption-desorption kinetics. In the course of this discussion, we present the molecular description, taking a nonuniform adspecies distribution into account, for the following experimental data: CO/Ru(001), Hg/W(100), and H2/Ir(110)2 × 1.

1. Introduction The thermal desorption method originally proposed in refs 1 and 2 has made substantial progress and now is widely used for studying adsorption processes.3-6 In this method, a surface is heated in accordance with a desired program T(τ) (T is the temperature and τ is the time), and the rate of desorbed molecules dθ/dT ) -UD is measured as a function of temperature T (θ is the surface coverage and UD is the rate of desorption), yielding the temperatureprogrammed desorption spectrum (TDS). To heat the surface, the linear law is typically used, T ) T0 + τb, where b is the rate of heating and T0 is the initial temperature (at τ ) 0). It can be easily proved that TDS from a homogeneous surface, provided there are no lateral interactions between adsorbed atoms or molecules (adspecies), shows one peak at a temperature Tp which is obtained from the Redhead’s equation:7,8

ED/RTp2 ) KD2 exp(-ED/RTp)ξm/b where the pre-exponential factor, KD, and activation energy, ED, are assumed to depend on the degree of * To whom the correspondence should be addressed. E-mail: [email protected]. Fax: 7-(095)-9752450. † Presented at the Third International Symposium on the Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland, Torun, August 9-16, 1998. (1) Becker, J. A.; Hartman, C. D. J. Phys. Chem. 1953, 57, 157. (2) Ehrlich, G. J. Chem. Phys. 1961, 34, 29. (3) Ehrlich, G. Adv. Catal. 1963, 14, 255. (4) Hansen, R. S.; Mimo, V. Experimental Methods in Catalytic Research; Anderson, R. B., Ed.; Academic Press: London, 1968. (5) Yakerson, V. I.; Rozanov, V. V. Resume of Science and Technics; Physical Chemistry Series, Kinetics Catalysis Series; VINITI: Moscow, 1974; Vol. 3. (6) Roberts, M. W.; McKee, C. S. Chemistry of the Metal-Gas Interface; Oxford: Clarendon, 1978. (7) Redhead P. A. Vacuum 1962, 12, 203. (8) Carter G. Vacuum 1962, 12, 245.

dissociation m only, ξ1 ) 1 and ξ2 ) θ0; θ0is the initial surface coverage (at τ ) 0) and R is the universal gas constant. The picture is more complicated for a heterogeneous surface, since the overall TDS spectrum now is a sum over desorption fluxes from the adsorption centers of different adsorption capacities. When the difference in the adsorption capacity is large, each desorption flux has its own peak in the overall TDS, and when this difference is small, the fluxes from the different centers are overlapping, so the spectrum gets broadened relative to the homogeneous case. Thus, an appearance of several peaks in the TDS is conventionally supposed to be due to a discrete distribution of adsorption sites over the activation energy, while a single broad peak is due to a continuous distribution. The first case allows a correlation between the number of peaks and the number of different types of adsorption sites,9,10 and the second leads to a task to predict the peak shape from the continuous distribution function max f(x) dx ) 1, where xmin f(x) over the adsorption heat, f xxmin and xmax are the smallest and the largest adsorption heats, respectively, and the difference, ∆Q ) xmax - xmin, is a measure of the energetic heterogeneity of the surface studied. The heterogeneous properties mentioned above are not the only information that one can extract from the TDS. There should generally be lateral interactions among adspecies, and they also affect the TDS shape. Thus, the proper quantitative description of TDS gives insight into the surface composition, the adsorption capacity for different centers and their mutual distribution, kind (attraction or repulsion) and the strength of the lateral interactions, and so forth. The thermal desorption method is essentially sensitive to each of these physical factors; however, the TDS itself gives integrated molecular (9) Tovbin, Yu. K.; Votyakov, E. V. Zh. Fiz. Khim. 1990, 64, 3024. (10) Tovbin, Yu. K.; Votyakov, E. V. Surf. Phys. Chem. Mech. 1991, No. 3, 111 (in Russian).

10.1021/la981492k CCC: $18.00 © 1999 American Chemical Society Published on Web 04/29/1999

Thermal Desorption Spectra/Heterogeneous Surfaces

information from the studied surface as a whole. All these physical factors are presented jointly on the TDS curve, and to get a proper molecular interpretation of the TDS is a complex problem, especially when the factors are almost of equal contributions to the curve. A convenient tool for describing TDS assumes the simplest molecular modelsthe lattice-gas model.11-13 In a natural way, such a model can include surface heterogeneity and adspecies lateral interactions.13 Taking the lattice model into a base, a theory of chemical kinetics for nonideal reaction systems has been developed, and the special cases of this theory are adsorption and desorption kinetic processes. This theory uses fundamentals of the transition-state theory14 and extends them to apply to any condensed matter, including interfaces between different parts of the system under consideration. This allows for various kinds of species interaction potentials and their influence on elementary kinetic processes. To include the composition and structure of heterogeneous surfaces, this model uses various distribution functions for adsorption centers or/and their clusters. As to the task of molecular interpretation for TDS, the presented theory has enabled us to give both a qualitative and quantitative description for various experimantal data.15,16 This paper considers fundamental questions of TDS interpretation for heterogeneous surfaces. These refer essentially to the case of heterogeneous distribution of adsorbed species over the surface, for instance, up to what degree the usual conceptions described above about continuous and discrete distribution functions are valid. Can the continuous distribution of adsorption centers over desorption activation energy be responsible for the broad TDS peak only or may this result also in splitting? Is it true that several TDS peaks relate to the same number of discrete adsorption centers of different desorption energy, or can the number of peaks be larger than the number of heterogeneous types of adsorption centers? These concepts were used in phenomenological approaches for describing TDS5,8,17-29, and we will show that the TDS splitting can also be due to continuous distribution; and the discrete distribution function, provided adspecies repulse each other, can result in additional peaks in the TDS. From this point of view, we discuss also the validity of molecular interpretations of the TDS by means of other phenomenological approaches, for example, the widely (11) Hill, T. L. Statistical Mechanics; Izdatelstvo Inostrannoy Literatury: Moscow, 1960 (in Russian). (12) Fischer, M. E. The Nature of Critical Points; University of Colorado Press.: Boulder, CO, 1965. (13) Tovbin, Yu. K. Theory of Physical Chemistry Processes at a GasSolid Interface; CRC Press Inc.: Boca Raton, FL, 1991. (14) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. (15) Tovbin, Yu. K. Prog. Surf. Sci. 1990, 34, 1. (16) Tovbin, Yu. K. In Equilibrium and Dynamics of Gas Adsorption on Heterogeneous Surfaces; Rudzinski, W. A., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997; p 201. (17) Kislyuk, M. U.; Sklyarov, A. V.; Dangyan, T. M. Isvest. Akad. Nauk SSSR, Ser. Khim. 1975, No. 10, 2161. (18) Schmidt, L. D. Catal. Rev.-Sci. Eng. 1974, 9, 115. (19) Chan, C. M.; Aris, R.; Weinberg, W. H. Appl. Surf. Sci. 1978, 1, 360. (20) Konvalinka, J. A.; Scholten, J. J. F. J. Catal. 1978, 52, 547. (21) Winterbottom, W. I. J. Vac. Sci. Technol. 1972, 9, 936. (22) Pisani, C.; Rabino, G.; Ricca, F. Surf. Sci. 1974, 41, 277. (23) King, D. A. Surf. Sci. 1975, 47, 84. (24) Falconer, J. L.; Madix, R. J. J. Catal. 1977, 48, 262. (25) Taylor, J. L.; Weinberg, W. H. Surf. Sci. 1978, 78, 259. (26) Habenschaden, E.; Kuppers, J. Surf. Sci. 1984, 138, L147. (27) Yokoro, Y.; Uchijima, U.; Yoneda, Y. J. Catal. 1979, 56, 110. (28) Forzatti, R.; Borghesi, M.; Pasquon, I.; Tronconi, E. Surf. Sci. 1984, 137, 595. (29) de Jong, A. M.; Niemantsverdriet, J. W. Surf. Sci. 1990, 233, 355.

