J . Phys. Chem. 1985, 89, 2151-2160 The overall interaction between the surface and beam moieties is approximated by a simple mechanism, eq 1-4, which neglects several potentially important channels. Among these we mention the impact dissociation of impingent cations, radiative and Auger deexcitation of neutral molecules, and surface sputtering. In addition, the model disregards surface-cation/molecule polarization and dynamic effects. It bypasses evaluation of the electronic components of the cation neutralization probabilities by treating
2757
their state-averaged values A, and A, as adjustable parameters. To test the validity of these approximations requires more extensive experimental data on interactions between solid surfaces and low-energy active ion beams. Acknowledgment. This material is based upon work supported by the National Science Foundation under Grant No. DMR8304658.
Rotational-State-Resolved Sticking Probabilities for a Diatom-Surface Coillsion Model Ralph J. Wolf* and Ricardo C. Davis Department of Chemistry, University of Arkansas at Little Rock, Little Rock, Arkansas 72204 (Received: December 10, 1984)
The stochastic classical trajectory method was employed to calculate the sticking probability of a diatomic molecule on a solid surface. The model uses a one-dimensional generalized Langevin oscillator to represent surface temperature and energy dissipation. The molecule-surface interaction was varied to study the effect of the adsorptive well depth. Rotationalstate-dependent sticking probabilities were calculated for a range of surface temperatures. The sticking probabilities were found to correlate with a function which depends on the adsorptive well depth, surface temperature, and rotational energy of the diatom.
Introduction Recent studies in molecul-urface interactions have revitalized interests in state-resolved collision processes.'-8 To the dynamist, state-resolved information begs for details on a microscopic level. Most importantly, one is concerned with uncovering the casual relationship between the potential energy function and the dynamical outcome. The generalization of these relationships to other systems has been highly successfull in gas-phase dynamics and recently in molecule-surface interactions. Dynamics calculations have recently been employed to describe the scattering of diatomic molecules from surfaces.*'5 For simplicity many of these studies employed the rigid-lattice ap~
(1) A. W. Kleyn, A. C. Luntz, and D. J. Auerbach, Phys. Reu. Lett., 47, 1169 (1981); A. C. Luntz, A. W. Kleyn. and D. J. Auerbach, J. Chem. Phys., 76,737 (1982); A. C. Luntz, A. W. Kleyn, and D. J. Auerbach, Phys. Rev. E, 25,4273 (1982); A. W. Kleyn, A. C. Luntz, and D. J. Auerbach, Surf Sci., 117, 33 (1982). (2) F. Frenkel, J. Hager, W. Krieger, H. Walther, C. T. Campbell, G. Ertle, H. Kuipcn, and J. Segner, Phys. Rev. Lett., 46, 152 (1981); F. Frenkel, J. Hager, W. Krieger, H. Walther, G. Ertle, J. Segner, and Vielhaber, Chem. Phys. Lett. 90, 225 (1982). (3) G. M. McClelland, G. D. Kubiak, H. G. Rennagel, and R. N . Zare, Phys. Rev. Lett., 46,731 (1981); G. D. Kubiak, J. E. Hurst, H. G. Rennagel, and R. N. Zare, J . Chem. Phys., 79, 5163 (1983). (4) J. W. Hepbum, J. F. Northrup, G. L. Ogram, J. C. Polanyi, and J. M. Williamson, Chem. Phys. Lett., 85, 127 (1982); D. Ettinger, K. Honma, M. Keil, and J. C. Polanyi, Chem. Phys. Lett., 87, 413 (1982). (5) J. P. Cowin, C. F. Yu, S. J. Sibener, and J. E. Hurst, J. Chem. Phys., 75, 1033 (1981); J. P. Cowin, C. F. Yu, S.J. Sibener, and L. Wharton, J . Chem. Phys., 79, 3537 (1983). (6) M. Asscher, W. L. Guthrie, T.