Langmuir 1992,8, 1470-1474
1470
Size-Dependent Rate of Desorption of Molecules from Supported Metal Crystallites E. Ruckenstein' and B. Nowakowskif Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received October 29, 1991. In Final Form: January 21, 1992
It is demonstrated that, under the action of long-range physical interactions, the rate of desorption of a molecule from the surface of supported crystallites depends upon the size of the crystallite and on the position of the molecule on the surface of the crystallites. The rate is predicted by solving the FokkerPlanck equation using an interaction potential obtained by pairwise addition of London-van der Waals interactions between the desorbing molecule and the crystallite plus substrate. Supported metals contain small crystallites of 10-102 A dispersed over the large surface areas of refractory metal oxides, We consider adsorbed molecules that have only physical interactions with the crystallites and substrate such as the noble gases, and demonstrate that the rate of desorption of a molecule from the crystallite depends upon the size of the crystallite and the position of the desorbing molecule on the surface of the crystallite. Of course, this is because the range of interactions of the molecule is greater than the thicknesses of the crystallites involved. The next section contains the derivation of the basic equations regarding the rate of dissociation per molecule. This is followed by the numerical calculation of the interaction potential between the molecule and crystallite plus substrate assuming London-van der Waals interactions for the pair potential and pairwise additivity. The final section provides numerical calculations for the rate of dissociation as a function of the size of the crystallite and position of the molecule on the surface of the crystallite; it also contains comments regarding the size dependence observed in some catalytic processes.
Equation for the Rate of Desorption Let us consider a molecule adsorbed on the surface of a crystallite located on the flat surface of a supporting solid. This system is sketched in Figure 1. The crystallite is assumed to have the shape of a spherical cap that has a contact angle 6 with the surface of the supporting solid. In a layer in the close neighborhood of the surface of the crystallite, a potential well is generated by the attractive interactions between the adsorbed molecule and those of the crystallite and support. The molecule is considered as adsorbed as long as its thermal motion is confined within this well. This means that its energy E lies in the range 0 > E > a0,where a0is the energy level at the bottom of the well. The molecule can leave the well only when it acquires a sufficient amount of energy from the collisionswith the molecules of the system to overcome the potential energy of the well. The process of desorption from the crystallite can be related to the rate by which the molecule passes over the upper energetic boundary of the potential well, E = 0. The random motion of the molecule in the potential well can be described by the Fokker-Planck equation (called often the Kramers equation when it includes an interaction potential) for the probability density p ( x , v , t )
* To whom correspondence should be addressed. t On leave from Agricultural University, 02-528 Warsaw, ul. Rakowiecka 26/30, Poland.
Figure 1. Sphericalcap crystallite supported on the flat surface of the substrate.
of a position x and velocity v at time t . The FokkerPlanck equation will be simplified to its one-dimensional form with respect to the radial direction. The Fokker-Planck equation, for the one-dimensional motion along the radial coordinate, has the form'
*= at
-2vp ar
+ -yv + 1 d@ jP +ykTa2p av a( m dr m av2
(1)
where y is the friction coefficient, m is the mass of the molecule, k is the Boltzmann constant, and Tdenotes the temperature (K). The potential energy in eq 1 is a function of r but contains also the angle rC, as a parameter. The following qualitative considerations provide some insight regarding the motion of the molecule in the well. The molecule performs anharmonic oscillations in the radial direction, whose characteristic time can be estimated from the expression t , = X,(m/kT)1'2, where A, is the thickness of the well. In gases, the molecules exchange energy only as a result of the relatively infrequent molecular collisions. Consequently the time scale for the change of energy which can be characterized by the energy dissipation time tc = y-' is much larger than t,, and a long sequence of oscillations is performed nearly at constant energy. One can therefore differentiate between the fast changing velocity and position, and the much slower change in energy. This separation of time scales between fast changes in x and v and slow changes in the energy E can be used to obtain the asymptotic solution of eq 1 in the long time scale, characterized by the time of dissipation tc of the energy of amolecule. In order to generate this approximate solution, we rewrite the Fokker-Planck equation (eq 1) in (1)Risken, H. The Fokker-Planck Equation;Springer: New York, 1989.