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used way to describe TDS with effective activation desorption energy and pre-exponential factor,30 and as well, new treatments which invoke the so-called entropic factor into adsorption-desorption kinetics.31,32 The answers on all these questions, and the joint description of surface heterogeneity and lateral interactions for the H2/ Ir(110)2 × 1 system, are obtained with the use of the strict kinetic theory.13,15 (Examples of earlier application of this theory are also given in refs 9, 10, 13, 15, and 16.) This paper is composed in the following way. Section 2 gives briefly the main equations for the molecular description of the desorption process. Section 3 illustrates a failure of the traditional conception that the number of TDS peaks is always related to the heterogeneous surface composition. The first example is splitting of the TDS curve for the continuous energy distribution function, if the width of the energy distribution function is sufficiently large; another one is additional splittings in the TDS from the patchwise surface when there is a strong adspecies repulsion. For the latter case there is a competition of the lateral interactions, from one side, and a difference in bond adspecies energies on different patches, from another side. Section 4 discusses the validity of effective desorption energies and pre-exponential factors to characterize properties of the desorption system. We concern the physical meaning of these effective desorption characteristics and consider in detail a linear dependence of desorption activation energy on surface coverage, which is the most widely used. At the end of section 4, we describe the TDS in the Hg/W(100) system, comparing our results and those obtained before in a phenomenlogical way, to show that the molecular models are self-consistent in contrast to phenomenological models. Section 5 deals with a place and role entropic factor in kinetics of adsorption processes; the molecular and entropic approaches are compared together in considering the TDS in the CO/ Ru(100) system. Section 6 gives another example of molecular interpretation on the basis of the proposed model for TDS in the H2/Ir(110)2 × 1 system. Section 7 concludes the paper. 2. Equations of the Desorption Process It is assumed below that a surface is stable at any surface coverage (i.e., there is no surface reconstruction during desorption), the adspecies mobility is sufficiently high (i.e., there exists an equilibrium distribution over the adsorption centers), and the type of adsorption site does not affect adspecies lateral interactions. (To go away from these restrictions, we might use kinetic models, developed in refs 13 and 15.) The direct correlations in the distribution of interacting species are included with quasichemical approximation. If the number of types of adsorption sites on a surface is large, it is convenient to calculate macroscopic characteristics from continuous distribution functions. Let us define the Langmuir adsorption equilibrium constant a(x) for adsorption on the sites with the adsorption heat x as a(x) ) a0(x) exp(βx), β ) (RT)-1, a0(x) is the pre-exponential constant for adsorption on sites with energy x. (This factor very often is assumed to be independent of x, a0(x) ) a0, i.e., the Langmuir adsorption constant is assumed to depend only on the adsorption heat.33-35) (30) Seebaur, E. G.; Kong, A. C. F.; Schmidt, L. D. Surf. Sci. 1988, 193, 417. (31) Nagai, K. Surf. Sci. Lett. 1991, 244, L147. (32) Ward, C. A.; Findlay, R. D.; Rizk, M. J. Chem. Phys. 1982, 76, 5599. (33) Temkin, M. I. Zh. Fiz. Khim. 1941, 15, 296. (34) Rogynsky, S. Z. Adsorption and Catalysis on Nonuniform Surfaces; Academiya Nauk SSSR: Moscow, 1948.

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The desorption rate UD, for laterally interacting molecules on a heterogeneous nonreconstructing surface, where the equilibrium distribution of molecules over adsorption sites is assumed to be constant during the whole desorption process, has the following form:15,36

UD )

∫xx

min

max

f(x) KD(x) θ(x) SD(x) dx, KD(x) ) K0D(x) exp(-βED(x)) (1)

where KD(x) is a constant of the desorption rate for the molecules adsorbed on the sites of the adsoprtion heat x; K0D(x) and ED(x) are its pre-exponential factor and activation energy, respectively; f(x) dx is a share of the adsorption sites having energy between x and x + dx, θ(x) is the coverage of these sites. M. I. Temkin supposed the properties of different sites are similar in their physical nature,33 so a difference in the activation energies for two arbitrary chosen sites can be assumed to be proportional to that of between the adsorption heats for the corresponding sites: ED(x1) - ED(x2) ) R(x1 - x2), R is a constant, 0 e R e 1, and x1 and x2 are any values inside [xmin, xmax]. For many adsorption, catalytic, and electrode processes, the value of R is close to 0.5.35,37 This assumption leads to

UD ) K0D

∫xx

min

max

f(x) SD(x) θ(x) exp(-βRx) dx

(2)

Functions SD(x) takes lateral interactions into account: R

SD(x) )

SD(x|r), ∏ r)1

[1 +

∫yy

max

min

SD(x|r) )

d(x, y|r) t(x, y|r)[eβδ(r) - 1] dy]z(r),

t(x,y|r) ) 2θ(x)/(δ(x,y|r) + b(x,y|r)), δ(x,y|r) ) 1 + X(r)(1 - θ(x) - θ(y)), b(x,y|r) ) {[δ(x,y|r)]2 + 4θ(x)θ(y)X(r)}1/2, X(r) ) exp(-β(r)) - 1 (3) where R is the largest range of lateral interactions counted as a number of the coordination shell, z(r) is the number of neighbors in the rth coordination shell, d(x,y|r) is the pair distribution function describing the surface structure15,36 (it is defined as the conditional probability to find a site with the adsorption heat y at the distance r from a site with the adsorption heat x), δ(r) ) *(r) - (r), here (r) is the energetic constant of lateral interactions for the species in the ground state at the distance r from each other, and *(r) is the similar constant but for the case when one of the interacting species is in a transition state. To get an ability to calculate the desorption rate with the use of eq 2, we are obliged first to find the equilibrium distribution of the species over heterogeneous adsorption sites, that is, the partial coverages θ(x). These values should be found from the following equations: a(x)P ) θ(x)S(x)/(1 - θ(x)), where P is the equilibrium pressure max f(x) corresponding to the fixed total coverage θ, and ∫xxmin θ(x) dx ) θ. The function S(x) includes lateral interactions; it is written in the same way as SD(x) in eq 3, but with *(r) ) 0. Analytical expressions for UD can be obtained for the simplest case only,33,36 for example, for an ideal desorption (35) Kiperman, S. L. Fundumentals of the Kinetics of Heterogeneous Catalytic Reactions; Nauka: Moscow, 1964. (36) Tovbin, Yu. K. Teor. Eksper. Chem. 1982, 18, 417 (in Russian). (37) Frumkin, A. N. Z. Phys. Chem. 1932, 160, 116.

process from a heterogeneous surface being characterized by a uniform distribution function f(x) )(xmax - xmin)-1 at R ) 0.5 and 1. In the general case, eq 2 should be numerically integrated, including all the integrals inside SD(x). For this purpose, the range [xmin, xmax] is divided into N intervals; for each interval, the above expression for the equilibrium distribution is written to form a system of N equations. Then, this system can be numerically solved to find θ(x) at the fixed value of the total surface coverage θ; the latter quantity is given by each temporary step of numerical integration of the master equation for thermal desorption. This is not a simple task especially relative to a phenomenological approach; on the other hand, one might think that this requires a lot of computer power. Nevertheless, we found that to divide the heterogeneous range [xmin, xmax] into ∼30 intervals is quite sufficient for numerical integration; and a simple desktop computer (we used an IBM-compatible PC with a 486 processor) spent a few minutes to calculate an entire spectrum with all the integrations for the maximal initial surface coverage. When a heterogeneous surface can be characterized with a finite number of the well-defined types of adsorption centers (sites), which do not allow a continuous set over activation desorption energy, we can apply a discrete distribution function fq instead of continuous f(x). In more detail, the function fq (1 e q e t, t is the total number of site types) gives a share of adsorption sites of type q, t ∑q)1 fq ) 1, and for this case the rate of desorption is t fqUDq, UDq ) KDqθqSDq, where KDq0 written as UD ) ∑q)1 and EDq are the desorption pre-exponential activation energies of the desorption rate constant KDq for the site t fqθq ) θ and SDq q, θq is the local coverage of site q, ∑q)1 is the function to include lateral interactions. The full description of the discrete equations are given in refs 13 and 15. For a homogeneous surface, eqs 2 and 3 are the same as that deduced in refs 38-41 and the equations38-41 at * ) 0 and R ) 1 are Roberts’s equations.42 The latter ones were applied originally by Toya43 to show how lateral repulsions among adspecies on a homogeneous surface split TDS. 3. Heterogeneous Surface and the Number of TDS Peaks How the number of TDS peaks is related to the number of types of adsorption centers is one of the main issues in studies of heterogeneous surfaces. Section 3.1 deals with the case when there are a large number of types of adsorption sites. For this case it has been found, even for continuous distribution functions, that TDS can split, although the general opinion is that there should be a broadening of TDS without any splitting, if continuous distribution functions are applied to describe surface heterogeneity. Section 3.2 gives the opposite parity, when it is possible to get the number of peaks large than the number of types of adsorption centers. In the latter case, an increase in the number of peaks is due to the effects of strong lateral interactions. (38) Tovbin, Yu. K. Ph.D. Thesis, Karpov Institute of Physical Chemistry, Moscow, 1974. (39) Tovbin, Yu. K.; Fedyanin, V. K. Russ. Solid State Phys. 1975, 17, 1511. (40) Tovbin, Yu. K.; Fedyanin, V. K. Russ. Kinet. Catal. 1978, 19, 989, 1202. (41) Adams, D. Surf. Sci. 1974, 42, 12. (42) Roberts, J. K. Some Problems of Adsorption; Cambridge University Press: Cambridge, 1939. (43) Toya, T. J. Vac. Sci. Technol. 1972, 9, 890.


