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Spontaneous desorption of vibrationally excited molecules physically adsorbed on surfaces
ChemicaI Physics 58 (1981) 385-393 North-Holland Publishing Company
SPONTANEOUS PHYSPCAECY Donald
LUCAS$
DESORPTION ADSORBED and George
Depnrzment ofChemistry, Indiana
OF VIRRATIONALLY
EXCITED
MOLECULES
ON SURFACES” E. EWING
Unir;ersity,
Bbomingron,
IN
474M.
USA
Received 12 December 1980
The energy within 2 vibrationally excited phy-sisorbed molecu!e often exceeds that needed to break its bond to the surface. Energy transfer from the vibrating chemical bond. to the surface bond causes the surface bond to ruptore and the vibrationdly relaxed adsorbate is released from the surface. We present a theoretical model which allows an estimation of the residence time of a vibrationally excited adsorbate on a surface. Because of uncertainties in the nature of the surface bond. the lifetimes obtained from the analytical expressions presented have only qualitative significance. The results are interpreted in terms of Franck-Condon overlaps between the wavefunctions which describe the adsorbatesu’bstrate complex and the released adsorbate. Lifetimes are calculated for hydrogen isotopes adsorbed on sapphire surfaces. Guide-lines are given for estimating lifetimes of other systems in terms of a few easily calculated parameters.
1. Inrroduction A physically adsorbed molecule may be driven off a surface in a variety of ways. If the system (adsorbate and substrate) is heated the surface bond wiU eventually break in the process called thermal desorption [l-3]. A more specific release of the adsorbate occurs if light from a laser source drives the vibration of the adsorbate against its surface bond until rupture occurs [4]. In thii paper we will discuss a third type of desorption. Here a chemical bond within the adsorbate is vibrationally excited. The energy in the vibrating chemical bond usually exceeds that needed to break a van der Waals surface bond. After a time, energy transfer from the excited chemical bond to the surface bond causes the surface bond to rupture and the adsorbed molecule is flung off_ This is then an
* Contribution No. 3549 from the Chemical Laboratories of Indiana University. Supported by the National Science Foundation. i Present address: Energy and Environment Division, Lawrence Berkeley Laboratory, University of California at EerkeIey, Berkeley. CA 94120, USA.
0301-0104/81/0000-0000/$02.50
@ North-I-Iolland
example of vibrational prediisociation of an adsorbed mo!ecule. We may describe this spontaneous vibrational predissodation process by a simple equation: A-D*
. . . SL_4-D+S+AE_
(I)
The adsorbed molecule is called A-D* and the asterisk tells us it is vibrationally excited. The substrate, S, holds the excited molecule by a van der Waals bond represented by the dots.. After time r the surface bond breaks and the adsorbate, A-D, now vibrationally relaxed, flies away with translational kinetic energy AE. It is our task in this paper to offer a theoretical model which aUows an estimation of the residence time 7 of the vibrationally excited adsorbate. There are at least two ways in which au adsorbed molecule can become vibrationaily
excited. The adsorbate may be the product of a chemical reaction on the surface which has left the adsorbate in a vibrational@ excited state. A few surface reactions have been studied where the vibrational state of the product has been speciiied. One example [S] is the nitrogen atom
386
D. Lucas, GE. Ewing / Desorprion of aibrationally excitedmo!ecules
recombination on iron, wit!? the subsequent release of vibrationally excited Nz_ Another example is the recombination of hydrogen atoms on surfaces prepared in the laboratory [6] or on grains in interstellar space [7]. The product, H$ -. - S, could be released by a process such as that in eq. (1). Alternatively, the adsorbate may absorb an infrared photon to become vlbrationally excited. This IR induced release of an adsoibed molecule is an example of photodesorption. Radiation, frequency tuned to match vibrational levels of specific adsorbates, could be used to induce the selective release of different chemical species or even different isotopes. Because of its potential for separating molecules this application of the photodesorption process has been named photochromatography by Suslick [8]. He has performed preliminary experiments for separating hydrogen isotopes adsorbed on AlzOj by photochromatography and has written a patent [S] which describes the practical aspects of the process. Others have also considered the feasibility of isotopic separation by IR induced photodesorption [9, lo]. Most of the mathematical machinery for discussing spontaneous desorption of vibrationally excited mo!ecules has recently beenset down in a slightly disguised form. The systems that have been discussed are vibrationally excited van der WaaIs molecules such as HCl* - - - O(CHJz, r?j . . . He, or Hz - - - Hz, and their vibrational predissociation lifetimes have been theoreticaIIy described [11-13-J. The vibrational prediisociation in these excited van der WaaIs molecules is formally similar to spontaneous desorption of -4-V -. . S. For this reason we will be drawing freely on the theory of relaxation of excited van der Waals moIecules in setting up our theoretical model for vibrational predissociation of molecules on surfaces. A series of papers 1143 on rhe thermal desorption of atoms on surfaces wri?ten over 40 years ago and largely ignored by -ihose working on van der -Waals molecules also contains essentially the same mathematical methods. In section 2, to follow next, the theoretical model will be presented. A numerical example
for calculation of desorption rates of HT - - - A1203 will be given in section 3. A general guide to desorption rates of other systems will be offered in section 4 which we
hope will be useful to those designing experiments. Finaily a summary is gathered section 5.