0 1992 American Chemical Society
Supported Metal Crystallites
Langmuir, Vol. 8, NO.5, 1992 1471
terms of the energy:
2 at P(E,t) = y
E = ‘/2mv2+ @(r)
&[ (g - kT) P(E,t)] +
(2)
The resulting equation for the probability density p ( r J , t ) of position r and energy E at time t has the following f0rm:~.3 a p ( r J , t ) / a t= - 2 ( ~ a r ) [ ( ( 2 / m ) (~ p(r~,t)l+ 2-/[(a/aE)(E - @ - (kT/2)) p(r$,t) + kT(d2/aE2)(E - p ( r J , t ) l (3)
One can take advantage of the separation of time scales, by observing that the probability density for the position r relaxes rapidly (in the short time scale) to its steadystate solution ~ ~ ( $ 3 ) .The latter solution contains as a parameter the energy E, which can be considered constant in the short time scale. Therefore, the joint probability density can be written as the product p(r$,t) = p,(rIB p ( E , t )
(4)
The conditional probability densityp,(r@) can be obtained as the stationary solution of eq 3 in the short time scale. The second term in the right side of eq 3 containing the friction coefficient y is then negligible, and this yields the equation ( w w [ ( ( ~ / ~ )(E @)F2 p,(r(E)~= o
The rate of change of P(E,t) is related to the flux j ( E , t ) in the energy space via the equation
(a/&) P(E,t) -(d/dE) j(E,t) (11) After a short lapse of time, the steady state is achieved and the rate of desorption is provided by the stationary flux, which is obtained from the stationary solution of eq 10. In order to determine this flux, we assume that the molecule desorbs from the crystallite when it reaches the upper boundary E = 0 of the well. This yields the absorption boundary condition P(E)IE,o = 0 (12) The steady-state solution of eq 10 that satisfies the boundary condition (eq 12) has the form
P(E) = ( l / y k n j E &’(E)e-E/kTJ~~E’I[&(E’) e-E’IkT~ (13)
The desorption rate a per molecule a = j E is obtained using the normalization condition J P ( E )dE = 1. This yields
(5)
which has the solution p,(rF) = c(E)/(~(E - @)F2
(6)
The constant (with respect tor) C(E)in eq 6 is determined from the normalization condition. The joint probability density p(r$,t) is obtained by combining eqs 4 and 6: p ( r J , t )= P ( E , ~ ) W ( E )
-w)’/~I
(7)
where the normalization constant C(E)is expressed in terms of the function
R1 and R2 in eq 8 are the roots of the equation E = @ ( r )
for a given energy& they represent the two limits of motion of the molecule in the potential well. Introducing solution eq 7 into the Fokker-Planck equation (eq 3), one obtains
a 1 -P(E,t) = &’(E)(2(E - @ ) ) ‘ / 2 at
Since the time evolution of P(E,t) is slow compared to the time evolution of the motion of the molecule in the potential well, a closed equation for this probability density of energy is obtained by integrating with respect to the position r of a molecule between the limits R1 and Rz.This yields the equation (2) Ruckenstein, E.; Nowakowski, B. Langmuir 1991, 7, 1537. (3) Stratonovich, R. L. Topics in the Theory of Random Noise; Gordon and Breach New York, 1963.
Equation 14 is used in the next section to compute numerically the rate of desorption of molecules from the crystallite.
Results and Calculations The rate of desorption of molecules from the surface of a spherical crystallite (Figure 1) was calculated as a function of the size of the crystallite and the position (i.e., the azimuthal angle $1 of the desorbing molecule. The attractive intermolecular potential between the molecules in the well and crystallite plus substrate was assumed to be of the London-van der Waals form, hence proportional to the inverse sixth power of the intermolecular distance r:
where E is a characteristic interaction constant, and u is the diameter of the molecule considered here the same for all three species. Assuming pairwise additivity, the effective interaction potential acting on the adsorbed molecule was calculated by integrating over the contributions coming from the small, elementary portions of the volumes of the crystallite and of the substrate: @(r)=
sv
p,U(lr - r‘l) dr‘
(16)
where Pa is the number density of the molecules (Pa = ps in the substrate and pa= pc in the crystallite). The integral in eq 16 was performed assuming different constants in eq 15for the interaction with the crystallite (e,) and with the substrate (es). The constant e, for the interaction with the substrate can have values typical for the van der Waals interactions, which means that e$kT is of the order of unity. On the
1472 Langmuir, Vol. 8, No. 5, 1992 other hand, the physical interactions of the adsorbed molecule with the crystallite are expected to be stronger. For the interaction potential with the semi-infinite support, the calculation yields the following analytical result:
Ruckenstein and Nowakowski
i
-3.5
G,=6
G,= 1
--
N=ZOO N=5000
--
N=20000
,,
as= -(Teg,2/6)(g//X)3