3.1. Is It Possible To Get a Splitting in TDS with Continuous Distribution Functions? We considered the simplest case of the ideal adsorption system (without lateral interactions the problem is getting much more simple because of ) * ) 0, and hence, SD(x) ) 1 and S(x) ) 1) with three distribution functions: (a) uniform distribution function, f(x) ) ∆Q-1 (∆Q is the energetic range of surface heterogeneity); (b) exponential one f(x) ) exp(γx)/n (γ is the distribution parameter); (c) Gauss function, f(x) ) exp[-(x-x*)/2σ]/n, where x* (x* ) xmax b1∆Q, 0 < b1 < l) is a parameter to define the maximal position of the Gauss distribution, σ is the a width at half-maximum; for all the functions, n is the normalization factor. The calculations were performed with the following parameters: K0D(x) ) K0D ) 1013 s-1, b ) 50 K/s, ED(xmax) ) 126 kJ/mol, and xmax ) 126 kJ/mol. For the sake of simplicity, the values of R were set equal to 0.5 and 1.0 that are typical for CO desorption from the platinum metals. The TDS calculated for the parameters defined above are plotted in Figure 1. Figure 1a shows the spectra for the uniform distribution as a function of ∆Q. As the surface heterogeneity increases, more broadening of spectra is observed, and at critical ∆Q*, one can see an appearance of the low-temperature peak (curve 3). With decreasing R, the main peak becomes narrow; nevertheless, the specific low-temperature peak also can be found at enough large ∆Q, so this peculiarity remains. Figure 1b refers to the case of the exponential distribution function. Now, the main TDS peak is shifted to either the higher (at γ > 0) or lower (at γ < 0) temperatures relative to the uniform distribution function; and, again, at a sufficiently high ∆Q, a sharp low-temperature TDS peak is observed. Thus, the value of ∆Q* depends on both the shape of the distribution function and the parameter R. Since the low-temperature peak has a sharp maximum, therefore, we must integrate theoretical spectra with a small temperature step to determine its position accurately. The intensity of this peak depends on the share of the sites with a sufficiently low adsorption heat, and the value of the lowest adsorption heat is responsible mainly for its temperature location. The physical process behind such a splitting of TDS curves is the redistribution of adspecies due to their migration over the surface; as a result, the main flow of desorbed molecules goes through weak adsorption sites.9,10 A qualitative analysis of eq 2 makes it possible to relate the condition for the appearance of the specific temperature peak with properties of the continuous distribution function f(x). Provided that SD ) 1 and coverages are large, max f(x) exp(-βRx) dx. At fixed θ(x) ≈ 1, we have UD ≈ K0D ∫xxmin xmin, an increase in ∆Q leads to an increase in xmax. Hence, at the sufficiently high values of ∆Q and fixed xmin, the integrand vanishes at xmax, so that the main contribution to the integral gives the integration near the lower integration limit, xmin. In the limit ∆Q f ∞, applying the midpoints rule, we conclude that the integral can be expressed as UD ) K*exp(-βE*), where K* ) K0Df(x*)/(βR) and E* ) Rx*, and x* is close to xmin. This means (1) the value E* does not depend on the shape of the distribution function, that is, the position of the peak, which is related to E* via the Redhead rule,7,8 depends only on the parameter R and x* ≈ xmin; (2) the shape of the distribution function influences only the magnitude of the preexponential factor K*: the larger the f(x*) the higher the peak. Figure 1c supports these analytical conclusions: at the fixed values of R ) 0.5 and high ∆Q ) 73.2 kJ/mol, all the spectra show the specific low-temperature peak at

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Figure 1. TDS for a heterogeneous surface with continuous distributed functions. Field (a): uniform distribution function, ∆Q (kJ/mol) ) 21 (1 and 4), 38 (2 and 5), 56.5 (3 and 6); R ) 1 (1-3) and 0.5 (4-6). Field (b): exponential distribution function, R ) 1, ∆Q is the same as in (a), γ is taken as γ ) ln(γ′)/∆Q, where γ′ ) 10 (1-3) and 0.1 (4-6). Field (c): R ) 0.5, ∆Q ) 73.2 kJ/mol; continuous distribution functions: 1, uniform; 2,3, exponential at γ′ ) 10 (2) and 1 (3); 3, Gaussian at b1 ) 0.5 and σ ) 0.5.

the same temperature independently of the kind of distribution function, while the intensity of this peak is the higher for the function with the more shares near xmin. A further increase in ∆Q will result in an increase in the fraction of the sites with adsorption heats below ∆Q, x < ∆Q*; therefore, the intensity of the low-temperature peak will increase very fast, whereas the high-temperature part of the spectrum will be represented by a low-intensity tail. Thus, in contrast to the usual conception, the above calculations prove that it is possible to get a splitting in TDS with a continuous distribution function, if a width of surface heterogeneity is sufficiently large. 3.2. Is There Always a Correlation in the Number of TDS Peaks and the Types of Adsorption Centers? It can be easily proved, that for a heterogeneous surface

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A relation between TDS peaks and the number of types of adsorption sites is not evident on a heterogeneous surface when the lateral interactions are sufficiently strong. An example of the change in the number of TDS peaks is a system with adspecies c(2 × 2) ordering on a patchwise surface consisting of two types of adsorption sites, shown in Figure 2. This is the simplest case of competition between lateral interactions, from one side, and bond energies on different patches, from another side. An analysis of the number of peaks and nature of their shoulders as a function of surface heterogeneity, ∆Q, enables a conclusion that because of the fast surface mobility, there exists an adspecies redistribution between ordered sublattices on the different patches during thermal desorption; this fact is of main importance in forming the TDS shape. As the degree of surface heterogeneity increases, one can observe from two (almost homogeneous case, ∆Q ) 12.6 kJ/mol) up to four (essentially heterogeneous surface, ∆Q ) 71.4 kJ/mol) peaks. The dashed lines in Figure 2 are the TDS peaks found without including the adspecies ordering. One can see that even for this case lateral interactions change the number of the TDS peaks relative to the ideal case. However, dashed spectra do not reflect the thin structure that is wellpronounced when allowing for c(2 × 2) ordering. This model example shows that even a quite simple task of a qualitative interpretation of the number of TDS peaks relative to the number of types of adsorption sites on a heterogeneous surface should be considered, taking lateral interactions into account. Figure 2. TDS for a patchwise surface consisting of two types of adsoprtion sites: Q1 ) 147 kJ/mol; ) -12.6 kJ/mol; * ) /2; E1D ) 1.1Q1; ∆Q ) ∆E at ∆Q ) 71.4 (a), 29.4 (b), and 12.6 (c) kJ/mol. Solid spectra include the c(2 × 2) adspecies ordering; dashed spectra do not.