-H. Lin, and G. A. Somorjai, Phys. Rev. Lett., 49, 76 (1982); M. Asscher, W. L. Guthric, T.-H. Lin, and G. A. Somorjai, J. Chem. Phys., 78, 6992 (1983). (7) R. R. Cavanagh and D. S. King, Phys. Rev. Lett., 47, 1829 (1981); D. S. King and R. R. Cavanagh, J. Chem. Phys., 76, 5634 (1982). (8) R. P. Thorman and S. Bernasek, J. Chem. Phys., 74, 6498 (1981). (9) D. E. Fitz, A. 0.Bowagcn, L. H. Beard, D. J. Kouri, and R. B. Gerber, Chem. Phys. L t t . , 73,4397 (1980). (10) J. M. Bowman and S.C. Park, J . Chem. Phys., 76, 1168 (1982); J. M. Bowman and S. C. Park, J. Chemm.Phys., 77,5441 (1982); S.C. Park and J. M. Bowman, J. Cham. Phys., 80,2183 (1984), (11) J. C. Polanyi and R. J. Wolf, Eer. Eunsenges. Phys. Chem., 72, 356 (1982). (12) J. A. Barker, A. W. Klyen, and D. J. Auerbach, Chem. Phys. Lett., 97, 9 (1983).
0022-3654/85/2089-21S1$01.50/0
proximation. These models are somewhat limited since they do not allow for adsorption (sticking). They do allow for temporary trapping due to energy exchange among the degrees of freedom but are still energy conservative systems. The addition of dissipative effects, as in Langevin formalisms or molecular dynamics methods, allows for more appropriate definitions of sticking. Sticking can be defined by means of a total energy requirementi6 or a time correlation function method, as recently employed by Adams.I5 In this study we have employed a dynamical model" to investigate the effect of the rotational state on the molecule-surface sticking probability (Le., the probability that a molecule adsorbs on a surface). Knowledge of the factors which control the sticking probability will illuminate the possibility of selectively enhancing or diminishing adsorption by rotational excitation. The long-range goal is to understand the sensitivity of the sticking probability to all parameters and conditions. This is an extensive task, even for the model utilized in this investigation. Therefore, we have concentrated on the effect of the potential energy well depth responsible for adsorption and the surface temperature on the sticking probability.
Computational Methods Description of Model. Surface motion was characterized by a one-dimensional generalized Langevin oscillator which provides energy dissipation and a thermal perturbation. Motion is perpendicular to the plane of the surface. The model" is an extension of the atom-surface models of Adelman and Doll18 and Tullyig (13) J. E. Hurst, Jr., G. D. Kubiak, and R. N. Zare, Chem. Phys. Lett.,
93, 235 (1982). (14) J. C. Tully, Acc. Chem. Res., 14, 188 (1981); J. C. Tully, C. W.
Muhlaueen, and L. R. Ruby, Eer. Eunsenges. Phys. Chem., 72,433 (1982); C. W. Muhlausen, J. A. Semi, J. C. Tully, G. E. Bccker, and M. J. Cardillo, Isr. J . Chcm., 22, 315 (1982). (15) J. E. Adam, Chem. Phys. L i t . , 110, 155 (1984). (16) J. C, Tully, Surf.Scf., 111,461 (1982); E. K. Qrimmolmmn, J. C. Tully, and E. Holfand, J . Chem. Phys., 74, 5300 (1981). (17) J. C. Polanyi and R. J. Wolf, J . Chem. Phys., 82, 1555 (1985). (18) S. A. Adelman and J. D. Doll, J . Chem. Phys., 64,2375 (1976); B. J. Garrison and S. A. Adelman, S u r - Sci., 66, 253 (1977). (19) J. C. Tully, J . Chem. Phys., 73, 1975 (1980).
