256
Langmuir 1988, 4 , 256-268
groups. A deficit of carbonyl groups indicates that the films may also contain PMDA fragments that have lost carbonyl groups.
by the Office of Naval Research, the National Science Foundation Grant No. DMR-8403831, and by the Royal Society.
Acknowledgment. We gratefully acknowledge our discussions with C. Feger. This work was supported in part
Registry No. PMDA, 89-32-7; ODA, 101-80-4; ODA-PMDA polyimide, 25038-81-7; Ag, 7440-22-4.
Precursor-Mediated Adsorption and Desorption: A Theoretical Analysis? Douglas J. Doren and John C. Tully* AT&T Bell Laboratories, Murray Hill, New Jersey 07974 Received July 17, 1987. In Final Form: August 21, 1987 Existence of a weakly bound precursor to chemisorption has been proposed to explain a variety of kinetic observations in gas-urface reactions. We develop a theoretical approach to adsorption-desorption kinetics which unifies the different kinds of precursor states that have been invoked. Our approach employs a “potential of mean force” or free energy as a function of the reaction coordinate, widely used in liquid-state theories. This potential incorporates a rigorous measure of the thermally averaged effects of the ignored coordinates on the reaction rate. As a result, it is temperature dependent, and it can be quite different from any cut through the potential hypersurface. We define a precursor in terms of the potential of mean force and show that it coincides with the usual qualitative understanding of the term in several model problems. Specificallywe consider intrinsic precursors to associative and dissociative chemisorption on a clean surface and an extrinsic precursor model for a partially covered surface. A major conclusion of this study is that the notion of a precursor should not be equated with multiple minima of the gas-surface potential. The one-dimensional potential of mean force picture is sufficient to allow interpretation of equilibrium rates of desorption and general features of thermally averaged sticking probabilities. However, it may be necessary to consider multidimensional aspects of the gas-surface interaction to understand the origin of some features of the reduced potential and to predict detailed dynamical behavior.
Introduction The concept of a “precursor”-a weakly bound, mobile speciesthat precedes strong chemisorption-plays a central role in discussions of chemical reaction mechanisms a t surfaces.l-14 Several types of precursors have been invoked to explain a diverse variety of observed behaviors. A distinction has been made between an “intrinsic precursor”, which involves an isolated adsorbate on a clean surface, and an “extrinsic precursor”, which arises only at non-zero coverage due to adsorbate blocking of chemisorption sites. A “dynamical precursor”,which is mobile by virtue of being transiently energized, has also been suggested. However, a universal criterion for what actually constitutes a precursor state is lacking. In this paper we propose a definition of precursor based on a traditional concept in liquid-state theory, the “potential of mean force”.15 The definition is rigorously founded, it appears to correspond correctly to the desired physical picture of a precursor, and it applies to all of the commonly invoked types of precursors. We apply the potential of mean force analysis, first, to some simple models that represent both intrinsic and extrinsic precursors and both intact and dissociative chemisorption. The models illustrate the separation of adsorption and desorption rates into “equilibrium” and “dynamical” factors and show how precursors can alter sticking probabilities and thermal desorption rates. In Presented a t the symposium entitled “Molecular Processes at Solid Surfaces: Spectroscopy of Intermediates and Adsorbate Interactions”, 193rd National Meeting of the American Chemical Society, Denver, CO, April 6-8, 1987.
particular, we show that the Arrhenius prefactor for simple molecular desorption may be reduced by 2 orders of magnitude or more due to the presence of a precursor. Next we examine a sequence of realistic potential energy hypersurfaces constructed as possible alternative representations of the interaction of an intact CO molecule with a bare Ni(ll1) surface. We employ the potential of mean force technique to reduce these multidimensional interactions to effective one-dimensional potential curves, which clearly reveal the presence or absence of precursor states. These examples demonstrate that the ability or lack of ability to isolate a weakly bound adsorbed species at low temperatures does not necessarily signal the presence or absence of precursor-mediated kinetics at higher temperatures. (1)Langmuir; I. Chem. Reu. 1929,6, 451. (2) Lennard-Jones, J. E. Trans. Faraday SOC.1932, 28, 333. (3) Taylor, J. B.; Langmuir, I. Phys. Reu. 1933, 44, 423. (4) Ehrlich, G. J. Phys. Chem. 1955, 59, 473. (5) Kisliuk, P. J. Phys. Chem. Solids 1957, 3, 95. (6) Kisliuk, P. J. Phys. Chem. Solids 1958, 5, 78. (7) King, D. A.; Wells, M. G. Surf. Sci. 1971, 23, 120. ( 8 ) King, D. A.; Wells, M. G. Surf. Sci. 1971, 29, 454. (9) Grunze, M. J.; Fuhler, J.; Neumann, M.; Brundle, C. R.; Auerbach, D. J.; Behm, J. Surf. Sci. 1984, 139, 109. (10) Adams, J. E.; Doll, J. D. Surf. Sci. 1981,103,472; 1981, I l l , 492. (11) Harris, J.; Kasemo, B.; Tornqvist, E. Surf.Sci. 1981, 105, 1288. (12) Tang, S. L.; Beckerle, J. D.; Lee, M. B.; Ceyer, S. T. J. Chem. Phys. 1986, 84, 6844.
(13) Weinberg, W. H.In Kinetics of Interface Reactions; Grunze, M., Kreuzer, H. J., Eds.; Springer-Verlag: Berlin, 1987; p 94. (14) Auerbach, D. J.; Rettner, C. T. In ref 13; p 125. (15) Hill, T. L. Statistical Mechanics; McGraw-Hill: New York, 1956;
p 193.
0743-7463/88/2404-0256$01.50/0 0 1988 American Chemical Society
Precursor-Mediated Adsorption and Desorption
Langmuir, Vol. 4, No. 2, 1988 251
Reaction Rate Theory and the Potential of Mean Force Chemical processes at surfaces have aspects in common with both gas-phase and liquid-phase reactions. In situations where the reactants and/or products involve molecular species distant from the surface, a gas-phase scattering picture is appropriate. Questions concerning the efficacy of initial translational or internal energy in promoting reaction, the nonequilibrium disposal of energy in products, and the lifetime of the gas-surface "collision complex" are relevant. During the time the reacting molecules are in close contact with the surface, however, the surface plays a role similar to that of the solvent in solution chemistry. Free energies are mediated by the 'bath" atoms, and frictional or stochastic elements are introduced. We outline below a theory of the kinetics of adsorption and desorption at surfaces that is a synthesis of standard concepts from gas-phase and liquid-phase reaction rate theory. We begin by writing the potential of interaction of a molecule with a surface in the form V(q,s), where s denotes the position along the "reaction coordinate", which we will define to be the minimum energy path connecting the chemisorbed and desorbed configurations.16 For adsorption/desorption of an intact molecule, s is ordinarily accurately approximated by the distance z of the center of mass of the molecule from the surface plane. For dissociative chemisorption, a more complicated reaction coordinate must be employed analogous to gas-phase reaction rate theory.16 All of the remaining coordinates orthogonal to s, including the orientation angles of the molecule, its intemal coordinates, its lateral position along the surface, the coordinates of other adsorbates, and the coordinates of all surface atoms, are encompassed in the vector q. We now consider an equilibrium canonical ensemble of gas-surface systems interacting according to the potential V(q,s) at some temperature T. For this ensemble, we can define a position distribution function, g(s)ds, which is proportional to the probability that the molecule will be located between s and s ds. The function g(s) is entirely analogous to the "radial distribution function" in liquid state theory.15 For a classical mechanical canonical ensemble, g ( s ) is given by
+
g(s) = G - l S dq exp[-V(q,s)/k~T]
(1)
where kB is Boltzmann's constant and G-' is an arbitrary normalization factor. Following conventional theories of liquids,17J8we define the potential of mean force, W(s), to be the effective one-dimensional potential that would give rise to the distribution g ( s ) W(s) = -kBT In g(s) W o (2) where Wo defines the zero of energy. Usually we desire W(m)= 0, requiring Wo = kBT In g(m) (3) Note that with this choice of Wo the potential of mean force is unaffected by the choice of normalization of g(s). The potential of mean force is simply a free energy potential curve. It depends on temperature and pressure (or coverage). As T 0, W(s)approaches the minimum energy potential along the reaction coordinate. For non-zero
+
-
(16)Truhlar, D. G.; Isaacson, A. D.; Garrett, B. C. In Theory of Chemical Reaction Dynamics: Baer,. M... Ed.:. CRC Press: Boca Raton. FL, 1985; Vol. IV, p 65. (17) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976; p 266. (18) Hynes, J. T. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. IV, p 171.