(described with a discrete distribution function, provided there is no lateral interaction), the total number of TDS peaks can be equal to the number of the types of adsorption sites when the difference in activation desorption energies is sufficiently high. As this difference gets lower, there appears to be an overlapping of adjacent peaks resulting in a characteristic flexure. Below, we show that allowing for adspecies lateral interactions can qualitatively change this traditional picture. As is well-known for a homogeneous surface, lateral interactions change the TDS curves:13,15 repulsive interactions broaden the TDS peak and may eventually cause its splitting, whereas attractive interaction makes the peak high and narrow. For example, for a square lattice (z ) 4) and initial coverage θ0 ∼ 1, the main TDS peak will split into two if the repulsive strength exceeds -6.3 kJ/ mol (at * ) 0) and -8.9 kJ/mol (at * ) 6.67). As the value θ0 decreases to 0.6, the splitting will disappear, and now, the TDS peak just broadens at an increase of the strength of repulsion. Repulsion between the species on a surface results in the apparent saturation of the surface, which manifests itself in that the surface coverage larger than a certain value increases very slowly with exposure. To observe a detectable increase in θ, it is necessary to increase the pressure by 1-3 orders of magnitude. At such conditions, well-known from LEED experimental data,6 an ordered adspecies structure usually arises, besides a splitting (for z ) 4 it appears at θ0 > 0.5); it is possible to observe a thin structure of a low-temperature TDS peak.15 According to LEED data, at a platinum face (111) (z ) 6) this gives a splitting near θ0 ∼ 1/3.

4. Effective Characteristics of the Desorption Process The formal expression for the desorption rate via the effective activation energy and effective pre-exponential factor for desorption rate is written as follows:

UD ) K0(θ)θm exp[-βEef(θ)]

(4)

where Eef(θ) and K0(θ) are the effective desorption activation energy and pre-exponential factor, respectively, defined as

Eef(θ) ) - ∂{ln[Ud(θ)/θm]}/∂β, K0(θ) ) Ud(θ) exp[-βEef(θ)]/θm (5) These effective quantities as a function of coverage can be found experimentally by means of varying either the initial coverages θ0 or temperature program T(τ).30 Also, these can be calculated from kinetic models. Evidently, if we have a kinetic model, the more proper way is to use this model directly to describe TDS. So, below we discuss the validity of the effective desorption energy and preexponential factor to characterize molecular properties of desorption systems. 4.1. Linear Effective Desorption Activation Energy. There are a lot of papers (e.g., see refs 8, 18, and 44), where eq 4 was used to describe experimental TDS with a linear dependence of activation energy on coverage:

Eef(θ) ) E0 - Dθ

(6)

This is the simplest function for Eef(θ) and, in the first approximation, it can really describe some experimental systems. Formula 6 can arise from kinetic models resulting from the presence of either a small contribution of lateral (44) Pawela-Crew, J.; Madix, R. J. Surf. Sci. 1995, 339, 8.


interactions or surface heterogeneity presented with a uniform continuous distribution function. However, an application of eq 6 can yield qualitatively incorrect results. Let us consider how eq 6 can be inferred from the molecular models, taking into account these physical factors. M. I. Temkin was the first who showed more than 50 years ago that eq 6 can be applied to describe the characteristic activation desorption energy when the surface heterogeneity can be represented by a uniform distribution function.33 Nevertheless, it is incorrect to identify the characteristic activation desorption energy and the effective parameter Eef(θ) used in eq 4. The characteristic activation desorption energy is related to adsorption on sites of one type, while the parameter Eef(θ) in eq 4 is defined for desorption on an ensemble of sites of different types. Therefore, Eef(θ) used in eq 4 along with other factors reflects the redistribution of adspecies over the sites of different types when changing the thermodynamic characteristics of the system studied: (For our case, these are temperature and coverage.) As a result, eq 6 is valid for medium coverages only, but not for small and large coverages, and this fact has been noted in the original paper.33 It is worth mentioning that logarithmic adsorption isotherms that are based on a linear dependence of the adsorption heat on coverage are valid only for medium coverages; in the entire coverage range, Temkin’s quasilogarithmic isotherms should be used.33 Now, we turn to Figure 3 in order to consider dependencies of the effective activation energy and pre-exponential factor on surface coverage for the same continuous distribution functions as those in section 3. The calculations were performed with eq 5. The aim is to show that use of these effective characteristics in a large degree of surface heterogeneity leads to a disappearance of the specific TDS splitting that was shown in section 3. Indeed, at small values of ∆Q, the dependencies of Eef(q) for the uniform and Gaussian distribution functions are similar to a linear dependence that is described by eq 6. However, in the limits θ f 0 and θ f 1, eq 6 becomes incorrect, and the higher the ∆Q the larger the error. In all cases, the dependencies Eef(θ) and K0(θ) are similar, which can be considered as a consequence of a compensation effect. Thus, we should not think that eq 6 is always valid for a uniform distribution function because this excludes a possibility for TDS to be splitting at large ∆Q, as was shown in the previous section. For the kinetic models including lateral interactions, eq 6 can be obtained with the mean-field approximation, taking care about the interactions of the closest neighbors, and for this case, D ) z(* - ).13,38,40 As is well-known, the mean field approach disregards the correlation effects for interacting species. As D increases, the TDS peak becomes broad and flat (Figure 4), but it does not split, for example, for the very left spectrum, the value D ()100.8 kJ/mol) is almost a factor of 10 larger than the splitting condition for lateral interaction on a uniform surface. On the other hand, the value D ) 100.8 kJ/mol is almost twice as large as the value ∆Q* at which the TDS curve for the uniform distribution function splits. Thus, eq 6 does not hold at small and large coverages for the uniform distribution function. At small values of ∆Q, this equation does hold for the entire range of surface coverages, but the splitting of TDS curves is not observed because of low heterogeneity. Although it correctly gives the desorption rate at medium coverages, this does not mean that eq 6 can be used to describe TDS curves at the surface coverages θ0 ) 0.7-1.0 because it cannot reproduce the splitting.

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Figure 3. Effective activation energies (a) and pre-exponential factors (b) as a function of surface coverage at Q1 ) 126, ∆Q ) 50 kJ/mol, and R ) 1 (1,3-5) and 0.5 The continuos distribution functions: 1,2, uniform; 2,3, exponential at γ ) 1; 4,5, symmetric Gaussian and σ ) 0.5 (4) and (2).

Figure 4. TDS calculated with mean field approximation at z ) 4, the value of (* - ) from left to side is 0, 4.2, 8.4, 12.6, 16.8, 21, and 25.2 kJ/mol.

4.2. About the Physical Meaning of Equations 4 and 5. The effective kinetic desorption characteristics originally were suggested by means of eqs 4 and 5 in the same way as a heat (enthalpy) and pre-exponential factor are written down for chemical equilibrium. This is a traditional way to treat experimental data. Conditionally is supposed that the physical meaning of the experimentally obtained effective dependencies on concentration is

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Figure 5. Effective activation energies (a) and preexponential factors (b) as a function of surface coverage, for a homogeneous surface, calculated with eq 7, z ) 4, ED ) 126 kJ/mol, */ ) 0 (1), 0.2 (2), 0.4 (3), 0.6 (4), and 0.8 (5).

kept also for the desorption activation energy and preexponential factor. Nevertheless, the work15 does doubt the validity of this analogy. Let us consider the following simplest example: a desorption process from a homogeneous surface without association. For this case, the desorption rate takes a simple expression:

UD ) -Kθ[1 + txD]z, t ) 2θ/[b + δ], δ ) 1 + X(1 - 2θ), b ) {δ2+ 4Xθ(1 - θ)}1/2, X ) exp(-β) -1. (7) (7) Here, xD ) exp[β(* - )] - 1, X ) exp(-β) - 1, and * are lateral interaction constants for adsorbed adspecies in ground and activated state, and z is the coordinational number. We calculated with eq 7 the dependencies Eef(θ) and K0(θ) on coverage at different values of and *. These results are given in Figure 5. Although the spectra in Figure 5a are usual ones, one can see that the desorption activation energies at * * 0 behave somehow strangely: they increase at low coverages. In other words, such dependences of Eef(θ) mean that the desorption decreases with a rise of the surface coverage. Evidently, this should not be from the physical point of view. The only case of proper behavior is observed at * ) 0, when the activation energy coincides with the isosteric adsorption heat Q(θ); thus, this is a well-known dependence.42 This “light” anomaly in the dependence of Eef(θ) on coverage is not so harmless: to suppress an initial increase of Eef(θ), the pre-exponential factor may take unphysical large values. The physical reason for these “unphysical” results is due to an initial assumption in deducing the equation for the desorption rate. Namely, an elementary desorption act is supposed to be so fast that the surrounding of the desorbed particle does not change; that is, the desorption process occurs in the field of the fixed particles affecting the desorbed particle. Thus, a nonequilibrium state is