0 1985 American Chemical Society
2758 The Journal of Physical Chemistry, Vol. 89, No. 13, 1985
‘t
Wolf and Davis TABLE I: Anisotropic Measurements of Potential Energy Functions Employed t = 0.20 eV t = 0.15 eV t = 0.10 eV v L m i n / V,lmin 0.85 0.85 0.85 Z,/Zil
I
V = 0.0 eV V = 0.1 eV V = 0.2 eV V = 0.5 eV V = 1.0 eV
IR
1.073 1.066 1.068 1.067 1.068
1.073 1.066 1.068 1.074 1.08 1
1.073 1.066 1.068 1.064 1.069
where D and r, are the well depth and equilibrium position, respectively, provides the magnitude of the force at the exponentially repulsive wall (or adjusted to yield a force constant for vibrational analysis), and z is the separation between the gas molecule and the surface. To include a dependence of the diatom orientation angle, z , D, and r, become functions of x: Figure 1. Description of model.
to diatom-surface collisions (using a rigid-rotor approximation). The gassurface potential energy function is assumed to be smooth as in “cubelike” models. As such, linear momenta parallel to the surface plane and rotational angular momentum about the surface normal are strickly conserved. The coordinates of a rigid rotor interacting with a smooth plane are shown in Figure 1. Coordinates which have time dependences are R, x, 6, and s, which are the diatom center of mass, the diatom orientation angle, the surface atom position, and the generalized Langevin ”ghost” atom coordinate, respectively. The equations of motion are expressed as mass weighted forces by = -VRV(R,X,t)/ M % ( t )= - v , V ( R x , € ) / I
i(t)= -Q:t(t) + Ao1/2u@(t) - vV(R,x,f)/m
+
J ( t ) = -*02S(t) + Ao’/z*ot(t) - yS(t) W(At) where m, M,and I are the masses (or moment of inertia) for their respective degrees of freedom and V(R,x,t)is the diatom-surface potential energy function. The equations of motion for the surface model use the notation of T ~ l l ybut ’ ~ reduced to a single oscillator. The variance of the white noise source, W(At),is (2kT/mAt)’IZ where At is the numerical integration step size. The parameters Q,,wo, &,1/2q,, and y can be described in mechanical terms. The generalized Langevin system is composed of two coupled harmonic oscillators one of which is damped and also driven by a Gaussian random noise. The primary surface atom and ghost atom have harmonic forces given by the frequencies Q, and coo, respectively. The two oscillators are coupled by a constant &,1/2wo and the ghost atom oscillator is damped with a coefficient of y. T ~ l l yhas ’ ~ given a prescription for finding the above parameters from experimental data. We have employed the same set of constants used previouslyl7to describe a Ag( 111) surface: s2, = 114.6 an-’, uo= 140.0 cm-’, ho = 1892 cm-2, and y = 11 1.O cm-l. Potential Energy Function. The diatom-surface potential energy function employed was an exponential-3 function recently used by Polanyi and Wolf.I7 This allows the diatom to interact with the surface with an inverse z3 attraction and an exponential repulsion.20 The potential energy function, V(R,x,t),was written as a product of two terms: V W , x , t ) = V’W,x9€) Y”(R,x,E) The first term is the exponential-3 function while the second term is a switching function used to remove the long-range (>25 A) part of the i3attractive potential. The exponential-3 potential energy function, V(R,x,t),was modified to include the diatom orientation angle. Written in its usual form
exp[-j3(z
- r,)] -
(20) H. Hoinkes, Rev. Mod. Phys., 52,933 (1980).
