- Email: [email protected]
Hydrogen atom recombination on graphite at 10 K via the Eley–Rideal mechanism
25 May 2001
Chemical Physics Letters 340 (2001) 13±20
www.elsevier.nl/locate/cplett
Hydrogen atom recombination on graphite at 10 K via the Eley±Rideal mechanism M. Rutigliano a, M. Cacciatore a,*, G.D. Billing b a
CNR-Centro Studi Chimica dei Plasmi, Dipartimento di Chimica, Universita' di Bari. V. Orabona N.6, 70126 Bari, Italy b Chemistry Laboratory III, H.C. érsted Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark Received 12 February 2001; in ®nal form 23 March 2001
Abstract This work reports results on the recombination of hydrogen atoms on a graphite surface at 10 K obtained using a detailed semi-classical molecular dynamics method in connection with the use of recently proposed ab initio potential energy surfaces. The calculated recombination probabilities together with the vibrational distributions of the formed H2 molecules obtained assuming that the surface reaction proceeds via the Eley±Rideal mechanism can have an impact on the chemistry of H2 formation in interstellar space. Ó 2001 Published by Elsevier Science B.V.
1. Introduction Recently, the hydrogen atom recombination on grains as a possible eective catalytic source to the formation of H2 molecules in interstellar clouds has attracted considerable and increasing interest in astrochemistry. The reason for this is the persistent diculty in understanding the abundance of H2 , in particular the high ratio between molecular and atomic hydrogen, observed in dense clouds of the interstellar space. Complex kinetic modeling has, in fact, shown that such abundance can only be consistent assuming a high, close to unity, recombination probability of H on catalytic grain surfaces [1±4]. On the other hand, it is well known that atom recombination on surfaces can be an eective source of molecular species as, for instance, H2 on Cu, W and other metals [5±7], CO
*
Corresponding author. Fax: +39-080-544-2024. E-mail address: [email protected] (M. Cacciatore).
formation on platinum [8]. In these and other cases it has been demonstrated that molecular species are formed predominantly in vibrationally excited states, the extension of the vibrational excitation depending, among other things, on the dynamical mechanism through which the atom recombination takes place, i.e., whether it is direct or indirect [8]. Although somehow hidden due to the diculty of direct probing, atom interaction with solid surfaces can therefore have a great impact on the overall reaction kinetics of molecular gases under non-thermal, diusive conditions as those met in interstellar space. Indeed, for low-pressure, lowdensity gases under diusive conditions, atom recombination on surfaces can compete with the less ecient three-body recombination reaction in the gas-phase. There are now quite ®rm evidences, attested by the so called U-V bump at 217 nm [9,10], that grains in interstellar clouds are predominantly made of silicate and, in particular, of carbonaceous amorphous compounds (very likely
0009-2614/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 3 6 6 - 9
14
M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
covered by ice mantels and other condensable interstellar species) . These indications, in connection with recent developments in molecular dynamics techniques applied to gas±surface interactions, have reinforced experimental [11] and theoretical [12±14] eorts to get accurate estimates of the catalytic activity of such surfaces. It is the purpose of the present Letter to give collisional data relevant for the hydrogen±carbon system. By combining recent progress in ab initio and here especially in density functional theory (DFT) for the calculation of the interaction potential with the semi-classical method developed for a dynamical treatment of surface processes makes it possible to perform suciently accurate calculations of collisional data as: recombination probabilities, energy disposal to the formed H2 molecules and surface energy accommodation. These quantities are calculated for collisional conditions typically met in the interstellar and in plasma-chemical devices and can therefore be relevant for understanding the basic behaviours of such systems. The surface temperature eects, an important issue in reactive catalysis, and the extension to higher surface temperatures relevant for fusion plasma devices and non-thermal laboratory plasmas, has also been investigated and will be presented in a subsequent publication [15].
k
Z gk t; TS
The semi-classical model used in the present calculations is described in detail in [16] and numerous papers [17]. We shall therefore only brie¯y summarize the important points here. We assume that the dynamics of the gas-phase atom can be followed using classical trajectories, i.e., by solving the classical equations of motion using however an eective interaction potential consisting of three terms: 1
where the ®rst term is the interaction potential between the two hydrogen atoms (usually taken as a Morse potential) and r RH±H is the hydrogen bond distance.