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considered when the use of convenient equilibrium relationships generally are not valid. As a result, the effective desorption energy changes its sign, and this is, of course, not a physical fact but just due to wrong application of equilibrium characteristics to the process not being in equilibrium. (If we would not agree with the restriction that an elementary desorption act is so fast that it can occur in nonequilibrium conditions, then we are obliged to take especially into consideration how a relaxation of the surrounding to an equilibrium configuration affects the elementary desorption act, and then we obtain another expression that gives qualitatively other results; see for example, ref 45.) Thus, there should be a doubt about anomalous high (e.g., 1015-1016, see below) or low (1010-109) preexponential desorption factors.30 As a rule, these “unexpected” results are due to incorrect application of eqs 4 and 5 to the process under investigation. 4.3. Self-consistency of the Transition-State Theory. As shown in the previous subsection, the activation energy of desorption coincides with the adsorption heat if there is no lateral interaction of the activated desorption complex and its surrounding, * ) 0. This is the model that Jones and Perry invoked46,47 to describe thermal desorption in the Hg/W(100) system. They were able to reproduce the experimental data with eq 4, taking K0(θ) ≡ K0 ) const and Eef(θ) ≡ Q(θ); the adsorption heat Q(θ) was calculated with the quasichemical approximation for a homogeneous surface (the original equation for Q(θ) was given in refs 42 and 48). The calculated and experimental spectra were found to be in a satisfactory accordance with the following molecular parameters: K0 ) 1016 s-1, HgHg ) 5.85 kJ/mol, and E(0) ) 172 kJ/mol. However, the fitted strength HgHg of lateral attractions is not enough to be responsible for the two-dimensional condensation from Hg atoms at the observed temperatures, although the same authors experimentally established this fact when measuring the equilibrium isotherms and then supported it by a Monte Carlo simulation.46 So, this is an example of such a theoretical treatment when thermodynamic and kinetic characteristics are not selfconsistent with each other. The theoretical failure above occurred because of the desorption equation used that does not allow for the real distribution of the Hg atoms; that is, the macroscopic equation used does not describe the two-dimensional adspecies condensation appearing in the experiment. It should be taken into account that adspecies are in two coexisting phases, so the total desorption rate is a sum of the desorption from these phases.49,50 The coverage of each phase is calculated from the two-dimensional phase diagram at fixed total coverage; the share of each phase in desorption follows from the rule of a level, imposed onto the two-dimensional phase diagram. We included this factor with the equations of section 2 and obtained an excellent accordance with experimental TDS (Figure 6) for the following molecular parameters: K0 ) 1013 s-1, E(0) ) 136.5 kJ/mol, HgHg ) 10.40 kJ/mol, and *HgHg /HgHg ) 0.55. The only difference in calculated and experimental results is observed at θ0 ∼ 1, and this is perhaps due to the formation of the second layer of adspecies that was neglected in our calculations. The values of lateral interaction constants found are quite reasonable: the (45) Tovbin, Yu. K. Russ. J. Phys. Chem. 1995, 69, 214, 220. (46) Jones, R. C.; Perry, D. L. Surf. Sci. 1978, 71, 59. (47) Jones, R. C.; Perry, D. L. Surf. Sci. 1979, 82, 540. (48) Wang, J.-S. Proc. R. Soc. London 1937, A161, 127. (49) Tovbin, Yu. K.; Kuznetsova, O. A. Zh. Fiz. Khim. 1990, 64, 1202. (50) Tovbin, Yu. K.; Votyakov, E. V. Zh. Fiz. Khim. 1993, 67, 141.


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Figure 6. The TDS for the Hg/W(100) system found experimentally (dashed)46,47 and with calculation (solid) including twodimensional condensation for the Hg molecules, an initial coverage θ0 is in the range 0.45 e θ0 e 0.80 for all the spectra excluding the very right spectra, where θ0 ) 0.99.

strength of lateral attraction, HgHg, is sufficiently large to be responsible for the two-dimensional condensation in the experimental temperature range; therefore, the anomalously high K0 that was obtained before takes an usual magnitude. Another result is a considerable decrease of the activation desorption energy, E(0). So, the values of the parameters found with the transition-state theory for reactions in condensed matter13,15 do lead to a consistency in thermodynamic and kinetic adsorption properties. Other systems of selfconsistency kinetic equations constructed with a lattice model are presented in refs 13 and 15. The lattice model is self-consistent; hence, with this model, both equilibrium and kinetic characteristics can be found on the basis of the same set of the molecular parameters. Thus, to reduce the number of unknown parameters involved in the lattice model from kinetic data, the independent experimental data are to be used. This especially refers to the data on equilibrium adsorption properties, to estimate the heterogeneous distribution function and lateral interaction constants. 5. An Entropy Factor in the Kinetics of Adsorption Processes 5.1. About “Two Variants” of the Absolute Rate Theory. As has been written in Nagai’s paper,31 “there are two groups of transition-state rate theory employed in the study of thermal desorption spectra”. In these theories, the desorption rates are expressed as

U(1) ) K(0)θΛ, K(0) ) ν0 exp[-βEd(0)]

(8)

U(2) ) ν exp{β[µ - Ed(0)]}

(9)

where K(0) is the desorption rate constant at θ ) 0; ν0 is the frequency factor; Ed(0) is the activation desorption energy; Λ is the factor, including nonideality of the adsorption system (this is the same factor as that in section 2 and eq 7 of subsection 4.2); µ is the chemical potential of the adsorbate system. Equation 8 being a single case of the formulas of section 2 can be deduced with the usual conceptions of the absolute rate reaction theory.14,13 It is worth mentioning that instead of the frequency factor ν or ν0, the relation of statistical sums for adspecies should be put in the ground (FA) and transition (F/A) states:33 K(0) ) (kBTF/A)/(FAh); however, for the strong localization of adspecies, this replacement is valid.

As to another group of the transition-state theory (TST), Nagai wrote that eq 9 was derived from refs 51 and 52 According to ref 51, U(2) ) ν exp{-β[Ed(0) - θEint]}θ/(1 - θ), where Eint is an energetic constant of the model not depending on coverage θ. All quantities that are in eq 9 and depend on the coverage are included in the equilibrium characteristics µ; however, the example of the previous section shows that involving the equilibrium expressions in the kinetic model can give nonphysical results. Thus, we first discuss in more details the origin of these formulas, and then we will consider the TDS of the CO/Ru(001) system53,54 because these experimental spectra were cited in ref 31 to support eq 9. First of all, the statement31 that formula 9 “can be derived most simply from TST” (transition-state theory) is doubtful: both of the cited papers51,52 do not use the principal foundations of this theory. The work51 essentially contains the second layer above the adsorbate; this layer is supposed to be filled with the particles that are in the transition state. Such two-layer models are intended usually either to describe multilayer adsorption or to include essentially a precursor state that is being an additional stage of a complex (three stages) process, but not for the elementary stage of the desorption process. Since the flaws of the model51 have been discussed by Cassuto55 before, they will be not repeated in details here. Briefly, the approach51 represents a phenomenological model that deforms the fundamentals of TST, and it has no strict physical substantiation. The paper52 was cited because of a factor (1 - θ) in the formula 51 on page 89, for a lifetime τ of the localized adspecies, from an analogy to eq 9, but without the numerator θ. It should be noted now that this formula (51) was cited itself from the paper.56 That is, De Boer himself pointed out that his result in ref 52 arose because he used a definition of τ given before in ref 56; such a definition requires additional introduction of a real lifetime for adsorption, which might be derived from his formula (51) if we omit the factor (1 - θ); so the final result given by formula (53) does not contain this factor.52 Note, the correct result for a lifetime of the localized adspecies was published by Frenkel57 earlier, without the factor (1 - θ). It is usually assumed that the famous Frenkel expression, τ ) τ0 exp(bQ) (here Q is the heat of adsorption), is valid for mobile adsorption58 only. Frenkel proved that this expression takes place with any type of adsorption. A type of adsorption affects the value τ0 only. (Evidently, here we deal with a homogeneous surface without lateral interactions.) Frenkel supposed that the adsorption activation energy is zero; otherwise, τ ) τ0 exp(βEd(0)),6 or τ ) τ0 exp(βEd(θ)) when we include the lateral interactions on a homogeneous surface. Thus, there is no cause to state that eq 9 has any theoretical substantiation. Nevertheless, another question still remains: can the phenomenological eq 9 agree better with experimental data or not if we apply the rigorous theory to the same data. As an example, Nagai considered the experimental spectra of the CO/Ru(001) system, so (51) Nagai, K. Surf. Sci. 1986, 176, 193. (52) De Boer, J. H. Adv. Catal. 1956, 8, 89. (53) Pfnur, H.; Feulner, P.; Engelhardt, H. A.; Menzel, D. Chem. Phys. Lett. 1978, 59, 481. (54) Pfnur, H.; Feulner, P.; Menzel, D. J. Chem. Phys. 1983, 79, 2400, 4613. (55) Cassuto, A. Surface Sci. 1988, 203, L656. (56) Kruyer, S. Koninkl. Ned. Akad. Wetenschap. Proc. 1955, B73, 73. (57) Frenkel, Ya. I. Statistical Mechanics; Izdatelstvo Academiya Nauk SSSR: Moscow-Leningrad, 1948. (58) Adamson A. W. Physical Chemistry of Surfaces, 3rd ed.; Wiley: New York, 1976.