in
2. The theoretical model 2.1.
The golden rule
An energy level diagram for the desorption process of eq. (1) may be viewed in fig. 1. We begin, on the left, with the vibrationally excited system A-D* . . . S in one of the discrete levels labeled by the u, quantum number. These levels represent the stretching vibration of the adsorbed molecule against the van der Waals bond which holds it to the surface. Isoenergetic with A-D* - - - S is the continuum of A-D (now vibrationally relaxed) as it fries away with translational energy AE and velocity urn from the surface S. This picture, which represents energy transfer from a discrete level into a continuum, invites application of the golden rule or its equivalent [15]:
(2) In order to obtain r-l we need to specify the final and initial state wavefunctions, qgt and Yf’ and give a coupling term V,,. 2.2.
The wauefunchxzs
We will take the initial state wavefunction
to
be
WP’ = r-‘R,(r)&(x), which describes the state of A-D* . - - S. The geometry for the problem is shown in fig. 2. The distance between the center of mass.of A-D and the surface is given by r. We can provide R,(r), which describes the vibration bf the adsorbate against the surface, only after the surface bond potential function is specified. We
D_ Lucas, G.E. Ewing / Desorpfion of vibrationally excitedtmfecules
2IA OA
387
A-D---S
Fig. 1. Spontanwus desorption of a vibrationally excited moiecule adsorbed on a surface. The scale for the energy is that for Hz .__ ALO3 considered in the text.
will use the Morse function for this potentid, V(r)=D,{exp[-2a(r-rc)]-2
exp[-a(r-r,)l), (4)
with D, the depth at the bottom of the well at separation r,. The steepness of this potential energy function is given by the range parameter a. The analytical form of the Morse oscillator wavefunctions R,(r) for the discrete v, levels is given elsewhere [13,14]. An harmonic oscillator wavefunction &(x) describes the internal vibration of A-D with x the v&rational displacement coordinate and u the quantum level. The adsorbate may be generalized to a polyatomic molecule in which case A represents other atoms in the molecule. In this paper, however, we will imagine A-D as a diatomic moIecule with atom D directed toward the surface. The initial state for the harmonic oscillator is &I(X). The final state wavefunction, P, m = r-l&(r) defines A-D
&(X) , + S; The molecule A-D
(3 which has
been released from the surface is now vibrationally relaxed and &(x) describes this condition. As A-D flies far away from the surface with translational kinetic energy LIE its wavefunction resembles a plane wave. Near the surface the translational motion of A-D is moderated by its interaction with the surface. The wavefunction R,(r) describing the continuum behavior of A-D under the influence of a Morse potential is given elsewhere [13]. FortunateIy, we will not need the involved analytical forms of either R,(r) or R,(r) for this paper. 2.3.
The coupling term
The coupling term V,, must connect the coordinates of the vibrating chemical bond of A-D” with the surface bond of A-D’ .- - S. We will make a simple assumption to connect the x and r coordinates. We imagine that motion of the chemical bond of A-D moduiates the surface potential energy only because the distance between D and S changes. This coupling term
D. Lucas, G.E. Ewing j Desorption of uibrationallyexcited molecules
388
proposed many years ago for vibrational relaxation processes and discussed elsewhere [13, 16) is given by Vmn(x, r) = -2asaDCx{exp -exp
[-Za(r
C-a (r - rC)]L
where the fraction S is u = nlAliiI&D,
[13] yields
-I-,)] (61
of displacement
algebra given elsewhere
of D toward
(7)
with mA the mass of A and m~-o the adsorbate mass. The component of the vibrational displacement of A-D toward the surface is given
X { n$l ecn -w+q2_1]
x{[(d0-~~-~~~iq2m1/2d0~*exp
(8)
where 13is defined in fig. 2. 2.4.