dt0 hxk
1
d DEk DEk dqk
Ic;k t0 cos Hk t0 Is;k t0 sin Hk t0 3 where Hk t xk t
1 h
Z
2
dt0 Vkk
4
and DEk denotes the energy transfer to the solid associated with phonon creation (+) and annihilation ()) processes. We furthermore have Z 1 1 0 dt0 Vk R t0 c or sHk t Hk t0 : Ic or s;k t 1
5 is the static interaction potential calculated assuming the lattice atoms in their equilibrium 1 2 positions, Vk is the ®rst and Vk the second derivative of the interaction potential with respect to the k-th phonon mode coordinate. The eective potential is evaluated at each point of the gasparticle trajectory R t0 . Thus the eective potential is time-dependent and couples the excitation processes in the solid to the motion of the gasphase atoms in a self-consistent manner. By including a distribution of phonon states according ph to the surface temperature Ts , the Vadd term also becomes a surface temperature dependent term. Thus, the motion of the atoms and molecules in the gas-phase or at the surface is obtained by numerical solution of the Hamilton's equations of motion using the 'eective' hamiltonian Heff t; Ts given by 0 VI
2. The collisional method
ph Veff VHH r VHH=C Rij Vadd ;
The next term is the intermolecular potential for the hydrogen±solid interaction with the solid carbon atoms in their reference, i.e. equilibrium positions and Rij denote atom±atom distances. The last term is a potential which depends upon the excitation processes occurring in the solid, i.e. it depends not only on the positions of the surface atoms but also upon the degree of phonon exciph tation in the solid. Vadd is given in terms of the Fourier component of the forces exerted from the gas-phase atoms on the atoms of the substrate X 1 0 ph VI Vk gk t; TS ; 2 Vadd
M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
Heff t; Ts
2 X 1 ph PX2i PY2i PZ2i Vadd 2m i i1
DEint t; Ts ;
6
where the ®rst term is the kinetic energy of the gasphase atoms and the last term is the energy transferred to the solid. The phonons are treated simply as a set of normal mode oscillators. We obtain the normal mode eigenvalues (the spectrum) and eigenvectors by diagonalizing the force constant (dynamical) matrix for a ®nite size crystal. Our previous calculations have shown that the quantities we are interested in, converge with a crystal size of about a few hundred atoms. Thus we use a `crystal' consisting of three layers with 62 atoms in each layer (K62/62/62). In order to calculate the dynamical matrix associated with the vibrational normal modes of the graphite lattice (0 0 0 1) assumed in the simulation we use the most relevant central and angular forces between the nearest and next-nearest neighbours in plane and out of plane carbon atoms recently obtained from semi-empirical calculations [18]. The resulting phonon frequency spectrum [15] exhibits a general very good agreement with the phonon density distribution of [18]. It is worth noticing that classical trajectory calculations have the well known zero-point energy problem, that is the fact that this energy is not conserved classically. On the contrary, this energy can be converted into the translational (and internal) motion of the adsorbed atom/molecule thus facilitating the surmounting of the activation barrier. The potential energy surface for H2 formation on graphite does not display any energetic barrier to reaction (see next paragraph) and this makes a classical dynamics treatment feasible even for the lightest nuclei in a wide collisional regime from subthermal energies up to several eV. Aside from this and contrary to dissociative chemisorption processes with energetic barrier [19], the zero-point energy problem is not so serious in recombination reactions because of the availability of plenty of energy (release of the dissociation energy, 4.5 eV for H2 ). In fact, the semi-classical results obtained in this work when compared with the quantum calculations [13] performed using an
15
interaction potential very similar to that assumed here show that a semi-classical treatment is able to retain the overall behaviours typical of a quantum system. Similar agreement between classical and quantum results was found for H2 formation on Cu surfaces [6].
3. The interaction potential We express the full atom/surface interaction as a sum of two terms Vint VH2 =C FS VH=C 1
FS ;
7
where VH=C and VH2 =C are, respectively, the interaction potential between the atomic and molecular hydrogen in the gas-phase and the carbon atoms of the lattice sample. The FS function is de®ned by FS
0:5 tanh 2:65RH±H
2:65
1:
8
It smoothly switches the potential from one term to the other according to the interatomic separation RH±H between the two hydrogen atoms. The two terms in Eq. (7) were modeled on the basis of ab initio binding energy calculations recently reported in the literature. For H2 interacting with graphite we refer to the experimental results obtained from molecular beam experiments [20] subsequently con®rmed in semi-empirical calculations [21]. In this latter work it has been shown that, irrespective of the adsorption surface site, H2 is physisorbed on graphite in the perpendicular geometry at a distance from the surface of about with a small adsorption energy of 51 meV. In 2.8 A contrast to this atomic hydrogen is chemisorbed on graphite. This potential has been explored in several theoretical electronic structure calculations [13,14,22,23] where dierent semi-empirical electronic structure methods have been used to calculate the surface binding energy for a hydrogen atom interacting with carbon at dierent adsorption sites . We refer in the present study to the most recent works by Fromherz et al. [22] and by Jeloaica and Sidis [23]. In the former study, accurate MINDO-based calculations were carried out to obtain the binding energies of hydrogen
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M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
approaching the surface perpendicularly on top at three dierent sites. It turns out that hydrogen is chemisorbed on top of a carbon atom placed on a corner of the hexagonal ring (site A) with a bind ing energy of 1.3 eV and a Z-distance of 1.5 A from the surface. At the centre of the hexagon (site C) the interaction is repulsive, while the hydrogen is weakly physisorbed on the surface site B placed midway between two adjacent carbon atoms. More recently, DFT calculations using generalized gradient corrected PW91-GGA functionals for the exchange and correlation electron interactions have been performed by Jeloaica and Sidis [23] on the same system. A general qualitative agreement among the two works is found for the chemisorption and the repulsive site. The two potentials however have dierent quantitative features in that DFT calculations give a chemisorption well depth of nearly 0.62 eV, twice as small as the MINDO results. A further important factor concerns the shape of the potential which is, in the repulsive branch, steeper in DFT calculations. As it will be shown in [15] the dierence between the two potentials has important consequences for various aspects of the catalytic activity. A further not completely clari®ed question concerns the nature and the height of the lateral barrier for atom migration between adjacent surface sites [23,24], a question of basic importance for understanding the H2 formation between two adsorbed H atoms. All these issues make the determination of the H± graphite interaction potential not completely solved and, indeed,further investigations are currently carried out [13,24]. We assume for the atom±molecule/surface interactions the following parametric potentials: VH2 =C