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Figure 7. TDS calculated with eq 8 (a) and eq 9 (b), that were given in ref 31 to describe experimental spectra in the CO/ Ru(001) system; experimental data53,54 are in Figure 8b.

we turn to these data to compare his results and those which can be obtained with proper use of TST. The spectra found with the use of eqs 8 and 9 are shown in Figure 7a,b, respectively;31 experimental data for the CO/Ru(001) system53,54 are plotted in Figure 8b. For eq 8 (Figure 7a) Nagai used31 a hexagon model with infinite repulsion for the nearest neighbors; an interaction for other neighbors were neglected (this means Λ ) 1 in eq 8); for eq 9 (Figure 7b) he employed31 an exact value of µ. As evidently follows from comparison of these results and experimental TDS (the latter are in Figure 8b), neither of these formulas reflect the peculiarities of the experimental spectra; moreover, there are clear qualitative differences. For the first case, there is no splitting appearing in the experimental curves at initial coverage, θ0 > 0.35. For the second case, two peaks are around θ0 ) 0.3-0.333, although the experimental data show only one peak at the same coverage. On the other hand, the adsorbed CO molecules are under strong lateral repulsion; in particular, adsorption of these molecules on the Ru(001) face leads to an appearance of the R(x3 × x3) ordered structures in LEED patterns.53,54 This means absence of a uniform occupation of every adsoprtion site by the CO molecules; the CO molecules are accommodated in the different sublattices nonuniformly, and this is the typical case when lateral interactions are responsible for a nonuniform adspecies distribution on a homogeneous surface. Splitting of the experimental spectra is a signal of the R(x3 × x3) ordering; therefore, a kinetic molecular model to describe such spectra should be able to predict properly an adspecies ordering. Such a model can be derived from the equations outlined in section 2. Formally, to include adspecies ordering with

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Figure 8. CO/Ru(001) TDS found in this work including ordered structures (a) and experimental data53,54 (b). The parameters of calculated spectra are KD ) 1013 s-1, ED ) 132.3 kJ/mol, (1) ) -6.3, (2) ) 3.15 kJ/mol, *(r)/(r) ) -0.5; θ0 ) 0.38, 0.43, 0.49, 0.54, 0.58, 0.63, and 0.66.

quasichemical approximation, we should select, on an initial surface, a few sublattices that are matched with the desired ordered sketch, and then, we can consider the sites of the different sublattices as heterogeneous sites; thus, all the qausichemical equations for heterogeneous lattices (see section 2) can be applied in a formal way. (Insertion in Figure 9a shows a division of the hexagonal lattices into sublattices to take R(x3 × x3) ordering into account). Figure 8a gives our results with R(x3 × x3) ordering included as compared to experimental data in Figure 8b. The molecular parameters found describing the experimental spectra are quite reasonable: the first neighbors are under repulsion, (1) ) -6.3 kJ/mol, and the second neighbors are under attraction, (2) ) 3.15 kJ/mol; *(r)/(r) ) -0.5 for r ) 1 and 2. Such strengths of the lateral interactions provide a difference in the coverage of the sublattices (see below); so we can reproduce all the experimental features, including the TDS splitting and shape of the high-temperature peak as well. Differences at large initial coverages are evident because we used the simple variant of the model, taking care about R(x3 × x3) ordering; there are evidently far more lateral interactions that are of importance at such coverages. An account of the adspecies ordering is of the main importance for our model to reproduce the experimental results of the CO/Ru(001) system. To stress this fact, Figure 9a shows the coverages of different sublattices as a function of the temperature. The saturation coverage is around θ0 ) 0.66; this refers to the case when the R-sublattice, θR, is filled totally, and γ-sublattice, θγ, is empty. As the temperature increases, the total coverage θ of the surface decreases, and at θ ) 0.5, we see the beginning of the low-temperature peak and inversion of


Figure 9. For the CO/Ru(001) system, sublattice coverages as a function of temperature, and relevant spectrum (dots) (a) and TDS found at the same parameters as in Figure 8a but without R(x3 × x3) ordering (b). Insert is a sketch of a hexagonal lattice divided in such a way to include R(x3 × x3) ordering; closed circles, R; open circles, γ sublattices.

the sublattices. Below θ ) 0.5, energetically more favorable is the configuration when the R-sublattice is empty and γ-sublattice is filled. During this inversion, one observes the low-temperature peak, and then, the desorption rate quickly drops down; this is reflected in the TDS splitting. Further rise in the temperature at first increases the probability for the elementary act of desorption and then destroys the R(x3 × x3) ordered configuration. Both factors lead to the second peak in the TDS, and at the top of this peak, a difference in the coverages of the sublattices vanishes totally; thus, we can see the characteristic flexure in the high-temperature peak. In contrast to the above picture, the model without ordering gives a simple splitting at θ ) 0.5 (see Figure 9b), and this is a qualitatively wrong picture for the experimental data where the splitting begins at θ ) 0.33. This is a well-known property of the quasichemical approximation, provided we do not select sublattices for the ordered sketch.15 Considering a c(2 × 2) ordering on a square lattice, this provides the correct behavior, but for the case of a hexagonal lattice we should specially take care about R(x3 × x3) ordering. To finish our discussion, we point out that an application of either eq 8 or 9 to the experimental data does not enable their correct description. The reason is that eq 8 is too primitive (does not include adspecies ordering) and eq 9 is a phenomenological model. The system under consideration requires a more strict approach that takes nonideality of the heterogeneous systems into account; the general expressions for such systems are given in refs

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13 and 15. They include eq 8 as the simplest case, while eq 9 is not the consequence of these equations. As an example, we presented the results obtained by means of the TST approach that includes adspecies ordering, and one can see that these results are in much better agreement with experimental data than those for the phenomenological model, eq 9. The main reason is that our model properly reflects the physical processes observed in the experiment and is based on the correct physical basis. That is, if we develop an equilibrium model that takes a real adpecies distribution into consideration, and afterward, we apply the absolute rate theory to derive the kinetic equations, then we are able to reflect all the necessary entropy contributions in kinetic processes, including the configurational and nonconfigurational. Therefore, the absolute rate theory does not need any modifications due to an entropic factor; it is sufficient to use this theory in a proper way. 5.2. Absolute Rate Theory and Statistical Rate Theory of an Interfacial Transport. The work60 has presented so-called statistical theory of the reactions for an interphase transport32 (in short, the statistical theory of reactions) as applied to the problems of adsorption kinetics. As a conclusion, the authors60 stated that their theory is in better accordance with experimental data than transition-state theory. Though they noted separately that there were not equal conditions for the comparison because the approach proposed took special care about nonideal properties of the system studied (by means of a dependence of frequency for the harmonic oscillator on coverage), the TST was involved without such care. Therefore, below we consider the physical validity of the approach32,60,61 only. First of all, we might note that the name of the given approach can lead to a misunderstanding. For the last few decades, the theory of chemical kinetics uses the same name (statistical rate theory) for another scientific direction which is strictly proved. It was Landau who started this direction by the work,62 where he considered the rate of a monomolecular reaction for the gas-phase disintegration of large molecules. Afterward, the given theory got a wide development for reactions occurring in both the gas and condensed phase.63-65 In particular, a probability W of a large molecule to disintegrate can be written down with the theory considered as follows:

W ) Z exp{β[S(E - EA) - S(E)]}

(10)

where EA is the activation energy, E is the energy of the molecule, S(E) is the entropy of the molecule at uniform energy distribution over all the degrees of freedom, and Z is the normalization factor. When the energy EA is accommodated by a few degrees of freedom only, the entropy of the remaining part of the molecule should be equal to S(E - EA). The expression 10 is based on fluctuation theory66 being applied to form the transition state of the excited molecules (similar to the activated complex in TST). (59) Baxter, R. J. Exactly Solved Models in Statistical Mechanics; Academic Press: London, 1982. (60) Elliott, J. A. W.; Ward, C. A. In Equilibrium and Dynamics of Gas Adsorption on Heterogeneous Surfaces; Rudzinski, W. A., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997; p 285. (61) Ward, C. A. J. Chem. Phys. 1983, 79, 5605. (62) Landau, L. D. Phys. Zs. Sowjet. 1936, 10, 67. (63) Nikitin, E. E. Theory of Elementary Atomic-Molecular Processes in Gases; Khimiya: Moscow, 1970. (64) Eyring, H.; Lin, S. H.; Lin, S. M. Basic Chemical Kinetics; Wiley: New York, 1980. (65) Hangii, P.; Talkner, P. Rev. Mod. Phys. 1990, 62, 251. (66) Landau, L. D.; Lifshits, E. M. Theoretical Physics. Vol.5. Statistical Mechanics; Izdatelstvo Nauka: Moscow, 1964.

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The approach32,60,61 has no relevance to the statistical theory above. This approach considers a transition between different phases as an elementary process with use of the formalism from quantum mechanics. Then, assuming that the energy of the system is constant within an uncertainty ∆E, an expression for the transition rate from one phase of molecular configuration, li, to another phase of molecular configuration, lj, is written as

J(λi,λj) ) K(λi,λj)exp{β[S(λi) - S(λj)]}, K(λi,λj) ) (2π/p)ωi(E)|Vnm|2 (11) where ωi(E) is the microscopic-state density estimated at the mean energy of the system, |Vnm| is an average over absolute elements of the transition matrix of the perturbation potential operator V, for the transition from the microscopic states of the molecular distribution λn to those states of molecular distribution λm, and S(λi) is the entropy of the molecular distribution λi, p ) h/2π. The final result for the adsorption process is the net rate of the molecular transport from phase γ to phase R presented as

JγR ) J(λi,λj) - J(λj,λi) ) K(λi,λj){∆i - ∆i-1}, ∆i ) exp{β[µR(λi) - µγ(λj)]} (12) This can be calculated afterward with the chemical potential of the molecules in adjacent phases R and γ as well as K(li,lj), the characteristics of the process of redistribution for molecules between these phases in the equilibrium state. The most important assumption used in deriving eqs 11 and 12 is that the intermediate (or transition) state of the process is completely neglected. This contradicts all the theories of chemical kinetics, where an elementary stage is presented as overcoming the activation barrier. The only case when a similar assumption can be valid (for the first approximation only) can be related to the reactions running through zero activation barrier; for these reactions, the reaction rate is limited by the diffusion transport of reagents.67 These reactions are known as diffusioncontrolled reactions; they are practically irreversible and have no relevance to adsorption-desorption processes. If we would trace the way to derive eqs 11 and 1232,61 in more detail, we would see that these equations arise because of an inconsistent use of the formalism of quantum mechanics. Although the authors32,61 apply the quantum mechanical description for elementary stages, they do not use Pauli’s quantum-mechanical equation68 to calculate the net flow JγR for molecules moving from the initial discrete degenerated state to the final degenerated state and in the opposite direction; thus, this takes the degeneration of each state into account incorrectly. Therefore, these equations have no quantum-mechanical substantation, and thus, they represent a phenomenological model like eq 9. Moreover, another quite strong assumption that the magnitudes ωi(E) for initial and final states are equal should also be under doubt for gas adsorption on solids. To conclude, different variants of the approaches,31,32 where additional shares of the entropy factor are proposed to be included, are not connected with the fundamentals of chemical kinetics. Moreover, both approaches51,32 take care about lateral interactions with the mean-field ap(67) Ovchinikov, A. A.; Timashev, S. F.; Belyi, A. A. Kinetics of Diffusion-Controlled Chemical Processes; Khimiya: Moscow, 1986. (68) Pauli, W. Collected Scientific Paper; Kronig, R., Weisskopt, V. S., Eds.; Interscience: New York, 1964; p 549; Vol. 1.

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proximation, which as shown above fails to describe thermal desorption over the entire range of initial surface coverage. Like any phenomenological model, such approaches can be applied to describe experimental data in a formal way; however, such an application cannot give a guarantee that processes in the system studied will be correctly explained. Indeed, the adsorption theory needs development if we include in more exact way the real physical factors of the system studied in order to explore a connection between these factors and heterogeneous adspecies distribution. Then, the transition-state theory can be applied to study kinetic processes, and the results obtained will look quite reasonable. 6. Molecular Interpretation of the TDS for the H2/Ir(110)2 × 1 System The TDS for the H2/Ir(110)2 × 1 system, measured by Ibbotson et. al.,69 have the following distinctive features: (1) Two peaks in which the areas under them are approximately 2:1; (2) the low-temperature peak is appreciably broad (it moves toward low temperatures because of an increase of initial coverage, showing the order of desorption is considerably larger than 2; (3) the high-temperature peak is narrow, it is almost constant as a function of initial coverage, so the order of desorption is slightly larger than 1. Evidently, the first point is evidence of surface heterogeneity, while the others are due to lateral interactions. (In the absence of adspecies lateral interactions, the order of desorption for the system considered should be 2, because this is dissociative desorption.) To qualitatively explain the features above, the authors of the experimental work69 proposed an adsorption model where hydrogen molecules are accommodated into adsorption sites of two types that are related 2:1. The reconstructed Ir(110)2 × 1 face enables such a picture: there are parallel rows located in such a way where two rows of one type are alternated with one row of another type (see insert in Figure 10a). Therefore, a sufficient difference in bond energies for the hydrogen molecules accommodated in the top and bottom row would provide two peaks in the TDS at saturation coverage, and the relation of the areas under the peaks would be exactly the same as the numbers of adsorption sites in the rows. Then, peculiarities of the peaks can be understood with taking lateral interactions into account. Since the high-temperature peak shows the desorption order as almost 1, there should be lateral attractions for the adspecies being in the strong row; and, in a similar way, the opposite picture for the low-temperature peak is caused by lateral repulsion of the hydrogen molecules located in the weak row. We applied our kinetic theory to describe quantitatively this experimental system and to illustrate how an experimental qualitative hypothesis can be checked with a quantitative description. The equations to calculate the rate of dissociative desorption have almost the same structure as those given in section 2. (They are given elsewhere;13,15 here, we do not repeat them.) The main difference concerns the unary concentration (for nondissociative desorption) that must be replaced by a probability to find two adspecies near each other (for dissociative desorption) and, of course, the factor that takes into account lateral interactions for the activated desorbed complex consisting of two atoms. (69) Ibbotson, D. E.; Witting, T. S.; Wheinberg, W. H. J Chem. Phys. 1980, 72, 4885.


Figure 10. TDS for the H2/Ir(110)2 × 1 system. Field (a): the family of calculated TDS (solid) at θ0 ) 0.06, 0.27, 0.37, 0.48, 0.61, 0.86, and 0.99; experimental spectrum69 (dashed) at maximal θ0. Field (b): the family experimental TDS. Insert in field (a), the structure of the Ir(110)2 × 1 face; large circles, iridium; small circles, hydrogen atoms; two hydrogen atoms at the top are in a weak row and those at the bottom, in a strong row.