Desorptior:
rates
Fortunately the matrix element (T~‘\V,,\Yr(no’) of the golden rule using our expressions for the wavefunctions and the coupling term has a simple analytical result. The
(9)
Several terms need defining in this rate expression. The harmonic oscilIator matrix element for the chemical bond of the adsorbate is [l6] (10)
by s =cos 8,
(-~q~).
CL= m+_mJm+_-D,
(11)
is the reduced mass of A-D. The vibrational energy WA_,, is the difference between the o = 1 and o = 0 states of the adsorbate. The adsorbate trapped by the Morse potential [eq. (4)] can be represented by the dimensionless parameter d = (2mA_DD,)“‘/aii.
(12)
The vibrational energy levels of the adsorbate against its surface bond are
* -“,
ir
%
\ :, ;:
‘1, .D
:.: r : .
l3
W ,=-[ah(2d
-1-2r~Jj’/8m~_~+D,,
given relative to the bottom of the Morse potentia1 well. Since the Morse parameters for the surface bond are not well known we Iose little useful information by rounding off d to an integer and calling it do. This parameter which appears in the rate expression of eq. (9) may be interpreted as the number of bound states of A-D .a. S contained in the Morse potential well. The final translational energyhE of A-D now released from the surface is incorporated in the dimensionless term qm = (2mA-&E)“‘/a
Fig. 2. The coordinates for a physicallya&orbed molecule
on a surface.
(13)
h = p/a tr,
(14)
which is a reduced momentum scaled by the range parameter a. Since A = h/p the q,,, parameter is related to the de Broglie wavelength of A-D flying away from the surface
D. Lucas, G.E. Ewing f Desorption of L’ibmtima~ly by the expression A = 2sr/aq,.
(1%
The parameter q,,, appears in the rate expression of eq. (9). We obtain ti by finding the energy remaining after the adsorbate bond from the vv level has been broken, AE=
WA-D-D,+
W,.
(16)
We now have all the expressions needed to perform numerical calculations of 7:: by vibrational predissociation of the adsorbate on a surface. However, thermal desorption is a competitive mechanism for releasing molecules from a surface. In this case the adsorbate and substrate are in equilibrium at temperature T. The energy needed to dissociate the adsorbate (in its u,. = 0 level) from the substrate is Do = D,- W,. The probability of the adsorbate site on the surface having this energy is exp (-D,/kl). The adsorbate can pick up this energy at its collision frequency, v = We/h, with the surface so the resulting thermal desorption rate is (ro’)rr’
= Y exp (-Do/kT).
(17)
This crude estimate of the thermal desorption rate, Frenkel’s formula, has been known for decades Cl] and has been recast by others in more sophisticated forms [14, 171.
3. A numerical example One of the most thoroughly studied examples of physical adsorption is HZ on AlaOS. The properties of AlaO, for chromatographic separation of ortho- and para-Ha and its isotopes have been discussed [lS]. Moreover, the Hz a.. AIZOa system was selected for photochromatographic experiments [8]. It is particularly useful for our purposes that the surface bond for Hz - - - AlJ& has been described [19]. It is for these reasons that we select this system for our sample numerical calculation of desorption. King and Benson [19] view HZ polarized by the electric field from A13+ and O*- ions near
excited m&cute+
389
the A1203 surface. The resulting induced dipole of HZ orients the molecule so that it is held perpendicular to the surface from which the electric field emanates. {It is this induced dipole which enables HZ. to absorb infrared radiation easily [20] and becomes vibrationally excited in the photochromatographic experiments.) They offer a variety of potential functions for the adsorbate-surface bond. The one they prefer is HZ located over a vacancy on the surface and is of the form V(r) = &P(rJr)9
-9(relr151
(18)
with D, = 1.0 x lo-l3 erg and r, = 2.40 A. An ambiguity results in trying to fit the range parameter of the Morse potential of eq. (4) to the 9-5 King and Benson potential of eq. (18). The 9-5 potential energy goes to zero at r = u = 2.07 A and somewhat arbitrarily we adjust the Morse potential so that it goes to zero at this separation as well. The result is a = 2.09 X 10’ cm-‘. Using D, = 1.0 x lo-l3 erg and a = 2.09 x 10’ cm-’ together with rn*-n = rnsl = 3.32 x lo-” g we obtain d = 3.71 (da =4) by eq. (12). The zero point vibration of HZ against the surface is W, = 2.53 x 10-r” erg (128 cm-‘) from eq. (13). With the vibrational frequency of Hz, WA-u = W,, = 8.26~ lo-” erg (4130 cm-‘) [20], we obtain AE = 7.51 x 10-r’ erg (3795 cm-l) by eq. (16). We then find q,,, = 10.2 by eq. (14), which corresponds to a de Broglie wavelength of A =0.29 A by eq. (15). Finally, use of eqs. (10) and (11) gives (w) = 8.96 x lo-” cm. Application of eq. (9) reveals r, = 6.0 x lo-’ s as the lifetime of Hz - - - AlzOs from the v,=O level. The error limits that may be placed on this lifetime are large. An uncertainty in the range parameter of only LO% introduces a variation in -ru,by an order of magnitude. Variations in the range parameter by more than 20% result from different choices of potentials ofIered by King and Benson [19]. Reasonable values of a fall between 2 x 10’ cm-’ and 3 x 10’ cm-‘. We can then safely say only that the desorption lifetime is better described by a nanosecond time scale rather than microseconds or picoseconds.