T, W(s) can deviate substantially from the minimum energy potential, and this difference can have an important quantitative and qualitative influence on the reaction rate. The potential of mean force enters directly into the standard theory of reaction rates. The exact (classical mechanical) rate constant, k(T), of a chemical reaction is conventionally split into a dynamic factor, fs(T),and an equilibrium or thermodynamic factor, k,*( T)19,20 k ( T ) = fs(T)k,*(T) (4) The factor k,'(T) can be identified with the "transition state theory" (TST) rate constant, and f,(T) is commonly referred to as the "recrossing correction" (or "transmission factor"). The TST rate constant is defined to be the one-way equilibrium flux through a "dividing surface" that separates reactants from products. The dividing surface is defined by all possible values of the coordinates q but with the reaction coordinate fixed at a particular value, s. The position s is conventionally chosen to correspond to the position of maximum energy along the minimum energy path, but as we show below, it is preferable to choose s to be the position of maximum free energy, i.e., the maximum of the potential of mean force. The TST rate constant is an upper limit to the exact (classical mechanical) rate constant, k( T);19any trajectory which evolves from reactants to products must pass through the dividing plane at least once and thus contribute to the equilibrium one-way flux. But some trajectories may cross the dividing plane more than one time, so the flux through the dividing plane may exceed the reaction rate. The recrossing correction f,(T) is therefore always less than or equal to unity. I t may be calculated directly via forward and backward integration of multidimensional classical trajectories originating at the dividing surface, as shown by many w ~ r k e r s . l ~ - ~ ~ The TST rate constant k,*(T) can be expressed compactly in terms of the potential of mean force, W(s). The one-way equilibrium flux through a dividing surface located at s is simply the mean velocity of motion, 8, through the dividing surface multiplied by the equilibrium density at the dividing surface. The latter is proportional to g(s), as defined in eq 1. The former, for a Boltzmann distribution, is 8 = (kBT/2"p)'/'
(5) where p is the reduced mass associated with motion along the reaction coordinate. Thus k,*(T) = ( ( k ~ T / 2 " p ) ~exP[-W(s) '~ /k~q)/
(L
ds'exp[-W(s')/kBq)
(6)
where R indicates that the integral in the denominator is carried out over the range of s' that is defined to be the reactant state. This is an exact expression for the classical mechanical equilibrium one-way flux through the dividing surface, normalized to unit density of reactants; i.e., eq I is the exact TST rate constant for a dividing surface chosen at position s along the reaction coordinate. Thus the potential of mean force evaluated in the reactant region and (19) Keck, J. C. Adu. Chem. Phys. 1957, 13,85. (20) Pechukas, P. In Dynamics of Molecular ColZisior~~, Part B; Miller, W. H., Ed.; Plenum: New York, 1976: D 269. (21) Anderson, J. B. J. Chem; Phyd.-1973, 58, 4684. (22) Bennett, C. H. In Algorithms for Chemical Computation; Christofferson, R. E., Ed.; American Chemical Society: Washington, D.C., 1977: D 63. (23j Montgomery, J. A., Jr.; Chandler, D.; Berne, B. J. J. Chem. Phys. 1979, 70,4056. (24) Grimmelmann, E. K.; Tully, J. C.; Helfand, E. J. Chem. Phys. 1981, 74, 5300.
Doren and Tully
258 Langmuir, Vol. 4, No. 2, 1988
at the dividing surface determines the TST rate constant. The rate constant k * ( T ) defined by eq 6 has units of s-l. For first-order processes, k * ( T ) is the conventional firstorder rate constant. For higher order processes, k * ( T ) given by eq 6 depends on the volume of the system. The conventional volume-independent rate constant is given by VN-lk*(T)for a bulk process and AN-lk*(T)for a surface process, where N is the reaction order, V is volume, and A is surface area. Equation 6 does not look like the usual expression for the TST rate constant:25
where h is Planck’s constant, Q* and Qo are the transition-state and reactant partition functions, and E* and Eo are the energies of the transition state and reactant. The partition functions Q* and Qo are frequently evaluated by assuming separability of degrees of freedom and employing simple one-dimensional expressions for vibrations, rotations, translations, etc. But the full multidimensional classical partition functions could be computed exactly as phase-space integral~:’~
l
dq exP[-V(q,s)/kBTl (8)
where N is the total number of atoms in the system and pi is the reduced mass associated with the ith degree of freedom. Similarly,
Substituting eq 8 and 9 into eq 7 gives eq 6. The exact rate constant, k(V,does not depend upon the choice of dividing surface location, s. However, the individual rate factors f,( T ) and k,*(7‘) can depend strongly on s. It is important to make an intelligent choice of dividing surface, particularly when the dynamical factor f,(T) is to be computed only approximately or set equal to unity. In calculating the rate of desorption from surfaces, it is frequently convenient to take the dividing surface location, s, to be at a large distance from the surface, outside the interaction distance of the gas with the surface. Calculation of the TST rate constant then involves information about the bound adsorbate-surface properties and about the free surface and molecule only; no information about intermediate gas-surface separations is required.26 Furthermore, for this choice of s, the recrossing factor f,( T ) can be identified with the thermally averaged sticking probability S ( V . If the dividing surface is outside the interaction region, there w ill be no forces that could reverse an outward moving trajectory. Thus the only nondesorption trajectories that can cross the dividing surface in the outward direction d e those which initially were directed inward from the gas phase and then reflected from the surface without sticking. An alternative procedure for choosing the location of the dividing surface is “variational transition state theory” (VTST).Ig This approach has been employed very suc(25) Laidler, K. J. Theories of Chemical Reaction Rates; McGraw-Hill: New York, 1969; p 47. (26) Tully, J. C.; Cardillo, M. J. Science (Washington,D.C.) 1984,223, 445.
cessfully in gas-phase s t ~ d i e s . ~ ’ -The ~ ~ idea is simple. As stated above, the TST rate constant k,*(T) is a rigorous upper bound to the exact classical mechanical rate constant k(T). By varying the location of the dividing surface to minimize k,*(T),one obtains the least upper bound to k(7‘), Le., the closest approximation to k(T). This is the VTST rate constant. From eq 6, it is seen that the VTST location of the dividing surface corresponds to the position of the maximum of the potential of mean force W(s). Thus, for low temperatures VTST becomes essentially equivalent to conventional TST with the dividing surface chosen at the peak of the energy barrier. At higher temperatures, however, the maximum of the potential of mean force may deviate substantially from the maximum of the minimum energy potential. This will be illustrated below. The recrossing correction, f,(T), can be calculated exactly, within classical mechanics, by forward and backward integration of trajectories on the full-dimensional potential energy surface, initiating at the dividing surface s.19-24 A variety of approximations to f,(T) have also been proposed.16 One approximation which we will make use of here is the “unified statistical m 0 d e 1 , ” ~which ~ ~ ~ is applicable to systems that exhibit two local maxima in the potential of mean force along the reaction coordinate between reactants and products. This is outlined in Appendix A. With this preliminary discussion completed, we now propose a definition of a precursor: A precursor state is a secondary local minimum in the potential of mean force, W(s),that is located between the chemisorption state and the state of infinite separation of the molecule from the surface. Note that, according to this definition, for a particular system a precursor may exist for some ranges of temperature and/or coverage and may not exist for others. We apply this definition to some examples, below, to demonstrate the emergence of precursor states in different situations and the use of the potential of mean force analysis to determine the effects of precursors on adsorption and desorption kinetics.
