Application of Linear Response Theory to Nonlinear Processes
123
Application of Response Theory to Adsorption / Desorption Kinetics E. Bergmann*l and 9. F. Antonlni Eatrelle, Centre de Recherche de Geneve, CH-1227 Carouge, Geneve, Switzerland (Received June 13, 1974)
The linear response theory for nonmechanical forces is extended semiphenomenologically to nonlinear processes. When applied to gas-surface interactions it leads to a simple relation between the adsorptionldesorption flux and the applied chemical potential difference. It thereby allows the separation of equilibrium properties from the kinetic data. The results can be compared directly with reaction rate theory if the information about the rate-limiting step is available. The use of the theory is illustrated for the case of hydrogen adsorption on fresh nickel surfaces.
Introduction It seems that the analysis of the experimental results of gas-surface interaction studies has been restricted so far to two main approaches. On one hand is a global description of adsorption at submonolayer coverage by means of various empirical formulas, the most successful of which is the Elovich e q ~ a t i o n Equivalent .~~~ to this are the different uses of coverage dependent energies for adsorption and desorption reaction^.^-^ These formulas are related to several assumptions concerning the adsorption process itself and the interaction energy among the adsorbed molecules.7-9 They thereby combine unknown features of the equilibrium system with features of sorption kinetics. Historically the motivation for such a combination may have arisen from the kinetic derivation of certain isotherms.10 This however has been overcome by n ~ w ~ ~and - l *for most of the empirical expressions for isotherms used today derivations have been found in terms of true equilibrium theory formulated through an equation of state for the adsorbate. On the other hand, considerable efforts are devoted in the recent experi-ments to design “ideal” conditions where either adsorption or desorption can be neglected.15 In the second case, then, an analysis in terms of a dependence of the sticking coefficient on coverage is made.8 However, suppression of redesorption in adsorption measurements is always slightly ambiguous. In view of the difficulties involved in achieving such ideal conditions, a theory which can give the same information without the suppression of the counterreaction should be of considerable advantage. The present application of response theory serves both purposes: combination of adsorption and desorption, and separation of equilibrium properties from kinetics. It interprets sorption data in terms of a surface-gas admittance function which is directly related to the rate constant, i.e., to the thermal average over the reactive scattering cross section. The underlying concept is very simple. In most experimental set-ups the equilibrium properties are obtained as a by-product7J6 of the kinetic experiment. They can therefore be removed from the kinetic data by the use of the parameters of the isotherm, if it fits to an analytic form, or graphically. Since the notion of equilibrium must be used with some care in the case of chemisorbed adsorbates, the next section will start with a discussion of the assumptions of our theory and the consequent restrictions on its use.
* Battelle Institute, Advanced Studies Center, CH-1227 CarougeGeneva, Switzerland.
We will develop a nonlinear response theory for nonmechanical forces with the help of local equilibrium theoI-Y.~’J~A similar extension with mechanical forces has been used previously far the description of charge transfer on electrodes.19 As an illustration of the power of this new method, we analyze experiments -of H2 adsorption on ultraclean nickel surfaces produced by abrasion.20This example shows that the concept can be used for a reduction of the equipment in contact with the ultrahigh vacuum chamber and therefore for the reduction of potential surface impurity sources.