2 X N X j1
D exp p Rji
and
A VH=C
10 where is the interaction potential for the chemisorbed surface site A. Rji is the separation distance between the hydrogen atom j in the gasphase and the lattice atom i of the surface. N is the total number of surface carbon atoms. A repulsive exponential and a combination of Morse functions similar to Eqs. (9), (10) has been used to model the interaction potential for site C and B, respectively, but we have for simplicity used only the hydrogenatom potential for site A in the present calculations. A more elaborate procedure would use switching functions of the type described above in order to distinguish between the sites. Since this calculation would involve the introduction of further, at present undetermined, parameters we have not considered this aspect here. Although one can choose among dierent functionals to ®t the reference potential and speci®cally the more ¯exible LEPS potential, the Morse-like functions assumed in Eqs. (9), (10) represent the most direct way to built the phonon eective potential as a sum of atom±atom interactions [16]. The potential parameters for the H2 /surface interaction were ®tted to the potential obtained by Novaco et al. [21] (D 3:287 10 3 eV, 1 in Eq. (9)). a 2:375 10 5 eV, p 1:385 A Similarly, we have constructed two dierent potentials for the H/surface interaction ®tting parameters in Eq. (10) so as to reproduce the main A VH=C
3:255
i1
exp p Rji
3:255
2:
a
8 P2 PN 2:304 > j1 i1 a exp b Rji > > > exp b R 2:304 2: q > ji > > > > Z 6 1:58; > P PH > > < j i a1 exp b1 Rji 3:755 exp b1 Rji 3:755 2: q1 > > > ZH 2:0; > P P > > > > > j i a2 exp b2 Rji 2:300 > > > : exp b2 Rji 2:300 2: q2 1:58 < ZH < 2:0;
9
Table 1 Parameters for the H/C interaction potential of Eq. (10)a a a
b
q
2.108()1) 0.108 1.350()2) 1. a and q in units of eV, b in A
a1
b1
q1
a2
b2
q2
4.310()3)
1.085
2.73()5)
0.614
1.015
0.094
M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
Fig. 1. H2 ±graphite interaction potential as a function of the molecule±surface normal distance. Dashed line: data from [21].
Fig. 2. H±graphite interaction potential with H interacting on top of a C atom placed at the corner of the exagonal ring. Dashed line: data from [23].
features of the theoretically predicted potentials given in [22,23]. However, due to the superiority of the DFT calculations, we report in Section 4 results relevant to this potential. In Table 1 we report the parameters for the potential ®tted to the DFT data (Pot-I). Figs. 1 and 2 show the one-dimensional VH2 =C and VH=C potentials as a function of the distance from the surface. A comparison between the calculated and the reference potentials is also shown. We notice that the complete potential energy surface, Vint VH2 , does not exhibit any energy barrier for H2 formation (a small barrier of about 0.1 eV is found for the potential given in [22]). 4. Results and discussion Making use of the semi-empirically determined potential, which however should be considered
17
acceptable as qualitative only, also due to the insucient or contrasting data available for the H/graphite interaction, the semiclassical dynamics of hydrogen impinging on a graphite surface was then developed aiming at determining at least some of the most basic features of the processes taking place at the surface. We in particular focus on the H2 formation after hydrogen atom adsorption at the surface. As it is well known, this reaction can occur through two extreme mechanisms. According to the Langmuir±Hinshelwood mechanism the reaction takes place between two adsorbed hydrogen atoms which migrate on the surface and then recombine. Consequently, the simulation of such complex reaction mechanism would require the knowledge of the full topology of the interaction potential, in particular the height and position of the energy barriers from site to site. For the case under study this information is, at the moment, not completely available yet, although the calculations by Jeloaica and Sidis [23] show the existence of a large lateral energy barrier that would prevent the diusion of the chemisorbed atom along the surface plane. The in¯uence of H atom diusion on graphite surfaces has been explored in the early work by Aronowitz et al. [14] and an energetic barrier for atom migration was also found of about 0.4 eV, very close to the value suggested by Jeloaica and Sidis [23]. The transmission coecient due to `static' tunneling through the barrier gives a rough estimation of the probability for H2 formation after atom migration as low as 10 5 (TS 10 K). (A larger value is however expected when the `dynamical' tunneling eect is considered.) Due to the lack of information on the potential energy surface corrugation, we have therefore considered in the present study the Eley±Rideal reaction mechanism, according to which the recombination takes place through two elementary steps: ®rst a hydrogen atom is chemisorbed, then the ad-atom, possibly scattered from the surface, reacts with the atom approaching the surface from the gas-phase to form molecular hydrogen in a speci®c vibrational and rotational state Hgas graphite ! Had graphite
11
18
M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
Hgas Had graphite ! H2 v; j graphite DEexo 12 We consider in the dynamical simulation the rate determining step (12) of the E±R recombination reaction. Therefore we assume that Had is chemisorbed on site A of graphite (0 0 0 1) surface at the from the surface, just equilibrium distance of 1.5 A at the bottom of the chemisorption well, and in thermal equilibrium at the surface temperature TS , while the hydrogen atom in the gas phase strikes the graphite surface at a given kinetic energy perpendicularly with respect to the surface plane (polar angles (h; u 0; 0). For each impact energy of Hgas a batch of about 500 trajectories were computed which assure a statistical error bar for the calculated probabilities of about 10±15%. Fig. 3 shows the calculated E±R recombination probability PE±R Ekin at TS 10 K as a function of the kinetic energy of the impinging atom. We notice that there is a rapid increase of the recombination probability with increasing impact energy in the interval 0.01±0.04 eV, then it stabilizes around 1 up to 0.1 eV. A relevant aspect of the recombination process concerns the total energy sharing among the internal and translational motions of the formed H2 molecules and the surface phonons. The fractional energy distribution in the reactive collisions is reported in Fig. 4 as a function of the impact energy
Fig. 3. Recombination probability for the E±R reaction Hgas Had ! H2 v; j as a function of the kinetic energy of the gas-phase H atom. The surface temperature is TS 10 K.