Simple Examples 1. An Intact Molecule on a Clean Surface. We now apply the concepts of the previous section to some exactly solvable models of adsorption/desorption. The models are grossly oversimplified and are intended as illustrations only. A more realistic example is addressed in the next section. The first model we consider represents the interaction of an intact molecule with a clean (zero-coverage) surface. We define a gas-surface interaction potential &,e) that depends on the distance z of the center of mass of the molecule from the surface plane and the orientation angle 6’ of the molecule axis with respect to the surface normal. The surface is assumed to be flat and structureless. Specifically, we write V(z,O)= V,(Z) + V,(z,6’)
V2(z,S)= A sin2 (8/2) exp[-a(z - zO)]
(lOc)
(27) Truhlar, D. G.; Garrett, B. C. Acc. Chem. Res. 1980, 13, 440. (28) Garrett, B. C.; Truhlar, D. G.; Grev, R. S. In Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G., Ed.; Plenum: New York, 1981; p 587. (29) Hase, W. L. “Properties of Variational Transition States for Association Reactions”, to be published. (30) Miller, W. H. J . Chem. Phys. 1975, 65, 2216. (31) Pollack, E.; Pechukas, P. J. Chem. Phys. 1979, 70, 325. (32) Garrett, B. C.; Truhlar, D. G. J. Chem. Phys. 1982, 76, 1853.
Langmuir, Vol. 4, No. 2, 1988 259
Precursor-Mediated Adsorption and Desorption
-1I
b)
-2
0
1
2
3 2
4
6)
0
6
04
0 00 2: 0
\I ,
,
1000 T (K)
6)
::::I
12
1 ;:I, ,
2000
6
1.3
wo 1 1
,
4
1.4
v
v,
2
2
>,
3
5
0
,
,
1000 T MI
,
,
~
101
2000
0
1000 T (K)
2000
Figure 1. Example 1,intact molecule. (a) Equal energy contours as a function of distance, t,of molecule center of mass from surface and orientation,8, of molecular axis with respect to surface normal. Energies are in eV. (b) Potentials of mean force for five different temperatures. (c) Unified statistical model sticking probability, eq A2. (d) Solid curve: Arrhenius activation energy for desorption, E,, as a function of temperature from unified statistical model, eq A3, using eq 16. Dashed curve: E, from TST with dividing plane at z m. (e) Arrhenius desorption prefactor, v , from eq 17, using unified statistical model (solid curve) and TST with Z * m (dashed curve).
-
-+
. VI(%)is a 3-9 potential describing the attraction of the molecule to the surface when in its most favorable orientation, t9 = Oo. V2(z,t9)vanishes when t9 = Oo and becomes repulsive for 0 # Oo, reaching a maximum for t9 = a,i.e., when the molecule is oriented upside down. Thus V2(z,t9) encompassesforces that orient the molecule perpendicular to the surface, with the favorable bonding end toward the surface. The parameter D in eq 10b is the binding energy of the molecule to the surface. The parameter zo, the minimum energy separation of the molecule from the surface, can be determined from the molecule-surface stretching vibrational frequency, a,: zo = (270/Ma,2)1/2
(11)
where M is the total mass of the molecule. The parameter A of eq 1Oc can be determined from the bending frequency, W b l , of the molecule on the surface A = 21ab2
(12)
where I is the moment of inertia of the (linear) molecule. The parameter CY determines the falloff of the orientational potential as the molecule recedes from the surface. Assuming this falloff is proportional to the falloff of the electron density of the surface, a can be estimated from the work function CY
= 1.025#'/2
(13)
where is the work function in electronvolts and CY is in angstroms. We have assigned parameters appropriate for the interaction of a CO molecule with Ni(ll1): D = 1.3 eV, M = 28 amu, a, = 359 cm-l, a b = 480 cm-', I = 8.72 amu.A2, and = 3 eV. We reemphasize that this is a very un-
realistic model, employed only as a illustration of the concepts outlined in the previous section. Equal energy contours of the potential V(z,t9)are shown in Figure la. There is no hint of a precursor minimum in the two-dimensionalpotential; it is downhill all the way to the chemisorption minimum. However, there is a constriction of the energy contours as the binding site is approached, due to the very strong orientational forces introduced in the potential. For this potential the minimum energy path connecting the adsorption and desorption states is simply s = z,t9 = Oo. The potential of mean force W(s)can be computed analytically. From eq 1 and 2
W(Z)= -kBT ln
l)!2xT
dt9 sin 6 eXp[-V(Z,6)/kBnl (14)
where t9 is the only coordinate q and sin 0 is the volume element. Substituting eq 10 into eq 14 and carrying out the integration yields
-
This potential of mean force is plotted in Figure 1b for several different temperatures. At T 0, W(z)converges to the bare 3-9 potential. However, as T increases, the constrained geometry of the adsorbate is manifested as an increase in its free energy relative to the free desorbed state. At temperatures above 700 K for this choice of parameters, an actual potential barrier and weak secondary potential minimum emerge; i.e., by our definition a precursor state exists for this simple model at temperatures above 700 K. For temperatures above 1800 K, the max-
260 Langmuir, Vol. 4, No. 2, 1988
Doren and Tully
m
,20
1
2
3 2
4
5
6
0
2
4
6
(A)
10
3
10
0 6
.
--
.
Y
cn
0 4
w" 0 9
0 2
08
00
0
1000
2000
0.7 0
1013
1000
2000
0
1000
2000
T (K) T (K) T (K) Figure 2. Example 2, intact molecule with explicit bump in potential curve. Plots are same as for Figure 1.
imum in the potential of mean force rises above the asymptotic zero of energy. The sticking probability, approximated by eq A2, is shown in Figure IC.For temperatures below 700 K, where the potential of mean force is monotonic in energy, the uniform statistical model sticking probability is unity. However, at higher temperatures the small barrier and precursor minimum result in a significant lowering of the sticking probability. Arrhenius parameters are shown in parts d and e of Figure 1. The activation energy is defined to be (16) where k is the desorption rate constant. The prefactor, v, is defined to be v ( T ) = k exp[&(T)/kBTl
(17)
For Figure ld,e, the solid curves employ the uniform statistical model rate constant, eq A3. The dashed curves represent standard TST, i.e., eq 6 with the transition state chosen to be the molecule at large separation from the surface. TST with this choice of transition state is equivalent to assuming unit sticking probability. Thus the dashed and solid curves of Figure Id,e coincide at low temperatures where the uniform statistical model sticking probability is unity. The prefactors, v, are shown in Figure le. With our choice of parameters, the desorption rate falls in the experimentally convenient range 10°-106 s-l for temperatures between 400 and 600 K. In this temperature range, the prefactor is large, of order 1015s-l. This arises from the reduced entropy of chemisorption associated with the alignment of the molecule. This effect has been well d o c ~ m e n t e d . ~ An ~ . ~additional ~ ? ~ ~ increase in v of at least (33) Pfnur, H.; Feulner, P.; Engelhardt, H. A.; Menzel, D. Chem. Phys. Lett. 1978, 59, 481.