Nonlinear Response Theory for Nonmechanical Forces There are two ways to approach sorption phenomena by response theory, One way would be to consider adsorption/ desorption reactions as recombinationldissociation between gas molecules and surface adsorption sites. In this picture one can apply the response theory as formulated for gas reactions.21 The other route is to follow the statistical theory of chemical reactions and consider sorption as a transport process in phase space.22This is much more general as it does not involve an ab initio choice of the ratelimiting step which can, e.g., be energy transfer. In fact the assumption of a single rate-limiting step is not even needed. The assumption of equilibrium in the “final” adsorbed phase is less restrictive than might appear. It requires the existence of a characteristic time scale on which the following processes are very slow and preceding processes very fast. This time has to be chosen as the measurement time in the experiment and some care must be taken that the measured equilibrium properties refer to the same state. This is normally the case if they are taken from the same experiment. Note also that equilibrium layers must not necessarily require high surface mobility, since exchange is always possible via the gas phase. One should however exclude systems where the rate-limiting step may be island growth. If we denote in the followingthe reaction coordinate by x it should be kept in mind that this can be identified with the same molecule-surface distance only if the reaction coordinate in the rate-limiting domain coincides with it. The use of this coordinate is also not in contradiction with the high multidimensionality of the relevant phase space in the case of surface reactions.22 We start with the definition of a “reaction density opera. tor.” Take the total Hamiltonian of the system The Journalof PhysicalChemisfry, Vol. 79, No. 2, 1975
E. Bergrnann and J. F. Antonini
124
The “driving part” of the density operator is
D =
Jo* d t J d x p(x)J(x,t’)V,p(x,t’)
(6)
J being the particle flow operator. Since we assume stationarity, Le., constant flow along the reaction coordinate
with H M the free molecule Hamiltonian, V the intermolecular interaction, H s the free solid Hamiltonian, and U the gas-solid interaction. We use italic letters to denote simultaneously a set of molecular quantum numbers; they also denote the molecule the wave function is centered on. Greek letters denote solid quantum numbers. These can, for example, be just occupation number vectors for the excitation of the solid one believes dominant for U.23For simplicity we made the following two easily removable assumptions: (1) the intermolecular forces are additive; (2) we have only one atomic or molecular species present. We do not claim that the eq 1 is a convenient starting point for a microscopic calculation, where it may be useful to separate out part of the perturbation by a renormalization procedure. Take, e.g., the case of dissociative adsorption of a diatomic molecule such as Hz. In this case H M would be the single atom hamiltonian and V would also contain the HL interaction. With (1) we have the “reaction density operator” for the molecules P =
V x P ( x , t ’ )= 4 p / l (7 1 where AMis the chemical potential difference between bulk gas phase and adsorbed layer and 1 the phase space distance. Noting that because of the present bath P ( x ) is constant, we see that the only remaining phase integral is the one over chemical potential and density i s m p ( x k ( x ) d x= ( p G + a 4 p ) N
The last equation serves as a definition of Ho, the effective hamiltonian of a system of molecules in contact with a solid which is strongly coupled to a heat bath. Such systems have been generally discussed by McLennan.ls He showed that systems approaching equilibrium can be described in the hydrodynamic phase by a local stationary density operator, which is an approximate solution of (3). His discussion is classical. The transcription to the quantum case is however straightforward. Note, however, that McLennan’s eq 212 contains an error. Its correct form can be obtained immediately from a comparison with eq C.3 of ref 24, namely B
p =
fL
exp[-i
D(-iEX) dX]
(41
with the local equilibrium density operator fL
=
rie x p [ - J b
p ( x ) ~ ~ (-x )p(x)n(x)I
(5)
2 being the partition function, p the chemical potential, n the one-particle density operator, and Ho the hamiltonian density. The Journal ot Physical Chemistry, Vol. 79, No. 2, 1975
CY 5
1 (8 1
This is our Br6nsted r e l a t i ~ n . ~a ~can -~~ be obtained in principle from eq 8 by evaluating the integral for a representative set of initial state weighted trajectories. The central hypothesis in our nonlinear response theory is the assumption of a small drift of the system over the hypersurface. This replaces the original assumption of ACT that the system is in equilibrium along the reaction path.22 In our case, it means that the flow and therefore D is small, so that it is sufficient to take the linear part. The validity of such an assumption is well established, except for extremely fast reaction^.^^,^^ We therefore have
We therefore have after some rearrangement the expectation value of the particle flow operator as AP ( J ) = T r p J = -Z-’ k T1
This operator satisfies the Liouville equation for a system in a bath
0 5
- 4 ( H O - ( G + a A G )N )
Tre /.R
Jo-dh
X n *
J
dtJ(t)J(-iEX) (IO) 0
We wish to evaluate this in the independent molecule limit. There we have to distinguish two situations. ( a ) Negligible Activation Energy. In that case the shift in the chemical potential of (8) is a mere change of scale. If we take the bulk gas as idea130 we therefore obtain the particle flow