(2)’)
I
5 1)
+ rd( - cos x - 5 D = Do + D ~ P ~ ( c X) os r, = ro + r2P2(cosx)
=R
where Do,4,ro,and r2are new potential parameters that control the anisotropy of the potential and rd,p, and b are the diatom internuclear separation, reduced mass, and the mass of the heavier atom within the diatom. Such a choice of functions restricts boths ends of the diatom to have exactly the same interaction with the solid plane, but shifts the center of mass when considering a heteronuclear diatom by rd[(F/b)- (1 /2)]. The switching function, V”(R,x,[), which accelerates the convergence of the long-range term in V’(R,x,t)to its z = m asymtopic value of zero, was
V”(R,x,t) =~XP[-(I/~~I where z, = 25 A and n = 12. At distances beyond 25 A, where very little coupling occurs, the potential quickly converges 0 while below 25 A the potential rapidly approaches V’(R,x,t). To approximately represent a diatom, NO, interacting with a silver surface, we have used the values m = 108.0 amu, M = 30.0 amu, p = 1.461 amu, b = 16.0 amu, rd = 1.51 A, ro = 2.935 A, and r2 = 0.1 2 A. The parameters Do and D2 were varied in order to generate potential energy functions with different adsorptive well depths, e. The parameter D2 was chosen to be 10% of the desired E in order to keep the anisotropy approximately the same for each case. The values (Do,D2) in units of electronvolts were (0.19, -0.02), (0.1425, -0.015), and (0.095, -0.01) which correspond to values of e = 0.20, 0.15, and 0.10 eV, respectively. Table I contains values that help characterize their anisotropic nature. The values of VLmin/V,lmin represent the anisotropy in the well depth while zJzI is the ratio of the classical turning points at a given energy for the two extreme orientations, perpendicular or parallel. Initial Conditions and Trajectory Methods. Initial conditions for the classical trajectories were chosen (Monte Carlo) for a given rotational state, thermal translation, and thermal surface motion. Translational temperatures were equal to surface temperatures in all calculations. Initial rotational states werej = 0, 8, and 16. The generalized Langevin equations were sampled by using”
Diatom-Surface Collision Model
The Journal of Physical Chemistry, Vol. 89, No. 13, 1985 2759 1.0-
TABLE II: Calculated Sticking Probabilities i
TSUR
=
TSUR
200 K
=
TSUR
400 K
=
600 K
TSUR
=
0.8 -
800 K
0 8 16
0.90f 0.02 0.80f 0.03 0.56 f 0.04
c = 0.10 0.63 f 0.03 0.52 f 0.04 0.33 f 0.03
eV 0.38 f 0.03 0.31 f 0.03 0.16 f 0.02
0.20 f 0.03 0.12 f 0.02 0.05 f 0.02
0 8 16
0.99 f 0.01 0.93 f 0.02 0.89 f 0.02
c = 0.15 eV 0.89 f 0.02 0.62 f 0.03 0.74 f 0.03 0.59 f 0.03 0.65 f 0.03 0.45 f 0.03
0.46 f 0.03 0.45 f 0.03 0.29f 0.03
= 0.20 eV 0.93 f 0.02 0.74 f 0.03 0.90 f 0.02 0.73 f 0.03 0.83 f 0.03 0.63 f 0.03
0.71 f 0.03 0.62 f 0.03 0.55 f 0.03
0.6 -
PS
0.4
-
O o.oo
200.
400
* 600
800L
e
0 8 16
0.99 f 0.01 1.00 f 0.01 0.99 f 0.01
where k is Boltzmann's coristant, G is a freshly generated Guassian random number of unit variance, and is the effective frequency of the surface oscillator, (Q; - &)I/*. The rigid rotor was sampled by R(0) = 30 A R(0) =
(gy2G
x(0) = r F j = 0 = 2rF j # 0 X(0) = hU0'
+ 1)]'/2/Z
TSUR (K) Figure 2. Ps(j)vs. surface temperature, with e = 0.15 eV.
j - t o I. TSUR=600 K
-
0.4
0.2
o.oo
0.10
0.15 0.20 E (eV) Figure 3. P,(j) vs. adsorptive well depth, with TsUR= 600 K.