an order of magnitude would be expected if constrained lateral motion of the adsorbate were included in the model. 2. Intact Molecule with Precursor. The second simple model we consider is identical with the first, except that a small potential barrier has been added along the minimum energy path, thereby introducing a weak precursor minimum. Specifically v(z,e) = VAZ) + v2(z,e) + v3(Z)
(18)
where Vl(z) and V2(z,e)are given by eq 10b and lOc, with the same choice of parameters as example 1. The new term V&) is simply a Gaussian bump V ~ ( Z=) 0.5 exp[-0.6(z - 2.2)2]
(19)
where distances are in angstroms and energies in electronvolts. The addition of this term and the choice of parameters were completely arbitrary, guided only by the desire to produce a potential with qualitatively reasonable double-minimum behavior. Figure 2a is the equal energy contour plot for this model potential. It is similar to Figure la, except for small differences due to the additional term, V~(Z). Figure 2b shows potentials of mean force calculated for this model at different temperatures. The T = 0 K curve is the minimum energy potential, exhibiting the small added potential barrier. The magnitude of the barrier is enhanced at higher temperatures. Figure 2c shows the sticking probability, S,(T), calculated from eq A2 for this model. The sticking probability decreases rapidly in the range 250-500 K, as the maximum in the potential of mean force rises above the asymptotic zero of energy. Thus a small barrier in the minimum energy path that does not rise above the E = 0 level can produce a dramatic decrease in the sticking probability. It can also cause a dramatic change in the Arrhenius desorption parameters, as shown in parts d and e of Figure (34) Ibach, H.; Erley, W.; Wagner, H. Surf. Sci. 1980, 92, 29.
Precursor-Mediated Adsorption and Desorption
Langmuir, Vol. 4, No. 2, 1988 261
b)
"" h
L
2000 K -1
0
1
2
3 z
4
5
1
1
l
I
6)
I Or
081 c ,
I
1
I
I
I
l
l
6
s (8) 0.6I
1
I-
I
O
U
10-4 10-5-
I000 xxx) T (K) T (K) T (K) Figure 3. Example 3, dissociative chemisorption. (a) Equal energy contours (eV) as a function of distance, z, of center of mass of molecule from surface and internuclear separation,r. (b) Potentials of mean force for five different temperatures. (c) Unified statistical model sticking probability. (d) Arrhenius second-order desorption activation energy. (e) Arrhenius prefactor.
0
2. As before, the solid curves are the uniform statistical model predictions, and the dashed curves are the standard asymptotic TST, assuming unit sticking probability. The dashed curves of Figure 2d,e have no knowledge of the presence of the precursor bump (except for the incidental 0.19-eV decrease of the binding energy resulting from the tail of the Gaussian bump). The major effect of the presence of the barrier/precursor on the desorption rate is the lowering of the prefactor Y by more than an order of magnitude. In the experimentally relevant temperature range 400-600 K, the prefactor is of order 5 X 1013 s-l, considerably reduced from the >1015 s-l value in absence of the potential barrier. This decrease in the prefactor arises from two factors. First, the lowering of the sticking probability, which multiplies the desorption rate, and second, the location of the transition state at a finite distance from the surface where some constraint in the orientational motion still persists, partially offsetting the constrained orientation in the chemisorption state. This prefactor lowering would likely be even more significant if constrained lateral adsorbate motion were included in the model. These considerations are general and suggest that ,systems that exhibit precursor states may produce "normal" Arrhenius prefactors of order 1014s-l, in contrast to the 1015s-l expected for strongly oriented adsorbates. This may provide a partial explanation for the large number of observations of desorption prefactors of order ' 1013-1014s-l for strongly oriented intact adsorbate^.^^ 3. Dissociative Chemisorption. The previous two examples addressed intrinsic precursors involving intact molecule adsorption and desorption. This example still examines an intrinsic (i.e., zero-coverage) precursor, but the chemisorption state corresponds to a dissociated (35)Morris, M. A.; Bowker, M.; King, D. A. In Comprehensiue Chemical Kinetics; Bamford, C. H., Tipper, C. F. H., Compton, R. G., Eds.; Elsevier: Amsterdam, 1984; Vol. 19, p 1.
1ooo2ooo
0
species, preceded by a molecular precursor state. As before, the model is grossly oversimplified and intended only as an illustration of the application of potential of mean force ideas to this situation. We define the reaction coordinate for this example by (z
- 1.878)(r - 1.128) = 2
(20)
where r is the internuclear separation of the diatomic adsorbate and z is the distance of the center of mass of the molecule from the surface. Distances are in angstroms. The distance s along the reaction coordinate is s =z
- r - 0.75
(21)
Large positive s corresponds to desorption, large negative s to dissociative chemisorption. The distance q locally
perpendicular to the reaction coordinate is
4 = [ r - r m i n ( s ) l [ z - zmin(s)l
(22)
where rmin(s) and zmin(s)are the values of r and z on the reaction path, i.e., values that satisfy eq 20 for a given value of s. The interaction potential is taken to be V(s,q) = Vl(4
+ V2(4)
Vl(s) = -[2 exp(-l/zs) - 3.2 exp(-s)
(23a)
+ exp(-2s)]/[l + 4 exp(-2s)] (23b)
V2k) = q2
(234
Vl(s) is a modified Morse potential that exhibits a local minimum between reactants and products. The parameters assumed are completely arbitrary. V,(q) simply introduces a harmonic restoring force prependicular to the reaction coordinate, assumed here to be unchanged as a function of s. Equal energy contours for this potential are shown in Figure 3a. The molecular precursor to dissociation and
Doren and Tully
262 Langmuir, Vol. 4, No, 2, 1988 12
b)
T = 1000 K 8 = 0,05,085,095, 0 99
10
a 6 4 2 0 0 2 4 6 0 1 0 1 2
I O
1.41
T=250 K
0
2
4
6
0.5
1.0
1
*
v)
0.4 02
0
0.5
io
0
8
0.5 9
1.0
0
e
Figure 4. Example 4, extrinsic precursor. (a) Schematic equal energy contours. (b) Potentials of mean force for different values of surface coverage, 8. (c) Unified statistical model sticking probability. (d) Arrhenius desorption activation energy. (e) Arrhenius prefactor. All as a function of coverage 8. Solid curves, 250 K; dashed curves, 500 K; dot-dashed curves, 1000 K.