Fig. 4. Total energy distribution among the vibrational (DEvib ), rotational (DErot ) and translational (DEtr ) motions of the formed H2 molecules. DEph is the fractional energy transferred to the graphite phonons.
of the impinging H atom. The results show that about 12% of the reaction energy is transferred to the graphite as phonon excitation energy, while the largest fraction is transferred to the vibrational motion with a still consistent translational and rotational excitation. From the displayed results the energy interplay between the molecular vibrations and the translation is also evident. The vibrational distributions reported in Fig. 5 at three typical collisional energies show that the distributions are hyperthermal with a broad peak between v 3 and v 6. It is worth noticing that the dynamics and the energetics of the recombination process are strictly connected with the interaction potential assumed in the dynamics, so that a model potential ®tted to that proposed by Fromherz et al. [22] exhibits dierent energy distribution and reaction probability [12,15]. The vibrational distributions predicted by Farebrother et al. [13] who used a dierent interaction potential, are in qualitative agreement with the present results in that in both cases H2 is preferentially formed in vibrationally excited levels. However, as a consequence of the slightly dierent interaction potential used in the two works, the statistical distributions found in [13] are much depressed with a peak shifted to v 2. A further expected dierence comes from the recombination probability which, according to the fully quantum calculations of [13] approaches
M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
19
Fig. 5. Vibrational population distributions of H2 formed in the E±R reaction. Ekin is the kinetic energy of the impinging hydrogen atom.
unity at about 0.02 eV, that is much faster compared to our semi-classical results shown in Fig. 3. The results displayed in Fig. 4 show that a consistent fraction of the reaction exothermicity is transferred to the rotational motion of the formed H2 molecules. The calculated rotational distributions, not reported here for lack of space, show that H2 is formed in rotationally excited states with a maximum at around j 8±11 and jmax 20±22, according to the impact energy. At Ekin 0:1 eV the rotational distribution is very similar to the results reported in [12] at Ekin 0:132 and TS 70 K. As it will be shown in a subsequent paper [15], the temperature of the graphite surface has, for this system, a signi®cant in¯uence on the interaction dynamics. (The surface temperature eect for H±graphite surfaces has been observed in laboratory experiments.) Thus, at TS 100 K, the surface temperature limit in astrochemistry, the recombination probability decreases while the vibrational distributions of H2 are less pumped compared to those at TS 10 K reported in Fig. 5
with a sharp peak at v 2; 3 (according to the kinetic energy).
Acknowledgements This research was supported by the Italian Space Agency (ASI) under contract Thermodynamics and Kinetics of Hydrogen Plasmas for Space Repulsion, the MURST (contract no. 9903102919-004) and EU TMR grant (contract no. HPRCN-CT- 1999-00005).
References [1] [2] [3] [4] [5] [6] [7]
J.H. Black, A. Dalgarno, Ap. J. 203 (1976) 132. W.W. Duley, D.A. Williams, MNRAS 260 (1993) 37. M.J. Barlow, J. Silk, Ap. J. 207 (1976) 131. A. Dalgarno, J. Chem. Soc. Faraday Trans. 89 (1993) 2111. M. Cacciatore, Pure Appl. Chem. 71 (1999) 1809. M. Persson, B. Jackson, J. Chem. Phys. 102 (1995) 1078. R. Hall, I. Cadez, M. Landau, F. Pichou, C. Shermann, Phys. Rev. Lett. 60 (1988) 337.
20
M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
[8] G.D. Billing, M. Cacciatore, Chem. Phys. Lett. 113 (1985) 23. [9] J.S. Mathis, W. Rumpl, K.H. Nordsiek, Ap. J. 217 (1977) 425. [10] E.L. Fitzpatrick, D. Massa, Ap. J. S. 72 (1990) 163. [11] V. Pirronello, C. Liu, J.E. Roser, G. Vidali, Astron. Astrophys. 344 (1999) 681. [12] P. Parneix, P. Brechignac, Astron. Astrophys. 334 (1998) 363. [13] A.J. Farebrother, A.J.H.M. Meijer, D.C. Clary, A.J. Fisher, Chem. Phys. Lett. 319 (2000) 303. [14] S. Aronowitz, S. Chang, Ap. J. 293 (1985) 243. [15] M. Cacciatore, M. Rutigliano, G.D. Billing, work in preparation. [16] G.D. Billing, Dynamics of Molecule Surface Interactions, Wiley, New York, 2000.