-
where F is a freshly generated random number from a flat distribution (0 1). The orientation angle, x, was selected from a uniform distribution (a random 2-D vector) instead of a sin (x) distribution as in 3-D studies. We consider this choice more appropriate considering the reduced dimensionality of the model. Also, since the calculated sticking probabilities were found to be independent of the diatom orientation angle this weighting is of little significance. The classical equations of motion were integrated with a fixed step size integrator.*l A step size of 2 X s satisfied numerical accuracy tests of energy conservation (0.001%) and back integration (0.1%) when the system was made conservative (y = 0 and W(At) = 0). All calculations were performed in single precision in atomic units on a Honeywell DPS 8/44. Sticking was defined with a criterion similar to Tully's.16 If, during the course of a trajectory, the total energy of the rotor and the surface atom drops below some specified value (-kT below the desorption threshold) it is labeled as "stuck". The value -kT was determined by calculating the sticking probability, P,,using -'/dkT, -'l2kT, -lkT, -3/2kT, and -2kT for each potential energy function at TsUR = 800 K. The calculated P, values converged for definitions below -1kT. The shallow well case, E = 0.10 eV, at 800 K was the most difficult by this method because E is only 1.45kT. Convergence in this case was studied more thoroughly and found to yield consistent sticking probabilities below -3/4kT (reported values are calculated by using -kT). Trajectories were integrated until they either stick or scatter (Le., R becomes greater than its initial value of 30 A). The sticking probability and its standard error are given byz2
p, = NdNT
(21)A sixth-order integrator was employed for our stochastic equations with no adverse time correlations. The white noise source is mechanically removed from the gassurface collision zone, thus filtering the noise source. (22) D. G. Truhlar and J. T. Muckerman in "Atom-Molecule Collision Theory: A Guide for the Experimentalist", R. B. Bernstein, Ed., Plenum, New York, 1979,p 505.
0
2
4
6
8
Figure 4. Correlation of sticking probability with smooth curve is a functional fit, see text.
1012 (e
- Ej)/kTsuR.The
where NT and Ns are the total number of trajectories and the number that stick, respectively.
Results Sticking probabilities were calculated for three different gassurface potential energy functions by using masses corresponding to NO/Ag. The major feature which distinguished the potential energy functions was the absorptive well depth, E . Values of E were 0.20,0.15, and 0.10 eV. There were some minor differences in molecular anisotropy between the choice of potential energy functions, see Table I. Rotational-state-dependent sticking probabilities, P&), are presented in Table I1 vs. surface temperature, TSUR,and t. The results follow anticipated trends; P, decreases as surface temperature increases and P, decreases as e decreases. However, the dependence of the sticking probability on the initial rotational state, j , is more subtle. At low surface temperatures, Tsm = 200 K, P, is independent of j . As surface temperatures increase, P, is no longer independent of j . In Figure 2, we plot an example of this trend for the c = 0.15 eV case. A similar trend occurs with decreasing well depth, t. In Figure 3, P&) values are presented vs. E at a surface temperature, T ~ U=R 600 K. One obvious explanation of above trends is that rotational energy and surface temperature help
2760 The Journal of Physical Chemistry, Vol. 89, No. 13, 1985
overcome the attractive part of the potential and, thus, inhibit sticking. Our calculated sticking probabilities should correlate with some function of e , j , and TSUR. The form this function should take is not clear. After a number of attempts the correlation in Figure 4, P,vs. ( e - EJ)/kTSUR, was deemed the most satisfactory, where E, is the rotational energy of the state j . The smooth curve in Figure 4 is a least-squares regression using the functional form P , = N(a)" exp[-Au] where a = kTsuR/(t
- E,)
and N , n, and A are fitted parameters ( N = 1.964, n = 0.180, and A = 2.668). We consider this form as being flexible enough to fit our data and do not consider it a universal function. In particular, our calculations set the translational temperature of the impinging diatomic equal to the surface temperature. It seems reasonable that some other functional relationship may be suggested in the future which incorporates the initial translational energy of the diatom. The correlation of P, with (t - EJ)/kTsuR is nonetheless a significant observation. We view the (t - E J ) term as a reduced well depth, while the surface temperature term, kTSUR,scales the (reduced) well depth (and makes the term unitless).