390
D. Lucas, G.E. Ewing / Desorptiqn of cibrationally excited molecules
In setting up experiments to detect molecules released from a surface by vibrational predissociation the thermal desorption channel must be considered. We will use eq. (17) for H; on Al& and our original set of parameters. We find v = We/h = 3.8 x IO” s-l and DO = Y&T W0 = 7.5 x lo-” erg and consequently = 3 x IO-to s at 77 I< and 7;“’ =0.2 s at 70 20 K. Thus the thermal and vibrational desorption rates may be competitive at 77 K but vibrational desorption clearly dominates at 20 K. A large isotope effect is predicted for op - - - AlzOs vibrational predissociation. Working through the same type of arithmetic as before with WA_,, = W,, = 5.92 x lo-r3 erg (2990 cm-‘) we find qm = 11.8 and r,= 1.2 x lo-’ s from the u,. = 0 level.
4. Other systems, other channels Rather than provide more detailed numerical calculations of other systems we will examine the form of ihe rate expression of eq. (9) to learn what factors are primarily responsible for the efficiency (or ineEiciency) of the desorption process. Two Franck-Condon factors appear in eq. (9). The first deals with changes in the vibrational wavefunctions when A-D* is relaxed to A-D. The overlap of the dr(x) and &(x) wavefunctions through the coupling term yields the vibrational Franck-Condon factor a’~~a’(x)~. The energy scale for the coupling term is D,, the we:1 depth of the surface bond. This accounts for all the system parameters to the Ieft of the bracket of eq. (9). The remaining terms reflect the FranckCondon factor involving changes in the motions of the adsorbate while on the surface and after it is released. The wavefunctions R,(r) and R,(r) through the ccupling term of eq. (6) yield this translational Franck-Condon factor. The result involves only the reduced parameters do and qm. While the vibrational Franck-Condon factor contains on!y quadratic terms whose numerical range is rather limited, the trans-
lational Franck-Condon factor contains a product expression, a factorial term and an exponential. It is the translational FranckCondon factor then which really dominates the outcome of the desorption rate. We have plotted the desorption lifetime from the u, = 0 level given by eq. (9) as a function of q,,, for several values of & in fig. 3. The vibrational Franck-Condon overlap has been given a typical value (r2~2u2{.x)2 = 2 x 10m3 and the adsorbate weI1 depth is taken to be 2 x lo-r3 erg, a representative value for physisorption [2]. The desorption lifetime follows a simple exponentiai dependence for qrn > do_ There is a simple physical explanation for this behavior. The departing molecule is approximately described by a plane wave of de Broglie wavelength A = 27r/aq,. The adsorbed molecule in its u, = 0 level is (almost) an harmonic oscillator against the surface and its wavefunction resembles a gaussian. It is the overlap of the near plane wave and the gaussian functions within the matrix element of the golden rule which determines the efficiency of the desorption process. The magnitude of this overlap is dominated by the de Broglie wavelength of the -4-D molecule flying away from the surface. Algebraically the exponential dependence on the de Broglie wavelength appears in the final term, exp (-mQ, of eq. (9). We have referred io the logarithmic lifetime dependence of the sort shown in fig. 3 as the momentum gap correlation [Zl]. The momentum gap can serve as a useful guide in estimating the lifetiie of the vibrational desorption process. Because q,,, = (2rn+,_oE)“*/a~ we see that three system parameters are involved: a, AE, and mA_D. A linear change in a results in an exponential change in r,,_ Thus the uncertainty in this parameter because of the poorly defined surface bond potential energy makes quantitative theoretical calculations unreliable. The energy gap AE is seen from eq. (16) to depend on three terms. An adsorbate which has both a low vibrational frequency and a strong surface bond favors a small AE and a short vibrational predissociation. time. Adsorbates in excited vibra-