[17] See e.g. G.D. Billing, J. Chem. Phys. 112 (2000) 335, and references therein. [18] L. Lang, S. Doyen-Lang, A. Charlier, M.F. Charlier, Phys. Rev. B 49 (1994) 5672. [19] See e.g. C.M. Laursen, E. Christoersen, G.D. Billing, Chem. Phys. Lett. 290 (1998) 275. [20] L. Mattera, F. Rosatelli, C. Salvo, F. Tommasini, U. Valbusa, G. Vidali, Surf. Sci. 93 (1980) 515. [21] D. Novaco, J.P. Wroblewski, Phys. Rev. B 39 (1989) 11 364. [22] T. Fromherz, C. Mendoza, F. Ruette, MNRAS 263 (1993) 851. [23] L. Jeloaica, V. Sidis, Chem. Phys. Lett. 300 (1999) 157. [24] V. Sidis, private comunication.
Chemical Physics Letters 340 (2001) 13±20
www.elsevier.nl/locate/cplett
Hydrogen atom recombination on graphite at 10 K via the Eley±Rideal mechanism M. Rutigliano a, M. Cacciatore a,*, G.D. Billing b a
CNR-Centro Studi Chimica dei Plasmi, Dipartimento di Chimica, Universita' di Bari. V. Orabona N.6, 70126 Bari, Italy b Chemistry Laboratory III, H.C. érsted Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark Received 12 February 2001; in ®nal form 23 March 2001
Abstract This work reports results on the recombination of hydrogen atoms on a graphite surface at 10 K obtained using a detailed semi-classical molecular dynamics method in connection with the use of recently proposed ab initio potential energy surfaces. The calculated recombination probabilities together with the vibrational distributions of the formed H2 molecules obtained assuming that the surface reaction proceeds via the Eley±Rideal mechanism can have an impact on the chemistry of H2 formation in interstellar space. Ó 2001 Published by Elsevier Science B.V.
1. Introduction Recently, the hydrogen atom recombination on grains as a possible eective catalytic source to the formation of H2 molecules in interstellar clouds has attracted considerable and increasing interest in astrochemistry. The reason for this is the persistent diculty in understanding the abundance of H2 , in particular the high ratio between molecular and atomic hydrogen, observed in dense clouds of the interstellar space. Complex kinetic modeling has, in fact, shown that such abundance can only be consistent assuming a high, close to unity, recombination probability of H on catalytic grain surfaces [1±4]. On the other hand, it is well known that atom recombination on surfaces can be an eective source of molecular species as, for instance, H2 on Cu, W and other metals [5±7], CO
*
Corresponding author. Fax: +39-080-544-2024. E-mail address: [email protected] (M. Cacciatore).
formation on platinum [8]. In these and other cases it has been demonstrated that molecular species are formed predominantly in vibrationally excited states, the extension of the vibrational excitation depending, among other things, on the dynamical mechanism through which the atom recombination takes place, i.e., whether it is direct or indirect [8]. Although somehow hidden due to the diculty of direct probing, atom interaction with solid surfaces can therefore have a great impact on the overall reaction kinetics of molecular gases under non-thermal, diusive conditions as those met in interstellar space. Indeed, for low-pressure, lowdensity gases under diusive conditions, atom recombination on surfaces can compete with the less ecient three-body recombination reaction in the gas-phase. There are now quite ®rm evidences, attested by the so called U-V bump at 217 nm [9,10], that grains in interstellar clouds are predominantly made of silicate and, in particular, of carbonaceous amorphous compounds (very likely
0009-2614/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 3 6 6 - 9
14
M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
covered by ice mantels and other condensable interstellar species) . These indications, in connection with recent developments in molecular dynamics techniques applied to gas±surface interactions, have reinforced experimental [11] and theoretical [12±14] eorts to get accurate estimates of the catalytic activity of such surfaces. It is the purpose of the present Letter to give collisional data relevant for the hydrogen±carbon system. By combining recent progress in ab initio and here especially in density functional theory (DFT) for the calculation of the interaction potential with the semi-classical method developed for a dynamical treatment of surface processes makes it possible to perform suciently accurate calculations of collisional data as: recombination probabilities, energy disposal to the formed H2 molecules and surface energy accommodation. These quantities are calculated for collisional conditions typically met in the interstellar and in plasma-chemical devices and can therefore be relevant for understanding the basic behaviours of such systems. The surface temperature eects, an important issue in reactive catalysis, and the extension to higher surface temperatures relevant for fusion plasma devices and non-thermal laboratory plasmas, has also been investigated and will be presented in a subsequent publication [15].