(17) where x is the distance of the molecule from the surface of the solid, and ps is the number density of molecules in the substrate. For the crystallite, for which we assume a sphericalcap shape (Figure l),no closed,analyticalformula for the interaction potential was obtained; the integration was performed analytically with respect to two variables and numerically with respect to the third variable (see Appendix). Parts a-c of Figure 2 present the dependence of the desorption rate a/? on the azimuthal angle # of the desorbing molecule for various values of the interaction parameters C, = eJkT and 6,= e$kT and for crystallites containing various numbers of molecules N. The contact angle fl was taken equal to 6 = d 6 . One can note that, for the values of the interaction constants considered, the rate of desorption is the highest for a molecule close to the contact edge of the crystallite with the substrate, and the lowest for the molecule located at the top of the crystallite. This dependence follows that of the depth of the potential well 9,. The dependence of 9,lkTon the angle and number of molecules in the crystallite is presented in Figure 3a-c for the corresponding values of 6, and The depth of the well is the greatest at the top of the crystallite and decreases monotonically with increasing angle #. When the two interaction constants become equal, the minimum desorption rate and the maximum depth of the potential well are located at the edge (Figures 2d and 3d). This can be explained by noting that the interactions of the adsorbed molecule with the substrate become stronger and those with the crystallite weaker as the distance to the surface of the substrate decreases. Because the size of the substrate is much larger than the size of the crystallite, the maximum depth of the potential well is displaced from the top of the crystallite when ec and eE become comparable. The rate of a catalytic process is expected to be higher when the desorption rate constants are larger. The present calculations indicate that, when the interactions with the crystallites are stronger than those with the substrate, the rate of desorption per unit exposed surface area of the crystallite decreases as the size of the crystallitesincreases. When the interactions with the substrate are comparable or stronger than those with the crystallites, the rate of desorption increasesas the size of the crystallitesincreases. The size dependence is, however, significant only for sufficiently small clusters, since clusters containing 5 X 103or 2 X l@ atoms have almost the same desorption rate constants (see Figure 2a-c). Experiment has shown that the rate of a catalyticprocess is not necessarily directly proportional to the area exposed by the metal crystallites to the chemical atmosphere. For instance, Zahradnik et al.4 measured the rate per gram of active metal catalyst (the global rate) as well as the rate per square meter of exposed metal (specific rate) for the oxidation of CO over a Pt-a-alumina catalyst. The catalysts had different particle sizes for the same loading, hence different exposed metal surface areas. They ob-
+
~~~~~
~~
Zapradnik! J.; McCarthy, E. F.; Kucynski, G. C.; Carberry, J. J. In Matenab Science Research; Kunynski, G. C., Ed.;Plenum Press: New York, 1975; Vol. 10, p 199. (4)
1
I
0.0
0.1
-4.5
I
0.1
0.0 -4 C
0.2 0.3 $[rod1
I
0.4
0.5
I
I
0.4
0.5
I
I
,
I
--
G,=ZO
-6
I
0.2 0.3 *[rod]
I
-6
I
I
1
N=200
N-5000
-
s,= 1
,;
- N=20000
-12 0.0
0.1
0.3 *[rad1
0.2
0.4
0.5
-2.60
-2.80
1
I
0.0
0.1
I
I
0.2 0.3 $[rod]
,
I
0.4
0.5
Figure 2. Dimensionless desorption rate a/yfrom a crystallite consisting of N molecules, as a function of the azimuthal angle $. The interaction constants are (a) 6,= eJkT = 6 and 6,= cdkT = 1; (b) gC= 10 and 6, = 1; (c) 6, = 20 and 6,= 1; and (d) 6,= 1.5 and 6,= 1.6. The densities pc and p, which are taken equal and the diameter u are considered related via the expression p d = 1.2.
Langmuir, Vol. 8, No. 5, 1992 1473
Supported Metal Crystallites a
-4.5
L
5 -6.0 8
-7.5
-9.0 I 0.0
I
I
0.1
0.2
I
I
I
0.3
0.4
0.5
*[rad1
-7
--
6,=10
'
6,= 1
~
'
-
N-200 N-3000 N=20000
served that the global rate decreased with increasing average size of the crystallites. This decrease was not, however, linearly related to the decrease in surface area, since the specific rate increased with increasing particle size. Similar observations have been made by Carballo and Wolfsduring the catalytic oxidation of propylene over Pt-yalumina. The specific rate is again greater on the larger crystallites. Hassan et al.s and den Otter and Dautzenberg' also observed that the catalytic process is sensitive to the size of the crystallites. However, crystallites used in the described experiments are not small enough for the size effect predicted in the present paper to be significant. The size dependence in those cases is a result of the fact that the reaction prefers as "landing areas" for the reactant molecules on the surface of the crystallites combinations of neighboring sites which are more likely to occur on the larger crystallites (for instance, terraces).+lo
Conclusion
;
The long-range physical interactions between the desorbing molecule and the crystallite plus substrate of a supported metal induce a dependence upon the size of the crystallite for the rate of desorption.
Appendix
'
-13
0.0
I
I
I
0.1
0.2
0.3
1
I
0.4
0.5
@[rod]
k-