D. Lucas, GE. Ewing I DesorpTionof uibracionnllyexcited molecules
391
we now explore. We have ignored energy flow channels which may leave either the released adsorbate or the surface excited. We have treated the substrate, S, as a homogeneous mass. This bulk mass absorbs the momentum needed to balance that of the departing adsorbate but accepts negligible energy. However, let us now imagine S to be an atom or ion (or several atoms or ions) at the surface bound directly to A-D. This portion of the surface is in turn coupled to the rest of the lattice. The disposition of the energy and momenta now changes significantly. We can take care of this in the simplest way by replacing rnA_D [in eqs. (12)-(14) and the abscissa of fig. 33 by the reduced mass of A-D and S, 9,,,=(2n1~_~
AE) “2,(1x
Fig. 3. The lifetime of a vibrationally excited mo!ecule adsorbed on a surface.
tional energy states (IJ,= 1,2 . ..) against the surface with the energy W, will be released with a smaller energy gap and tend to have shorter residence times. Isotopic changes affect both m&u and dE (through WA-, and W,) and may result in either an increase or decrease in TO. In regions where do> qm, the released adsorbate has relatively low translational momentum and its effective de Broglie wavelength near the surface is dominated by the shape of the adsorbate-substrate potential well. The lifetime for desorption then depends exponentially on do with milder variations with q,,,. As an application of fig. 3 [or eq. (9)] let us estimate the lifetime of CH8 ... AlzOa. We take the molecular excitation to be [22] w&u = P& = 6 x lo-r3 erg (3010 cm-‘) and 0, = 2 x lo-” erg (1000 cm-l). We find do= 12 with Q ~2.5 X IO* cm-’ and appioximate CHZ by CH3-H* and take & before ~2s2~2(~)2 = 2 X lo-‘. Ignoring the zero point vibration against the surface we find AE = WA-D-D== 4 X 1013 erg (2000 cm-‘) which gives qm = 18. Fig. 3 then reveals r. = lo-’ s. The model we have presented leading to eq. (9) or fig. 3 contains critical limitations which
&A-D5 =
mA-_DmS/(mA-D+
mS.),
(19)
where ms is the effective mass of the surface beneath the adsorbate. The picture that emerges is that of A-D flying away from the surface and the rebounding vibrationally excited S* colliding with its neighbors which take up its translational ener-v and distribute it to the remainder of the solid. The substrate will retain energy WS into the localized vibrational modes of the surface which supports the adsorbate. This process is A-D*
..-S+A-D+S*+AEv-s
(20)
with AEv_s=
W..-D-D,+
W,-
WS.
(21)
We call this the V-S channel. Again referring to the CHZ -0 - Alr03 example, the substrate has fundamental vibrations up to 870 cm-’ [231 and strong optical absorption to ~1600 cm-’ [24]. The en:rgy gap may then be AEv-s = 400 to 1100 cm . If we imagine methane bound directly to A13+ then the effective reduced mass wil1 be J.Jc~.~ = 1.7 x 10eZ3 g. Using o = 2.3 x 10’ cm-l we have do = 10 and a resulting qEs = 6.3 to 10.4. Desorption leading to a vibrationally excited released adsorbate may be represented as A-D*
. ..S-+A+-D+S+~E~X..