k
Z gk t; TS
The semi-classical model used in the present calculations is described in detail in [16] and numerous papers [17]. We shall therefore only brie¯y summarize the important points here. We assume that the dynamics of the gas-phase atom can be followed using classical trajectories, i.e., by solving the classical equations of motion using however an eective interaction potential consisting of three terms: 1
where the ®rst term is the interaction potential between the two hydrogen atoms (usually taken as a Morse potential) and r RH±H is the hydrogen bond distance.
dt0 hxk
1
d DEk DEk dqk
Ic;k t0 cos Hk t0 Is;k t0 sin Hk t0 3 where Hk t xk t
1 h
Z
2
dt0 Vkk
4
and DEk denotes the energy transfer to the solid associated with phonon creation (+) and annihilation ()) processes. We furthermore have Z 1 1 0 dt0 Vk R t0 c or sHk t Hk t0 : Ic or s;k t 1
5 is the static interaction potential calculated assuming the lattice atoms in their equilibrium 1 2 positions, Vk is the ®rst and Vk the second derivative of the interaction potential with respect to the k-th phonon mode coordinate. The eective potential is evaluated at each point of the gasparticle trajectory R t0 . Thus the eective potential is time-dependent and couples the excitation processes in the solid to the motion of the gasphase atoms in a self-consistent manner. By including a distribution of phonon states according ph to the surface temperature Ts , the Vadd term also becomes a surface temperature dependent term. Thus, the motion of the atoms and molecules in the gas-phase or at the surface is obtained by numerical solution of the Hamilton's equations of motion using the 'eective' hamiltonian Heff t; Ts given by 0 VI
2. The collisional method
ph Veff VHH r VHH=C Rij Vadd ;
The next term is the intermolecular potential for the hydrogen±solid interaction with the solid carbon atoms in their reference, i.e. equilibrium positions and Rij denote atom±atom distances. The last term is a potential which depends upon the excitation processes occurring in the solid, i.e. it depends not only on the positions of the surface atoms but also upon the degree of phonon exciph tation in the solid. Vadd is given in terms of the Fourier component of the forces exerted from the gas-phase atoms on the atoms of the substrate X 1 0 ph VI Vk gk t; TS ; 2 Vadd
M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
Heff t; Ts
2 X 1 ph PX2i PY2i PZ2i Vadd 2m i i1
DEint t; Ts ;
6
where the ®rst term is the kinetic energy of the gasphase atoms and the last term is the energy transferred to the solid. The phonons are treated simply as a set of normal mode oscillators. We obtain the normal mode eigenvalues (the spectrum) and eigenvectors by diagonalizing the force constant (dynamical) matrix for a ®nite size crystal. Our previous calculations have shown that the quantities we are interested in, converge with a crystal size of about a few hundred atoms. Thus we use a `crystal' consisting of three layers with 62 atoms in each layer (K62/62/62). In order to calculate the dynamical matrix associated with the vibrational normal modes of the graphite lattice (0 0 0 1) assumed in the simulation we use the most relevant central and angular forces between the nearest and next-nearest neighbours in plane and out of plane carbon atoms recently obtained from semi-empirical calculations [18]. The resulting phonon frequency spectrum [15] exhibits a general very good agreement with the phonon density distribution of [18]. It is worth noticing that classical trajectory calculations have the well known zero-point energy problem, that is the fact that this energy is not conserved classically. On the contrary, this energy can be converted into the translational (and internal) motion of the adsorbed atom/molecule thus facilitating the surmounting of the activation barrier. The potential energy surface for H2 formation on graphite does not display any energetic barrier to reaction (see next paragraph) and this makes a classical dynamics treatment feasible even for the lightest nuclei in a wide collisional regime from subthermal energies up to several eV. Aside from this and contrary to dissociative chemisorption processes with energetic barrier [19], the zero-point energy problem is not so serious in recombination reactions because of the availability of plenty of energy (release of the dissociation energy, 4.5 eV for H2 ). In fact, the semi-classical results obtained in this work when compared with the quantum calculations [13] performed using an
15
interaction potential very similar to that assumed here show that a semi-classical treatment is able to retain the overall behaviours typical of a quantum system. Similar agreement between classical and quantum results was found for H2 formation on Cu surfaces [6].