There are two groups of molecular parameters in the model proposed by Ibbotson et al.:69 the first one is mainly responsible for surface heterogeneities; these are the difference in bond energy for the strong and weak sites, ∆Q, and rate constants of dissociative desorption, Kqp, where qp ) 11, 12, and 22 (subscript “1” refers to the strong and “2” to the weak sites.), Kqp ) K0qpexp(-βEqp); another group includes lateral interaction parameters qp(r). The latter were assumed to be dependent on the type of the adsorption site because the desorption order is different for the low- and high-temperature peak. Although unknown parameters should be jointly fitted in the general case, a prior analysis has shown that in the first steps it is possible to distinguish the contribution to TDS from surface heterogeneity and lateral interactions. For the system studied, the surface heterogeneity is basically responsible for TDS splitting, while lateral interactions affect the order of desorption. So, at the first moment we can assume these quantities are independent of each other, and after we roughly estimated these factors, we can fit them together. Below is an explanation for the procedure to fit unknown parameters. Indeed, we easily found the parameters that are responsible for the surface heterogeneity. They were estimated as follows: a difference in bond energy in the strong and weak row, ∆Q, is around 21 kJ/mol; and activation desorption energies are E11 ) 90.3, E12 ) 81.9, and E22 ) 69.3 kJ/mol and all the pre-exponential factors were assumed to be of an usual value, K011 ) K012 ) K011 ) 1013 s-1. These estimates enable the relevant splitting of the TDS to be calculated, as well the relative heights of high- and low-temperature peaks; however, the width of

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the spectrum at saturated initial coverage was too small to reproduce experimental data. There were two ways to get a satisfactory accordance for this factor: either to increase the degree of surface heterogeneity, ∆Q, as well the range between the maximal and minimal Eqp or to assume a larger strength for lateral repulsions of the nearest neighbors. However, model simulations have shown that the former way resulted in a splitting that was too deep relative to the experimental TDS, while the latter caused additional peaks. So, to solve the problem, we have considered in more detail a nature of lateral interactions. It is supposed that, for chemisorption, the first neighbors are under lateral repulsion, while the second and further neighbors can have lateral interaction parameters of various signs.70,71 We considered interactions for the first, second, and third neighbors (these were included with the quasichemical approximation), whereas the long-range lateral interactions (further than the third neighbors) have been estimated with the mean field approximation.13,15 (In the same manner, the spectra for the Si(100)2 × 1 system have been explained72). The involved lateral constants allowed us to reproduce the peculiarities of the experimental peaks: broadening of TDS and the order of desorption for low- and hightemperature peaks. The long-range lateral interactions are summed over all the neighbors excluding those where the quasichemical approximation is applied, and for the case of repulsion, they provide a broadening of the TDS, not changing its qualitative details. The fitted value of this repulsion for the H2/Ir(110)2 × 1 system was found to be 12.6 kJ/mol. Since the nearest neighbors are assumed to be repulsive in chemisorption, to fix the location of the high-temperature peak as a function of initial coverage, we should employ an attraction of third neighbors in the strong row. Otherwise, for the order of desorption at coverages responsible for forming the high-temperature peak, it is impossible to be slightly larger than 1, because the dissociative desorption itself is of the desorption order 2; moreover, the first neighbor repulsion should evidently increase the order of desorption. (A similar conclusion was made originally by experimentalists.69) As to the order of desorption for the low-temperature peak, since it is larger than 2, this effect can be essentially simulated with the repulsion of the first neighbors. The lateral interaction constants were estimated as follows (minus is repulsion): qp(1) ) -2.1 (the same for all the nearest neighbors), 12(2) ) 2.1, and 11(3) ) 4.2 kJ/mol, and other parameters are zero. Now, we appeal to Figure 10 where the calculated (Figure 10a) and experimental (Figure 10b) spectra are plotted for comparison. One can see a satisfactory accordance, the calculated TDS reproduce all the quantitative peculiarities of the experimental data, that is, a degree of splitting, relation in heights of the low- and high-temperature peak (and as a consequence, a relation in areas under these peaks), the order of desorption larger than 2 for the lowtemperature peak and almost 1 for the high-temperature peak. Thus, we quantitatively tested the adsorption model for the H2/Ir(110) 2 × 1 system that was proposed originally in the experimental work.69 All fitting parameters are quite reasonable, apart from the strength of attraction for the third neighbors in the strong rows. Indeed, in absolute value, this is twice as high as the repulsion of the first neighbors. Nevertheless, (70) Muscat, J. P. Prog. Surf. Sci. 1987, 25, 191. (71) March, N. H. Prog. Surf. Sci. 1987, 25, 229. (72) Tovbin, Yu. K.; Votyakov, E. V. Surf. Phys. Chem. Mech. 1993, No. 11, 31 (in Russian).

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this result can be understood if we consider a competition of lateral interactions for the adspecies accommodated in the strong row: without lateral interactions, the desorption order is 2, and the numbers of the first and third neighbors in the strong row are the same; hence, to suppress the repulsion of the nearest surrounding neighbors and to reach the desorption order 1, the strength of the third attraction should be sufficiently higher than that of repulsion. To find a physical explanation for this effect, either a quantum mechanical method to calculate adspecies lateral interaction should be applied or there are additional physical processes that were not taken into account in our model. For instance, these can be a reconstruction of the Ir(110)2 × 1 face during thermal desorption or a penetration of hydrogen molecules under the iridium atoms, and so forth. 7. Summary This paper has considered fundamental questions of TDS interpretation that refer to the case of heterogeneous distribution of adsorbed species over the surface. A use of strict kinetic theory, which includes jointly surface heterogeneity and lateral interactions, enables a test to find the limits of the validity of phenomenological conceptions and approaches for heterogeneous surfaces.We checked traditional conceptions that explain TDS peaks only as a manifestation of the distribution function type for a heterogeneous surface. It has been shown that this is wrong for the general case, in particular, in contrast to the usual opinion that a continuous distribution function can be responsible for a broadening TDS only; we demonstrated that this factor can lead to a splitting in TDS, if a difference in maximal and minimal adsorption heat of the surface is enough. On the other hand, a discrete distribution function, provided adspecies strongly repulse each other, can result in additional peaks in the TDS so the total number of the peaks is larger than that of the types of adsorption sites. Moreover, we discussed the validity of molecular interpretations of the TDS using effective activation desorption energy and effective preexponential factor as well as the new treatments which invoke so-called entropic factor into adsorption-desorption kinetics. Our conclusion is that such approaches are not based on a hard theoretical backbone; these are

Tovbin and Votyakov

essentially phenomenological and so cannot be applied to treat experimental data in order to get correct molecular parameters. Statistic theory based on the theory of absolute reaction rates include all energetic and entropic factors into equations for reaction rates. The main problem of TDS interpretation is to elaborate on a physical model that reflects the real nature of heterogeneous distribution of adspecies over adsorption centers. The nonuniformity in adspecies distributions can be due to either 2D phase transitions (2D ordering or 2D condensation) or a difference in adsorption strength of various adsorption centers. In the course of discussion, we presented a proper description for TDS of the CO/Ru(100) and Hg/W(100) systems. For the first case, our model, taking adspecies ordering into account, is in satisfactory accordance with experimental data, the fitted molecular parameters being quite reasonable, and for the second case, our model, taking the adspecies condensation into account, is an illustration for self-consistency of models based on the transitionstate theory in contrast to a phenomenological approach. Moreover, the spectra for the H2/Ir(110)2 × 1 system have been reproduced in good accordance with experimental data, when we include jointly surface heterogeneity and lateral interactions. Thus, we have shown that any phenomenological method generally does not allow a proper molecular interpretation for the desorption process. To get such an interpretation, it is necessary to use a molecular model, for example, on the basis of a lattice model that is capable of jointly including both surface heterogeneity and lateral interactions. This model can describe the entire family of TDS curves at various initial coverages, reflecting all the quantitative peculiarities of experimental spectra. In contrast to any phenomenolgical theory, this model is selfconsistent; that is, it enables us to treat both equilibrium and kinetic characteristics on the basis of the same set of molecular parameters. This property can reduce the number of unknown quantities invoked to describe kinetic data, if we additionally use independent experimental data (e.g., on equilibrium properties containing information about heterogeneous distribution functions and lateral interaction constants). LA981492K

Analysis of Thermal Desorption Spectra of Heterogeneous Surfaces

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