the saddle point are apparent. As for the previous models, the potential of mean force can be computed analytically: W(s)= Vl(s) - kT In (27rr) kT In ( 2 ~ ~(24) )
+
where ro is the equilibrium diatomic separation. The second term on the right-hand side of eq 24 arises from the increased configuration space associated with large r (dissociated atoms). It provides an entropic driving force for dissociation at higher temperatures. The final term in eq 24 simply shifts the zero of energy so that W(m)= 0. Potentials of mean force are plotted in Figure 3b for several temperatures. The effect of the entropic driving force is seen to be significant but not dominant. Sticking probabilities computed from eq A2 are shown in Figure 3c. The sticking probability behaves as expected for an activated adsorption process with an apparent threshold energy of about 0.03 eV, compared to the 0.04 eV that the minimum energy path potential rises above E = 0. The slight lowering of the apparent threshold energy is due to the reduction of the barrier at finite temperatures, Figure 3b, due to the entropy term of eq 24. The presence of the precursor well does not have a substantial effect on the sticking probability, within the approximation of the unified statistical model. The second-order area-independent desorption rate constant for this model is given by k2(T) =
(k,T/ 2 ~ ) ' / ~ [ 2 ? r r , ~=~m () IsS ( T ) exp[EAT) / ks?l (25) where Ei(T)is the energy of the chemisorption state, -0.25 eV in this example. The factor 27rrmin(s=m) arises from the integration weighting r dr in the adsorption region. Parts d and e of Figure 3 show the activation energies and prefactors obtained by applying eq 16 and 17 to eq 25, using the uniform statistical model sticking probability for S(T) in eq 25. The activation energy is nearly constant and approximately equal to the sum of the 0.25-eV chemisorption energy and the 0.04-eV barrier height. The
prefactor is also roughly equal to expectations for a second-order rate process. No surprises emerge from this example, except perhaps for the lack of influence of the precursor state. Nevertheless, the example illustrates how the potential of mean force analysis can be applied to dissociative chemisorption. Indeed, for more realistic interactions complicated by inclusion of many degrees of freedom, the analysis presented here may prove quite valuable in producing a simple but meaningful one-dimensional representation of the reaction process. 4. Extrinsic Precursor. Our final example of this section is a simple model of an extrinsic precursor, Le., a weakly bound molecule that is blocked from strong chemisorption by another adsorbate. The potentials of mean force and Arrhenius rate parameters we derive for this example are defined for constant coverage, 8. The interaction potential we examine is
V(z)= V1(z) = V,(z)
(empty site) (filled site)
where V,(z) = A[exp[-2a(z - zl)]- 2 exp[-a(z - zl)]]
"[ (s)
V&) = 2
(264
- (3
s)]
(26b) (26c)
The Morse potential parameters describing strong binding were taken to be A = 1.3 eV, a = 2.22 A-1, and z1 = 2.0 A. The Lennard-Jones parameters describing the precursor state were taken to be D = 0.15 eV and z2 = 4.5 A. The fraction of filled vs unfilled states is determined by the coverage, 8. The arrangement of filled sites (ordered, islands, random) does not enter into the simple analysis presented here. An illustration of the model is shown in the contour plot of Figure 4a. This plot is only schematic and is not in-
Langmuir, Vol. 4, No. 2, 1988 263
Precursor-Mediated Adsorption and Desorption
tended to quantitatively represent the potential of eq 26 for some assumed adsorbate arrangement. (This is in contrast to Figures la, 2a, and 3a, which are quantitative representatives of the respective interactions.) For a specified coverage 0 , O I0 I1, the potential of mean force for this model is given by 0
W ( z )= -kBT 1n {Jd0 eXp[-v&)/k~T]
+
where, as in the earlier examples of intact molecule adsorption, we have identified the reaction coordinate with the distance z of the center of mass of the molecule from the surface. Thus
W(z)= - ~ B In T (0 exp[-Vdz)/kBl + (1 - 6) exp[-Vl(z)/kBTIlJ (28)
W(z)is plotted in Figure 4b for several different coverages, with temperature equal to 1000 K. At 0 = 0 there is no precursor; the potential of mean force is monotonic. At any finite coverage a precursor state emerges, as expected. More interestingly, at high coverages the maximum of the potential of mean force rises above the asymptotic zero of energy. Thus there is a free energy barrier to entering the chemisorption state, even though the potential energy surface defined by eq 26 exhibits no barrier. The Arrhenius activation energies and prefactors derived from this model are relatively weakly dependent on temperature and coverage, as shown in Figure 4d,e. The sticking probability is more interesting, however. At low temperatures, e.g., 250 K, the uniform statistical model sticking probability is non-unity and remains relatively constant for coverages less than about 0.6. This behavior is quite similar to that arising from the standard Kisliuk model for an extrinsic ~ r e c u r s o r However, .~~~ in the present model there are no assumptions about the detailed kinetic pathways, e.g., diffusion across adsorbate islands to find an unfiied site, pushing aside of adsorbate atoms, etc. The constancy in sticking probability does not require the existence of a precursor over unfilled sites, with equal trapping probabilities and binding energies of the two types of precursors. There is only one precursor here, and it does not exist over unfilled sites. In the present model, the constancy of sticking with coverage results from the dominance of energy over entropy factors at low temperatures. The potentials of mean force, at T = 250 K, are nearly identical for coverages between 0.01 and 0.6. In eq 28, kBTIn 0 and kBTIn (16) are relatively unimportant terms compared with those involving V&) or Vl(z), respectively. The non-unity sticking arises, in the uniform statistical model invoked here, from the fact that at any (except extremely low) coverage the shallow Lennard-Jones term dominates the potential of mean force at distances of order z = 4.5 A. Thus there is a peak in the position distribution function in this region and competition between chemisorption and desorption for these precursor molecules as specified by eq A2. At higher temperatures, entropy begins to exert a larger influence, and the sticking probabilities no longer show a constancy with coverage. kBTh 6 and ~ BhT(1 - 6) terms become more significant, and the potentials of mean force change with coverage, as shown in Figure 4b for T = 1000 K. Experimental observations of temperature and coverage dependences of sticking probabilities have been reported that are quite similar to those exhibited by this
model, Figure 4c. One example is for CO adsorption on ~i(100).36 This concludes our discussion of some simple models that exhibit precursor behavior. We reiterate that these models are grossly oversimplified and are intended only to illustrate qualitative behavior. A more realistic sequence of assumed interaction potentials is examined in the next section.
Realistic Examples As an illustration of these methods on a realistic multidimensional potential, we have constructed a model of the interaction of a CO molecule with a clean Ni(ll1) surface. The potential takes a form employed previously in numerical simulations of other sy~tems.~'It is described fully in Appendix B. Since the vibrational spectrum of CO chemisorbed on Ni is well characterized experimentally, the model potential has been chosen to match the available data in the deep well. However, there are no direct measurements of the interaction potential in the region where a precursor well or a barrier might occur. To investigate the effects which such a structure could have, we have postulated an additional contribution to the potential which introduces a barrier between the deep chemisorption well and the weakly attractive region at large distances from the surface. We shall discuss three versions of this system, differing only in the nature of the precursor barriers. The first is the unadulterated potential, with no added barrier. There is a strong angular dependence of the attractive terms, so it is comparable t? the first example of the last section. A contour plot of this model is shown in Figure 5a. The full model has many degrees of freedom not shown here, but there is relatively little variation in the potential as a function, for example, of lateral position over the surface (i.e., the surface is quite flat). Comparison with Figure l a shows that the present example has a much larger attractive region and is not so strongly repulsive in unfavorable orientations. As a result, the potentials of mean force (Figure 5b) show no barrier and secondary minimum. (A description of our calculation of these mean force potentials is included in Appendix B.) In order that the potential of mean force be repulsive at a given z , the full potential must be repulsive for a significant fraction of the accessible orientations at that z. As in the earlier examples, we can calculate sticking probabilities and Arrhenius parameters as a function of temperature. Since there is only a single minimum in the potential of mean force, the unified statistical theory reduces to transition-state theory; the TST results are shown as triangles in Figure 5d-e. The sticking probability in TST, with dividing plane chosen at large z, is always unity. We have also calculated an accurate estimate of the true sticking probability by direct stochastic classical trajectory simulations on the full multidimensional potential, as described in detail e l s e ~ h e r e . ~Where ~ the results are distinguishable from TST, they are shown in Figure 5 as solid circles. At all temperatures studied, the sticking probability is so large that there is no significant modification of the transition-state theory results for the activation energy or prefactor. The next interaction potential is the same as the last except for the addition of a Gaussian bump; thus it is (36)DEvelyn, M. P.; Steinruck, H.-P.; Madix, R. J. Surf. Sci. 1987, 180. 47.