Discussion Sticking probabilities were found to correlate with a function of ( e - EJ)/kTsm,see Figure 4. We feel the term (t - E,) should be interpreted as a reduced well depth, resulting from the efficient use of the incideqt rotational excitation. In a case where the absorptive well depth is large, the correlation implies that sticking probabilities will be independent of j as long as t >> EJ. Two recent studies that report sticking probabilities for large t cases agree with this observation. Polyanyi and Wolf,I7 using the same model as used here but with e = 0.58 eV, found sticking probabilities to be independent of j unless EJ approached t. By reducing the well depth, it was found they could alter the sticking probabilities dependence on j . Adams15 has observed a similar dependence for a large absorptive well depth ( e = 1.2 eV).23 It is interesting that the sticking probabilities calculated in these recent studies correlate with ( e - EJ)/kTSuR.However, the functional form and parameterization of the correlation is different than presented for our data, Figure 4. In the case of Adam's work,15 the similarities were not expected because the potential energy function is extremely different (topologically) and the model for surface motion is highly detailed. (23) The molecule surface potential energy function employed in ref 15 is due to J. C. Tully, J . Chem. Phys., 73, 6333 (1980).
Wolf and Davis The anisotropy of OUT potential energy functions are rather weak compared to the study by A d a m ~ . I The ~ + ~adsorption ~ of the rotor in our potentials can be described as slightly hindered with a preference for the diatom to be oriented parallel to the plane of the surface. However, the potential employed by Adams is strongly hindered and the diatom, carbon monoxide, prefers a perpendicular orientation with the carbon atom down. The diatom was also allowed to vibrate in Adams' study. The results differ between our study and Adams' in that we observe no initial diatom orientational dependence of the sticking probability and Adams does. He attributes the dependence to the long-range torque applied by the large anisotropy of the adsorptive well (same as hinderance to rotation). The long-range torque aligns the molecule prior to its collision with the repulsive part of the moleculesurface potential. Given these effects, Adams attributes the reduction in sticking probabilities at large j values to a competition between the alignment force (torque) and the rotational period. Large j values yield rotational periods short enough (given the collision frequency with the surface) to overmme this torque and, thus, sticking is inhibited. This is a reasonable explanation for the strongly anisotropic case. However, we observe no orientation dependence, thus, our explanations are mainly energetic at this time and consist of a competition between energy transfer processes. In order for a trajectory to be classified as "sticking", the internal energy of the molecule must be removed before the molecule leaves the vicinity of the molecule-surface potential well. Rotational energy can be efficiently converted into diatom translational energy (R T), which inhibits sticking. Countering this effect, diatom translational energy can be used to excite surface vibration which is dissipated to the bulk (T VsUR bulk), which promotes sticking. By stating only these two processes we do not imply others are nonexistant (e.g., R VsuR bulk) but we assume they have smaller energy-transfer coefficients. We believe that our interpretations and those of Adam's may not be as different as they appear a first glance. Adams has provided a specific microscopic mechanism to describe the energy transfer within the highly anisotropic case.15 The work done to date provide some clues to connecting state-resolved moleculesurface studies with features of the potential energy function. Furture studies along these lines should produce an illuminating picture of molecule-surface interaction.
-
+
+
+
-+
Acknowledgment. We thank John E. Adams, University of Missouri-Columbia, for helpful discussions concerning this work. The research reported here was supported by Research Corporation under a Cottrell Research Grant. We also acknowledge the financial assistance of the UALR, College of Science, Office of Research in Science and Technology (ORST) and the UALR Donaghey Faculty Research Fund.