3. The interaction potential We express the full atom/surface interaction as a sum of two terms Vint VH2 =C FS VH=C 1
FS ;
7
where VH=C and VH2 =C are, respectively, the interaction potential between the atomic and molecular hydrogen in the gas-phase and the carbon atoms of the lattice sample. The FS function is de®ned by FS
0:5 tanh 2:65RH±H
2:65
1:
8
It smoothly switches the potential from one term to the other according to the interatomic separation RH±H between the two hydrogen atoms. The two terms in Eq. (7) were modeled on the basis of ab initio binding energy calculations recently reported in the literature. For H2 interacting with graphite we refer to the experimental results obtained from molecular beam experiments [20] subsequently con®rmed in semi-empirical calculations [21]. In this latter work it has been shown that, irrespective of the adsorption surface site, H2 is physisorbed on graphite in the perpendicular geometry at a distance from the surface of about with a small adsorption energy of 51 meV. In 2.8 A contrast to this atomic hydrogen is chemisorbed on graphite. This potential has been explored in several theoretical electronic structure calculations [13,14,22,23] where dierent semi-empirical electronic structure methods have been used to calculate the surface binding energy for a hydrogen atom interacting with carbon at dierent adsorption sites . We refer in the present study to the most recent works by Fromherz et al. [22] and by Jeloaica and Sidis [23]. In the former study, accurate MINDO-based calculations were carried out to obtain the binding energies of hydrogen
16
M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
approaching the surface perpendicularly on top at three dierent sites. It turns out that hydrogen is chemisorbed on top of a carbon atom placed on a corner of the hexagonal ring (site A) with a bind ing energy of 1.3 eV and a Z-distance of 1.5 A from the surface. At the centre of the hexagon (site C) the interaction is repulsive, while the hydrogen is weakly physisorbed on the surface site B placed midway between two adjacent carbon atoms. More recently, DFT calculations using generalized gradient corrected PW91-GGA functionals for the exchange and correlation electron interactions have been performed by Jeloaica and Sidis [23] on the same system. A general qualitative agreement among the two works is found for the chemisorption and the repulsive site. The two potentials however have dierent quantitative features in that DFT calculations give a chemisorption well depth of nearly 0.62 eV, twice as small as the MINDO results. A further important factor concerns the shape of the potential which is, in the repulsive branch, steeper in DFT calculations. As it will be shown in [15] the dierence between the two potentials has important consequences for various aspects of the catalytic activity. A further not completely clari®ed question concerns the nature and the height of the lateral barrier for atom migration between adjacent surface sites [23,24], a question of basic importance for understanding the H2 formation between two adsorbed H atoms. All these issues make the determination of the H± graphite interaction potential not completely solved and, indeed,further investigations are currently carried out [13,24]. We assume for the atom±molecule/surface interactions the following parametric potentials: VH2 =C
2 X N X j1
D exp p Rji
and
A VH=C
10 where is the interaction potential for the chemisorbed surface site A. Rji is the separation distance between the hydrogen atom j in the gasphase and the lattice atom i of the surface. N is the total number of surface carbon atoms. A repulsive exponential and a combination of Morse functions similar to Eqs. (9), (10) has been used to model the interaction potential for site C and B, respectively, but we have for simplicity used only the hydrogenatom potential for site A in the present calculations. A more elaborate procedure would use switching functions of the type described above in order to distinguish between the sites. Since this calculation would involve the introduction of further, at present undetermined, parameters we have not considered this aspect here. Although one can choose among dierent functionals to ®t the reference potential and speci®cally the more ¯exible LEPS potential, the Morse-like functions assumed in Eqs. (9), (10) represent the most direct way to built the phonon eective potential as a sum of atom±atom interactions [16]. The potential parameters for the H2 /surface interaction were ®tted to the potential obtained by Novaco et al. [21] (D 3:287 10 3 eV, 1 in Eq. (9)). a 2:375 10 5 eV, p 1:385 A Similarly, we have constructed two dierent potentials for the H/surface interaction ®tting parameters in Eq. (10) so as to reproduce the main A VH=C
3:255
i1
exp p Rji
3:255
2:
a
8 P2 PN 2:304 > j1 i1 a exp b Rji > > > exp b R 2:304 2: q > ji > > > > Z 6 1:58; > P PH > > < j i a1 exp b1 Rji 3:755 exp b1 Rji 3:755 2: q1 > > > ZH 2:0; > P P > > > > > j i a2 exp b2 Rji 2:300 > > > : exp b2 Rji 2:300 2: q2 1:58 < ZH < 2:0;
9
Table 1 Parameters for the H/C interaction potential of Eq. (10)a a a
b
q
2.108()1) 0.108 1.350()2) 1. a and q in units of eV, b in A
a1
b1
q1
a2
b2
q2
4.310()3)
1.085
2.73()5)
0.614
1.015
0.094
M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
Fig. 1. H2 ±graphite interaction potential as a function of the molecule±surface normal distance. Dashed line: data from [21].
Fig. 2. H±graphite interaction potential with H interacting on top of a C atom placed at the corner of the exagonal ring. Dashed line: data from [23].