---7
(37) Muhlhausen, C. W.; Williams, L. R.; Tully, J. C. J . Chem. Phys.
1985,83, 2594.
(38) Tully, J . C. J. Chem. Phys. 1980, 73, 1975.
Doren and Tully
264 Langmuir, Vol. 4, No. 2, 1988 180 150
->
120 8
0.0
J -0.5-
90
3 60 -1.030
0 0
1
2
3 Z
4
5
6
0
1
2
3
(5
Z
4
5
6
(a
0.8 0.6 0
VI
v)
0.4
0.0 0
10’2
500
0
1000
1000
500
500
0
1000
T(K) T(K) T(K) Figure 5. CO on Ni(lll),no explicit barrier. Plots a and b are the same aa for Figure 1. (c) Sticking probability from classical trajectory calculations. Arrhenius activation energy E, (d) and prefactor Y (e) from TST with dividing plane at z m (triangles)and from trajectory calculations (circles, where distinct from the TST result). -+
0.5
180
150
120 I000 K
2 -0.5
la 90
Y
3 60
30 -1.5
0
0
1
2
3
4
5
0
6
1
2
3
0.6 0.4
. ..
2
0.0 0
1.2
1.o
0.2
500
1000
5
z (A)
z(A)
v)
4
0
1013
500
1000
0
500
1000
T (K) T (K) T (K) Figure 6. CO on Ni(lll), small amplitude barrier with slow angular cutoff. Plots are same as for Figure 5 except that plots c-e also contain predictions of unified statistical model (squares,where distinct from the TST result).
analogous to example 2 of the last section. The bump has the form v,(~,o)= 0.4 exp(-(z - 3.3)2/1.4) exp(-02/31 (29) Here Z is the height in angstroms of the center of mass of CO above the surface, measured from the instantaneous
average z-position of the nearest 36 surface atoms. The angle 0 determines the tilt of the CO axis away from the surface normal; the units are radians, and 0 = 0 rad when the carbon end is directed toward the surface. Enernv units are electronvolts. Again the parameter values are arbitrary, being chosen to provide a plausible potential
Precursor-Mediated Adsorption and Desorption
Langmuir, Vol. 4, No. 2, 1988 265
L
-1.5 0
1
2
3
4
A 5
6
(8) 0.8 m
0.6 0.4
0.2 0.0 0
O O O
0
.
ln
Y
w“
1.1
e
e
I000
0.9 0
500 I000 0 500 T (K) T(K) Figure 7. CO on Ni(lll), larger barrier with faster angular cutoff. Plots are same as for Figure 6. 500
1000
T (K)
surface. Because this barrier depends on 0 instead of the height above the equilibrium surface plane, there can be some weak energy exchange between the gas molecule and the barrier. The factor depending on 9 has been included so that the net repulsive effect is not too strong in unfavorable orientations. Without any angular cutoff, the sticking probability is nil. Figure 6a is a contour plot of this potential. There is now a large range of 9 for which the potential is always positive, but when 9 = O‘ it is still downhill all the way into the deep well. As shown in Figure 6b, this is sufficient to generate a secondary minimum in the potentials of mean force. Parts c-e of Figure 6 show the sticking probability and Arrhenius parameters as derived from transition-state theory (triangles), the unified statistical approximation (squares, where distinguishable from TST), and full classical trajectory simulation (circles). In this example the sticking probabilities are much less than unity, so the modifications to the TST activation energies and prefactors are substantial. The activation energy is increased with respect to TST at high temperatures; prefactors are reduced by half an order of magnitude. Our final example differs from the last only in the shape of the Gaussian bump, which now takes the form V&9) = 0.6 exp{(i- 3.3)2/1.4}exp{-e2/2) (30) The height of this bump at its maximum is greater than in the last example, but the angular cutoff is more rapid. Now the minimum energy pathway is not downhill all the way. At larger angles the contours of Figure 7a look qualitatively quite similar to those of the last example, though this surface is clearly more repulsive in general. Corresponding potentials of mean force are shown in Figure 7b; as expected, they are generally higher in energy than their counterparts in Figure 6b. The sticking probability is Substantially lower than in the last two examples. This modifies the prefactors by an order of magnitude or more relative to the TST values. The largest corrections to the prefactor in these models are not due to the entropic
effects described in the last section; the transition state is not very restricted. The large shifts are a dynamic effect due to barrier recrossing. In each of the last two examples the sticking probabilities decrease as a function of temperature until they reach a minimum and then level out or increase slightly as the temperature increases further. This feature is observed in both the unified statistical theory approximation and in the full trajectory simulations. Similar behavior has been observed experimentally; it suggests that there are two competing mechanisms for adsorption and this is taken as evidence for the existence of a precursor well and barrier.12J4 In the usual picture these potential features are static, but a similar interpretation in terms of the potential of mean force is at least as simple. At low temperature, gas molecules travel slowly and need not lose much energy in order to be trapped. As temperature increases, the inner barrier gets higher and the likelihood of crossing it must decrease. That is, the “precursormediated” mechanism leads to lower sticking probability with increasing temperature. On the other hand, the models presented in this section have an upper bound on the barrier height: the barrier grows with increasing temperature because more unfavorable orientations are sampled, but since the most unfavorable configuration in the barrier region has an energy slightly over 0.1 eV (1200 K), above 10oO K or so they will all be sampled effectively and the barrier will only grow slightly. Since typical gas kinetic energies will continue to increase with T, a “direct” mechanism takes over in which most of the molecules reaching the surface arrive directly from the gas phase, rather than by diffusing over the barrier from the precursor well. This description depends only on equilibrium velocities of the gas and the relative heights of the maxima surrounding the precursor well. As such, the unified statistical theory is sufficient to reproduce the behavior. It is apparent that the sticking probabilities obtained from full simulation can be substantially lower than the prediction from the unified statistical model. This illus-
266 Langmuir, Vol. 4, No. 2, 1988
trates a shortcoming of the unified statistical model for surface problems with a strong dependence on orientation. The model assumes that there are no recrossings between the two wells. However, when a gas molecule strikes the surface in an unfavorable orientation, it stays relatively far from the surface and energy exchange is weak. Hence many molecules will bounce back to the shallow well region. Since the likelihood of such recrossings is related to the true sticking probability, we would expect larger deviations from S,(T) as the true sticking probability decreases, and this is observed in all of our examples. The potential of mean force is defined for all values of the reaction coordinate while TST and the unified statistical theory only exploit the free energy in a few isolated regions. We could imagine trying to develop a more accurate theory of adsorption and desorption by running trajectories on this reduced one-dimensional potential, with the inclusion of some type of frictional and fluctuating forces. In contrast to the unified statistical theory, this would include effects of energy exchange at the repulsive wall at small z and would incorporate more general types of recrossing effects. Naturally the friction and fluctuating forces must be stronger near the surface. For a simple Markovian friction for which the frictional force is proportional to velocity in the z-coordinate,its effect will be greatest when the particle moves fastest, i.e., when the mean force potential is most negative. In these three examples, for any fixed temperature, the potential of mean force has a successively smaller region in which the potential is strongly negative, thus predicting less energy loss to the solid and hence a smaller sticking probability. A calculation similar to that suggested here has been performed by Adams and Doll.’O The potential was simply two square wells separated by a square barrier. A simple friction modulated as a function of distance from the surface was used to model the effects of dissipation. One conclusion of that work was that at high temperatures the sticking probability depended strongly on the width of the inner well and activation barrier, though not on the details of the secondary well. This suggests that the picture we have proposed may be qualitatively correct, but physical reasoning indicates that important dynamical effects would be neglected. Since the potential has a strong orientation dependence, only those molecules with a favorable orientation can sample the high friction part of the surface, so the molecules that do not stick are expected to be dominated by ones that hit the surface the wrong way. Reorientation occurs on a time scale comparable to motion in the reaction coordinate, so it cannot be treated reliably as another frictional effect. In other words, we expect at least two coordinates must be explicitly accounted for to accurately model the dynamics on these surfaces. Since the mean force potential averages over all orientations, it is unlikely that a dynamical model based on it can display these features. Construction of two- or three-dimensional potentials of mean force may be required. Conclusion The potential of mean force analysis provides a useful, precise means for reducing multidimensional potentials to a one-dimensional potential in a single reaction coordinate. It is applicable to any anisotropic potential regardless of the physical origin of the various maxima and minima. It could even apply, in principle, to electronically nonadiabatic systems involving multiple potential energy surfaces. Since it is obtained through an equilibrium average, it depends on entropic effects and can be substantially different from the minimum energy pathway. In