n density of reactants, Le., a linear relationship between sorption flow and chemical potential difference. ( b ) Appreciable Activation Energy. In processes involving an activation energy which is considerably higher than the thermal energy, even after the quantum corrections for tunneling, we have an energy level i, of the system such that for levels below the matrix element of the flow operator is zero. In that case the chemical shift is not cancelled and we have
Application of Linear Response Theory to Nonlinear Processes
125
Usually for dominant adsorption we will have first-order kinetics and therefore n proportional to the gas-phase density. Relation to ACT The linear case, eq 11, should clearly not differ from ACT since we made essentially the same hypotheses, even although we avoided in our general formalism some specifications necessary for the application of ACT. We see that the Green’s function in eq 11 is essentially the same as the one defined previously in the response theory for gas react i o n ~ . Simple ~ ~ , ~comparison ~ gives us the rate constant for adsorption or desorption
Note that our procedure has some advantages over Yamamoto’s, insofar as it uses the flow operators and therefore by-passes the difficulties connected with a correct definition of reactant and product state projection operators.32 It seems worthwhile noting that, despite the general validity of eq 11and 12, an evaluation of the Green’s function requires a laborious microscopic analysis similar to the one necessary in ACT. However, in any case they are very useful phenomenological equations. To ,demonstrate how the increased versatility can be used to make improvements on experimental conditions we give an analysis of sorption on ultradean surfaces produced by abrasion.16 Experimental Set-Up. Experimental Procedure A detailed description of the experimental system has been given elsewhere.16,20The main characteristics of the apparatus are the following. The adsorption data have been obtained in an ultraclean ultrahigh vacuum system with a nominal pumping speed of 3 l h e c for H2; the ultimate pressure after bake-out is below Torr for all gases except H2, due to H2 outgassing out of the stainless steel wall of the chamber. The pressure is measured by means of an Omegatron type mass spectrometer, fitted with an oxide-coated cathode (for operation at low cathode temperature). The 5.8 N nickel sample is a cylindrical tube which protrudes from the ultrahigh vacuum chamber. It is surrounded by an electrical furnace which allows the temperature of the tube to be changed or maintained with great stability. The inner surface of the nickel tube can be scratched by means of a diamond tool, manually controlled from outside oia metallic bellows. This simple device for abrasion is the only adjunction to the inevitable constraints of an ultrahigh vacuum chamber. Therefore ultraclean conditioning is easily achieved and maintained. The adsorption curves are obtained by creating a new fresh surface by a short abrasion under a dynamic equilibrium pressure p o (both H2 leak-in valve and pumping valve open). The experimental parameters are chosen in such a way that the resulting pressure change ( p0 - p 0 ( t )) is well below 5%of the initial pressure P O . Analysis of t h e Experimental Results in Terms of Response Theory The experimental data provide straightforwardly the dependence of the surface coverage 0 as a function of time
-0
I
0.5
Ap [kcal Mo1e-q
Figure 1. Adsorption flow at 293’K as a function of applied chernical potential for different equilibrium gas pressures: (a) V, 8.7 X Torr; (c) A,3.8 X Torr: (d) A, 2.8 Torr; (b) 0, 5 X X lob7 Torr; (e) 0,1.7 X Torr; (f) 0 ,0.97 X Torr.
(for different gas pressures and surface temperatures) from which the dependence of the rate of change of 0, dsldt, as a function of time can be easily obtained. The asymptotic values of the curves 0 ( t ) , when plotted as a function of po, give a family of isotherms, 0 ( p 0 ) .From ~ ~ these isotherms the applied chemical potential difference as a function of time in each experiment can easily be derived by the following steps -
& T o ) -fi(t,TO)
+
The almost negligible time dependence of p 0 ( t ) was taken into account. To see if the adsorption is activated according to eq 12, a plot of In q / A p as a function of Ap was made. The resulting straight lines were horizontal. This was to be expected since H2 adsorption on nickel is usually found to proceed without activation at room temperature. The first figure shows the net adsorption flow q as a function of A p for several equilibrium gas pressures. We considered only the region where 0 ( t ) can be obtained with sufficient accuracy. The agreement with the prediction of eq 11is very satisfactory as practically all points lie on straight lines through the origin. The domain is constrained on the side of small A p by the limited precision of the coverage record near saturation, amplified by the fact that we take there the logarithm of the quotient of two big numbers differing by a small amount. The maximum possible values of the flow are determined by the response time of the experimental apparatus. Unfortunately, the original measurements were only made with varying surface temperature, the bulk gas at room temperature. While this does not invalidate the above analysis (since the isotherm temperature must be the surThe Journal of Physical Chemistry, Vol. 79, No. 2, 1975
E. Bergmann and J. F. Antonini
in adsorption. It therefore allows an unambiguous determination of the reaction rate as a function of temperature by experimental systems with considerably less sophistication than needed hitherto. One can avoid the search for ideal experimental conditions with negligible adsorption or desorption. Acknowledgment. The application of response theory to adsorption/desorption kinetics was originally suggested to us by Professor €I. Suhl. We have benefited also from many discussions with Dr. L. A. PBtermann.