features of the theoretically predicted potentials given in [22,23]. However, due to the superiority of the DFT calculations, we report in Section 4 results relevant to this potential. In Table 1 we report the parameters for the potential ®tted to the DFT data (Pot-I). Figs. 1 and 2 show the one-dimensional VH2 =C and VH=C potentials as a function of the distance from the surface. A comparison between the calculated and the reference potentials is also shown. We notice that the complete potential energy surface, Vint VH2 , does not exhibit any energy barrier for H2 formation (a small barrier of about 0.1 eV is found for the potential given in [22]). 4. Results and discussion Making use of the semi-empirically determined potential, which however should be considered
17
acceptable as qualitative only, also due to the insucient or contrasting data available for the H/graphite interaction, the semiclassical dynamics of hydrogen impinging on a graphite surface was then developed aiming at determining at least some of the most basic features of the processes taking place at the surface. We in particular focus on the H2 formation after hydrogen atom adsorption at the surface. As it is well known, this reaction can occur through two extreme mechanisms. According to the Langmuir±Hinshelwood mechanism the reaction takes place between two adsorbed hydrogen atoms which migrate on the surface and then recombine. Consequently, the simulation of such complex reaction mechanism would require the knowledge of the full topology of the interaction potential, in particular the height and position of the energy barriers from site to site. For the case under study this information is, at the moment, not completely available yet, although the calculations by Jeloaica and Sidis [23] show the existence of a large lateral energy barrier that would prevent the diusion of the chemisorbed atom along the surface plane. The in¯uence of H atom diusion on graphite surfaces has been explored in the early work by Aronowitz et al. [14] and an energetic barrier for atom migration was also found of about 0.4 eV, very close to the value suggested by Jeloaica and Sidis [23]. The transmission coecient due to `static' tunneling through the barrier gives a rough estimation of the probability for H2 formation after atom migration as low as 10 5 (TS 10 K). (A larger value is however expected when the `dynamical' tunneling eect is considered.) Due to the lack of information on the potential energy surface corrugation, we have therefore considered in the present study the Eley±Rideal reaction mechanism, according to which the recombination takes place through two elementary steps: ®rst a hydrogen atom is chemisorbed, then the ad-atom, possibly scattered from the surface, reacts with the atom approaching the surface from the gas-phase to form molecular hydrogen in a speci®c vibrational and rotational state Hgas graphite ! Had graphite
11
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M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
Hgas Had graphite ! H2 v; j graphite DEexo 12 We consider in the dynamical simulation the rate determining step (12) of the E±R recombination reaction. Therefore we assume that Had is chemisorbed on site A of graphite (0 0 0 1) surface at the from the surface, just equilibrium distance of 1.5 A at the bottom of the chemisorption well, and in thermal equilibrium at the surface temperature TS , while the hydrogen atom in the gas phase strikes the graphite surface at a given kinetic energy perpendicularly with respect to the surface plane (polar angles (h; u 0; 0). For each impact energy of Hgas a batch of about 500 trajectories were computed which assure a statistical error bar for the calculated probabilities of about 10±15%. Fig. 3 shows the calculated E±R recombination probability PE±R Ekin at TS 10 K as a function of the kinetic energy of the impinging atom. We notice that there is a rapid increase of the recombination probability with increasing impact energy in the interval 0.01±0.04 eV, then it stabilizes around 1 up to 0.1 eV. A relevant aspect of the recombination process concerns the total energy sharing among the internal and translational motions of the formed H2 molecules and the surface phonons. The fractional energy distribution in the reactive collisions is reported in Fig. 4 as a function of the impact energy
Fig. 3. Recombination probability for the E±R reaction Hgas Had ! H2 v; j as a function of the kinetic energy of the gas-phase H atom. The surface temperature is TS 10 K.
Fig. 4. Total energy distribution among the vibrational (DEvib ), rotational (DErot ) and translational (DEtr ) motions of the formed H2 molecules. DEph is the fractional energy transferred to the graphite phonons.
of the impinging H atom. The results show that about 12% of the reaction energy is transferred to the graphite as phonon excitation energy, while the largest fraction is transferred to the vibrational motion with a still consistent translational and rotational excitation. From the displayed results the energy interplay between the molecular vibrations and the translation is also evident. The vibrational distributions reported in Fig. 5 at three typical collisional energies show that the distributions are hyperthermal with a broad peak between v 3 and v 6. It is worth noticing that the dynamics and the energetics of the recombination process are strictly connected with the interaction potential assumed in the dynamics, so that a model potential ®tted to that proposed by Fromherz et al. [22] exhibits dierent energy distribution and reaction probability [12,15]. The vibrational distributions predicted by Farebrother et al. [13] who used a dierent interaction potential, are in qualitative agreement with the present results in that in both cases H2 is preferentially formed in vibrationally excited levels. However, as a consequence of the slightly dierent interaction potential used in the two works, the statistical distributions found in [13] are much depressed with a peak shifted to v 2. A further expected dierence comes from the recombination probability which, according to the fully quantum calculations of [13] approaches
M. Rutigliano et al. / Chemical Physics Letters 340 (2001) 13±20
19
Fig. 5. Vibrational population distributions of H2 formed in the E±R reaction. Ekin is the kinetic energy of the impinging hydrogen atom.
unity at about 0.02 eV, that is much faster compared to our semi-classical results shown in Fig. 3. The results displayed in Fig. 4 show that a consistent fraction of the reaction exothermicity is transferred to the rotational motion of the formed H2 molecules. The calculated rotational distributions, not reported here for lack of space, show that H2 is formed in rotationally excited states with a maximum at around j 8±11 and jmax 20±22, according to the impact energy. At Ekin 0:1 eV the rotational distribution is very similar to the results reported in [12] at Ekin 0:132 and TS 70 K. As it will be shown in a subsequent paper [15], the temperature of the graphite surface has, for this system, a signi®cant in¯uence on the interaction dynamics. (The surface temperature eect for H±graphite surfaces has been observed in laboratory experiments.) Thus, at TS 100 K, the surface temperature limit in astrochemistry, the recombination probability decreases while the vibrational distributions of H2 are less pumped compared to those at TS 10 K reported in Fig. 5
with a sharp peak at v 2; 3 (according to the kinetic energy).
Acknowledgements This research was supported by the Italian Space Agency (ASI) under contract Thermodynamics and Kinetics of Hydrogen Plasmas for Space Repulsion, the MURST (contract no. 9903102919-004) and EU TMR grant (contract no. HPRCN-CT- 1999-00005).
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