Doren and Tully
particular, wells and barriers may appear in the reduced potential which are not evident from a simple cut through the hypersurface. We emphasize that the measure of energy which determines a reaction rate is the potential of mean force and not the bare interaction potential. As McQuarrie17has stated, “one enters the realm of rigorous statistical mechanics when this distinction is clearly appreciated.” The potential of mean force analysis appears particularly useful for describing precursor-mediated processes at surfaces. A simple definition of a precursor emerges that unifies the various types of precursors that have been invoked. As illustrated by the examples in the previous sections, a precursor-at least as defined here-can come or go in a particular system depending on temperature and coverage. The effect, or lack of effect, of a precursor on a rate is thus immediately evident. We note that this definition distinguishes between a true precursor that is a weakly bound intermediate to adsorption and an alternative weak binding site. For example, the interaction potential for CO on Ni(ll1) shown in Figure 5a exhibits a weak second binding site for upsidedown CO, 0 = 180°, with binding energy of 0.07 eV. This is only hinted at in the figure by the increasing spacing between the 0.0- and -0.05-eV contours, but a real local minimum exists. This local minimum appears to have no effect on adsorption or desorption kinetics at room temperature or above, and there is no sign of it in the potential of mean force. Thus it is not a precursor by our definition. However, if trajectories are calculated for collision of very low energy CO molecules (Ei < 0.01 eV) on a 0 K surface, a significant fraction of the trajectories, of order 3070,get trapped in this upside-down state. Thus isolation of a weakly bound state at low temperature does not necessarily demonstrate the existence of a precursor to chemisorption, a t least by our operational definition. In particular, the low-temperature freezing out of a weakly bound CO species on Ni(ll1) achieved by Shayegan et al.,39associated with a reduced work function change, might possibly be due to an admixture of normal and upside-down CO with oppositely aligned dipoles. In addition to producing a simple qualitative picture of the energetics of chemisorption, the potential of mean force analysis is useful for quantitative estimates of sticking probabilities and desorption rate parameters. Variational transition-state theory is based on the potential of mean force and frequently provides an accurate description of desorption. The examples discussed above show how the variational transition state can move discontinuously from an asymptotically large separation from the surface to a finite separation. This can have a significant effect on the rate parameters. We have shown that the presence of a precursor well and secondary barrier in the potential of mean force can reduce the Arrhenius prefactor by 2 orders of magnitude or more, through a combination of two effects. The first is a partial cancelling of the entropic driving force from orientational or binding site constraints in the adsorbate, due to similar, albeit reduced, constraints at the transition state. The second is the reduction of the sticking probability due to reflection off the secondary barrier of unfavorably oriented molecules. Thus a “normal” prefactor of 1013-1014s-l for desorption of an intact molecule from a surface might be indirect evidence for precursor-mediated desorption. The variational transition-state approach and the unified statistical model applied in this paper provide simple but (39)Shayegan, M.; Glover, R. E., 111; Park, R. L.J . Vac. Sci. Technol. A 1986, A 4 , 1333.
Langmuir, Vol. 4, No. 2, 1988 267
Precursor-Mediated Adsorption and Desorption
J
useful approximations to adsorption and desorption rates based on the potential of mean force. The one-dimensional description provided by the potential of mean force could provide the framework for more sophisticated approximations, analogous to frictional theories for liquid-state reactions.18 However, as discussed above, the strong distance and orientational dependence of energy transfer at surfaces will likely render introduction of a simple frictional force inadequate. We have not discussed such dynamical questions as whether an equilibrium distribution is achieved in the precursor well, whether gas molecules are likely to be trapped in this well, or how the presence of a secondary minimum can affect surface diffusion or the velocity and internal-state distributions of desorbed particles. These questions will be addressed in future work.
Appendix A The unified statistical model of reaction rates was developed to interpolate between two simple limiting cases? when there exists a dividing surface that is never crossed more than once, it reduces to transition-state theory; when there is a long-lived intermediate complex with competition between a process leading to reactants and one leading to products, it becomes unimolecular "phase space theory". The theory can be derived by postulating a "complex" region, containing a local minimum of the free energy curve and bounded by dividing surfaces at local maxima. It must be assumed31that (1)every trajectory entering this region crosses the minimum; (2) successive recrossings of the local minimum are random, uncorrelated events; and (3) trajectories which leave the region never return. Thus the unified statistical theory can predict corrections to TST (i.e., sticking probabilities less than unity) to the extent that they arise because molecules arrive in the precursor well and then decide whether to stick or desorb on a strictly statistical basis. It will not account for molecules that penetrate closer than the inner dividing surface and then bounce off the repulsive wall of the gas-surface interaction. For a canonical ensemble, the unified statistical model predicts a dynamic correction to TST of the form16
f,( T ) = exP IW*( s )/kB?l / b P [ W*(s) / k B r l + e x p W * ( s ) / k ~ m- exp[W*(s)/kBfll (-41)
*
where denotes the higher maximum in the potential of mean force, Le., the variational transition state, and denotes the lower maximum. The asterisk denotes the minimum that lies between the two maxima. For adsorption and desorption from surfaces, this provides an approximation, S,(T), to the sticking probability, for adsorption into the chemisorption state (not trapping into the precursor):
**
SJT) = (1+ exP[W'(s)/kBTl - exp[W*(s)/k,rlI-l (A2)
S,( T ) approximates the thermally averaged sticking probability for both gas and surface at temperature T. W'(s)is the potential of mean force at the inner barrier, if it exists, not the asymptotic s m state. W*(s)is the potential of mean force of the secondary (precursor) minimum, if it exists. It is assumed that the outer maximum is the asymptotic state at the zero of energy. If a secondary maximum or minimum does not exist, then the corresponding potential of mean force W1(s) or W*(s) is simply set equal to zero in eq A2.
-
Table I. Interaction Potential Parameters A 85.13 eV E 0.088 eV.A3 B 0.77 eV a 3.366 A-1 C 8.73 eV.A9 B 1.683 D 0.69 eV.A3 re 2.0 A
The thermal desorption rate constant ku(7'), in the unified statistical model, is
kJT) = ((kBT/27rd'/2su(7'))/( R ds'exp[-W(sl/kd'l)
(A31
Equation A3 is obtained by substituting eq A1 into eq 4 and 6, again assuming W ( m ) = 0.