References and Notes Note change of address. C. Aharoni and F. C. Tompkins, Advan. Cafal., 21, 1 (1970). V. E. Ostrovski and M. I. Temkin, Kinet KafaL, 10, 93 (1968). G. C. Bond, "Catalysis by Metals," Academic Press, London, 1962. (5) J. Lapujoulade, Surface Sci., 35, 288 (1973), and references to previous work therein. (6) D. L. Adams, Surface Sci., 42, 12 (1974). (7) H. P. Bonze1 and R. Ku, Surface Sci., 40, 85 (1973). ( 8 ) C. Kohrt and R. Gomer. Surface Sci., 40, 71 (1973). (9) P. R. Norton and P. J. Richards, Surface Sci., 41, 293 (1974). (IO) J. H. de Boer, "The Dynamical Character of Adsorption," Clarendon Press, Oxford, 1953. (11) W. A. Steele in "Solid-Gas Interface," Vol. 1, E. A. Flood, Ed., Marcel Dekker, New York, N.Y., 1967, p 307. (12) C. V. Heer, J. Chem. Phys., 55,4066 (1971). (13) E. Bergmann, J. Phys. Chem., 78, 405 (1974). (14) J. Appel, Surface Sci., 39, 237 (1973). (15)See, e.g., the review of L. A,, Petermann, frogr. Surface Sci., 3, 1 (1972). (16) J. F. Antonini, Thesis, Federal Institute of Technology, Lausanne, Switzerland, 1973. (17) D. N. Zubarev, Fortschr. Phys., 18, 125 (1970), and references therein. (18) J. A. McLennan, Jr., Advan. Chem. Phys., 5, 261 (1963). (19) E. Bergmann, Physica, BO, 499 (1972). (20) J. F. Antonini, Helv. Phys. Acta, in press. (21) T. Yamamoto, J. Chem. Phys., 33,281 (1960). (22) J. Horiuti and T. Nakamura, Advan. CataL, 17, 1 (1967). (23) Eg., T. B. Grimley. Ber. Busenges. Phys. Chem., 75, 1003 (1971). (24) L. P. Kadanoff and P. C. Martin, Ann. Phys. (New York), 24, 419 (1963). (25) J. N. Brdnsted and K. Petersen, Z.Phys. Chem., A, 108, 185 (1924). (26) P. R. Wells, "Linear Free Energy Relationships," Academic Press, New York, N.Y., 1968, Chapter 5. (27) R. R. Dogonadze in "Reactions of Molecules at Electrodes," N. S. Hush, Ed., Wiley-lnterscience, New York, N.Y., 1971, p 135. (28) 6.Shizgal, J. Chem. Phys., 55,76 (1971). (29) J. Popielawski, Mol. Phys., 17, 341 (1969). (30) Effects of deviations from ideality in adsorption studies have been calculated by D. H. Everett, Discuss. Faraday Soc., 40, 177 (1965). (31) R. D. Levine, "Quantum Mechanics of Molecular Rate Processes," Ciarendon Press, Oxford, 1969, Section 3.7. (32) S. Golden, "Quantum Statistical Foundations of Chemical Kinetics," Clarendon Press, Oxford, 1969. (33) The accuracy of the isotherm has actually been improved in this case by separate repeated measurements of saturation coverages. (1) (2) (3) (4)
Figure 2. Surface gas admittance as a function of equilibrium gas density derived from the slopes of Figure 1.
face temperature) it gives as a result a two-temperature Green's function comparable with two-temperature sticking coefficients.8 We could not devise a procedure to incorporate this in a simple way in the response theory. We hope, however, to perform the measurements with a homogeneous system temperature at a later stage. The slopes of the lines in Figure 1 can be used to determine the order of the adsorption reaction. Plotted double logarithmically as a function of density we found as best fit a line whose slope was one within error limits and therefore concluded that we have first order kinetics. Figure 2 shows therefore the direct plot of the slopes of Figure 1as a function of bulk gas.density. The slope of this line gives us the reaction rate, divided by kT1. 1 is the phase space distance of the rate-limiting step(s). It should be mentioned that the deviations in Figure 2 are not related to the deviations in Figure 1, which are much smaller, but are caused by the difference in the cross section of the scratch. Conclusions The presented kinetic theory of gas-surface kinetics allows separation of equilibrium and nonequilibrium effects
The Journal of Physical Chemistry, Vol. 79, No. 2, 1975