Appendix B The multidimensional interaction of CO with the Ni(111)surface has been modeled with a potential of the form V = CVl(rC,ro,ri)+ C Z - ~ (D+ E cos2 0 ) r 3 + Vb(z,O) i
(B1) where rc and ro are, respectively, the coordinates of the carbon and oxygen atoms; rl is the coordinate of the ith substrate atom; z is the height of the CO center of mass above the equilibrium position of the surface plane; 0 is the angle between the CO axis and the surface normal (0 = 0" when CO is in its most stable orientation, with the carbon atom toward the surface); and z is the height of the CO center of mass above the instantaneous average z-coordinate of the nearest 36 atoms on the surface. The first term on the right of eq B1 represents the strong, angle-dependent "chemical" interactions between CO and the individual Ni atoms. I t is pairwise additive, the s u m being carried out over a 6 x 6 slab of surface atoms and another 6 x 6 slab in the second layer of the solid. The interaction with an individual Ni atom is given by Vl(rc,ro,rl)= A exp(-alr, - rol) + B{exp[-2P(lrl - ~ c Ire)] - 2 cos2 v1 exp[-P(lr, - rcl - r e ) ] ) 032) The first term is a purely repulsive interaction between the oxygen and surface atoms. The second term is a modified Morse potential that describes the attractive interaction between the surface and the carbon end of the molecule. The attractive term depends on the angle q1 formed by the 0, C, and Ni atoms. With the values of the parameters given in Table I, this has the effect of making the molecule most strongly bound when in an atop site, oriented normal to the surface. The parameters for the oxygen interaction, A and CY, were chosen equal to values used in earlier calculations of NO scattering from metal surfaces.37 The Morse parameter P was taken equal to CY/^ so the short-range repulsions are the same for both atoms; B and re were chosen to produce a Ni-C stretch frequency of 400 cm-' and an equilibrium Ni-C distance of 1.8A in an atop site. The second term of eq B1 represents a short-range repulsion at the surface. The third term is a weak attraction due to image dipole and van der Waals forces. The coefficient C of the repulsive term was taken to be the same as for NO ~ c a t t e r i n g .The ~ ~ attractive term parameters were determined from the polarizability of CO and the plasma frequency of Ni. This relationship is well known for an atom interacting with a Drude model for the meta1.40t41 When the gas atom is replaced by a diatomic molecule, some additional geometric factors are introduced (40) Steele, W.A. The Interaction of Gases with Solid Surfaces; Pergamon: Oxford, 1974; p 68. (41) Mavroyannis, C . Mol. Phys. 1963, 6, 593.
268
Langmuir 1988, 4, 268-276
and both the parallel and perpendicular polarizability of CO are needed.42 The final term in eq B1 is a barrier to chemisorption described more fully in the text. The CO molecule itself is treated as a rigid rotor of length 1.13 A. The internal vibratiohs of CO are so high in frequency (~1800 cm-l) compared to the surface Debye frequency of Ni ( ~ 1 5 cm-’) 0 that there is little coupling between them. Table I lists all the potential parameters used in this study. One notable feature of this potential surface is the relatively small variation in energy for different orienta(42) Harris, J.; Feibelman, P. J. Surf. Sci. 1982, 115, L133.
tions or lateral positions for a fixed height of the molecule above the surface. This and the fact that there are only two lateral and two orientational degrees of freedom means that the integral of eq 1 can be calculated for a fixed surface by direct Monte Carlo sampling; that is, Metropolis sampling is unnecessary. In calculating the potentials of mean force in Figures 5-7, we have evaluated the integrand for each z a t 1000 randomly chosen points in the fourdimensional space of CO configurations. The results appear to be converged to within a few percent, though we have made no quantitative error estimates. These calculations are quite fast, and there is little difficulty in calculating the free energy at all values of z.
Adsorbate-Induced Reconstruction of p(2X2)X Adlayers on Ni( 100)f Jay Benziger,* Gregory Schoofs,f and Andrea Myers Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544 Received July 17, 1987. In Final Form: August 24, 1987 The interactions between CO and Ni(100)-p(2X2)X,X = C, N, 0, S, C1, have been examined experimentally with LEED, AES, temperature-programmed desorption, and reflection absorption infrared spectroscopy and modeled with a semiempirical tight-binding model. Adsorption at 170 K was reduced on all the surfaces with adlayers relative to clean Ni(100). The adsorption enthalpies, as estimated from TPD desorption energies, were in the order clean < 0 < C1< C < S < N. Infrared spectroscopy found that at 170 K both bridge and on-top CO species were found on the surfaces with adlayers of C, N, and C1; only bridge-bonded CO was found on surfaces with 0 and S adlayers. Above 300 K only on-top bonded species were seen on surfaces with 0, S, and C1 adlayers. No CO remained on the p(2X2)N surface above 300 K, and both bridgeand on-top-bonded CO persisted above 300 K on the carbided surface. The calculations indicated that CO adsorption in on-top sites is greatly inhibited by the presence of a p(2X2)X adlayer, with sulfur and chlorine being the most detrimental to CO adsorption. The effect of the adatoms on CO adsorption is dominated by the CO(5u) adatom p overlap, such that CO should be preferentially adsorbed on bridge sites on the surfaces with p(2X2)X adlayers. The experimental results have been explained by considering the reconstruction of the p(2X2)X adlayer into islands of C(2x2)X and clean surface. It is shown that this reconstruction makes energetically more favorable on-top binding sites available for CO, and the thermodynamic driving force for this reconstructionis favorable for all the adatoms except nitrogen. This model is able to account for the experimentally observed CO binding-site transformationsfrom bridge sites to on-top sites, and it also accounts for why nitrogen had the most deleterious effect on CO adsorption.
Introduction The effects of adatoms on the adsorption of carbon monoxide on metal surfaces have received a tremendous amount of attention. On metal surfaces the adsorption of carbon monoxide has generally been likened to the bonding of CO in metal carbonyls as suggested by B1yholder.l In this model CO adsorption occurs with the CO(5a) acting as a donor into the metal d-orbitals, while the C0(2n*) acts as an acceptor for back-donation from the metal d-orbitals. The resulting interaction is bonding with respect to the C-M bond and antibonding with respect to the C-0 bond. One of the implications of this theory is that adatoms that are more electronegativethan CO should deplete the metal surface of electrons, thereby weakening the M-C bond and strengthening the C-0 bond. Baerends and Ros have
* Author
to whom inquiries should be addressed. Presented a t the symposium entitled “Molecular Processes a t Solid Surfaces: Spectroscopy of Intermediates and Adsorbate Interactions”, 193rd National Meeting of the American Chemical Society, Denver, CO, April 6-8, 1987. Present address: Department of Chemical Engineering, Stanford University, Stanford, CA 94305.
*
attempted to quantify how the C-0 bond stretching frequency should vary with the population of the CO(5a) level: and Doyen and Ertl have calculated the dependence of the CO adsorption energy on the occupancy of the CO(2a*)0rbita1.~ We are attempting a systematic experimental test of the concepts outlined above using temperature-programmed desorption (TPD) and reflection absorption infrared spectroscopy (RAIS) of carbon monoxide on a series of well-defined Ni(lOO)-p(2x2)X surfaces with carbon, nitrogen, oxygen, sulfur, and chlorine adlayers. Presumably the behavior of CO on these different surfaces relative to the clean Ni(100) surface should reveal the chemical and steric influences of the various adatoms. Several other studies had been performed which examined the influence of adatom surface coverage on Ni(100) for various adatoms to try to identify whether the bonding effects were short-range (such as site blocking) or longer range elec(1) Blyholder, G. J. Phys. Chem. 1964,68,2772.
(2)Baerenlis, E. J.; Ros, P. J. Quantum Chem., Quantum Chem. Symp. 1978,12,169. (3) Doyen, G.; Ertl, G. Surf. Sci. 1974,43, 197.
0743-7463/88/2404-0268$01.50/0 0 1988 American Chemical Society