(22)
The energy gap now is AEvv = W-j,_,-,- DC + W, - WA*_*
(23)
D. iiicar, G.E. Ewing / Desorpticn of oibrationally excited molecules
392
and has been reduced from eq. (16) by the energy WArD in the vibration of A*-D. This process is anaIogous to V-V transfer in gas phase relaxation [16]. In the CHT *.- Al103 system just considered the released adsorbate may be excited in the u2 vibration at 1525 cm-’ [22]. The energy gap originally A.E=2000 cm-’ is now lowered to &XE,-475 cm-’ and the reduced momentum becomes q,” - 8.6. While either the V-S or V-V channels lower the momentum gap they also introduce an additional Franck-Condon factor involving the excitation of a vibrational mode of the substrate or the adsorbate. As we saw a typical value for this Franck-Condon factor in a rate expression is =10e3 (or lo3 in a residence time). Thus whiie values of q,, ~6.3 to 10.4 for the V-S channel of CHZ -a- Al203 applied to fig. 3 would suggest times from rO= lo-’ to lo-’ s, the vibrational Franck-Condon factors to excite the substrate modes increase these times to To== 10e6 to lo-” s. Likewise qm 28.6 expected for the V-V channel when the v1 mode of the released methane is excited yield r0 2 lo-’ s rather than ro= 10e8 s read from fig. 3. In any event all these channels are more efficient than the process in which the released CH, deposited no energy into either itself or the substrate_ A final relaxation channel may leave the released adsorbate rotationally excited A-D*
--.S+A-D’+S-i&5,,_~.,
with the ener,7
gap
&v-X.,,
-D,
= w,,
+ w,-
(24)
W&D?.
(25)
The energy partitioned into the tumbling motion of the released adsorbate is given by WA-D+. In the case of CHZ k-. Alz03 we might expect to find the released methane in a high J state. It should be noted that there are channels in which the adsorbate is not released from the surface. Transfer of vibrational ener-9 from one adsorbed molecule to another adsorbed molecule nearby is possible through long-range
.
coupling terms. Another possibility is that the surface modes are in resonance with the excited adsorbate, resulting in energy transfer to the substrate with no energy remaining to break the surface-adsorbate bond. Brus [25] has treated this relaxation channel. He considers the energy transfer of a vibrationally excited molecule to the surface of a dielectric solid as a classical electromagnetic phenomenon. When there is a match in their vibrational frequencies and their oscillator strengths are large, the vibrating electric dipole of the adsorbate couples strongly with surface polarons. The lifetime associated with this channel of resonance energy transfer from the adsorbate to the substrate is 7” 10-l's. However, the energy transfer rate drops rapidly when the adsorbate vibrational frequency is even slightly off resonance from the characteristic surface polaron frequency. If these frequencies are mismatched by only 10% the lifetime increases two orders of magnitude to t = 10M1’s. If the mismatch widens, the surface mode adsorption coetiicient at the frequency of the vibrating adsorbate vanishes and this channel for energy transfer is closed. For the molecules we have been considering at VA_*-3000 to 4000 cm-’ on typical dielectric solids with 3;, L- 1000 cm-‘, we expect therefore that this non-desorptive channel will be closed. We have been talking about desorption from dielectric solids. What about metals? Here the free electrons are able to take up energy in a variety of ways. The surface plasmons can be driven over a wide frequency range by a vibrationally excited adsorbate as Brus shows {25,26]. With such an efiicient channel for vibrational relaxation without breaking the surface absorbate bond, metals may be poor subsirates for the desorption experiments we have been discussing. Calculations for such systems have, however, been performed [27J Exploration of the various V-V and V-T,R channels has been initiated for vibrationally excited van dar Waals molecules [i3,28-311. Further theoretica investigation of the V-S channel should probably await the first experimental results of vibratitinally induced desorption.
D. Lucas, G.E. Ewing / Desorption of vibrationally excited molecules
5. SMlmary Let us summarize this guide to spontaneous desorption of physically adsorbed vibrationally excited molecuies. The most efficient desorption processes will occur for adsorbates with a small number of bound states (do small) and when released the adsorbate has small translational momentum (small qm). This momentum gap correlation is most succinctly revealed by fig. 3. Smaller translational momentum will be achieved if the adsorbate can take up energy into its internal motions. Absorption of energy into lattice modes of the substrate will also serve to reduce the translational momentum and provide for more e9icient desorption. However, if the vibrational frequency of the adsorbate is in near resonance with surface polarons or plasmons of the substrate, energy transfer to the solid will be so efficient that desorption will be quenched. A test of these possible relaxation channels awaits the first experimental measurements of desorption of vibrationally exdited molecules.
References [I] J.H. de Boer, The dynamical character of adsorption, 2nd Ed. (Clarendon, Oxford, 1968). [2] S. Ross and J. Oliver, On physical adsorption (Interscience, New York, 1964). [3] G.A. Somorjai, Principles of surface chemistry (Prentice-Hall, Englewood Cliffs, 1972). [4] MS. Slutsky and T.F. George, Chem. Phys. Letters 57 (1978) 474; J. Lin and T.F. George, Surface Sci. 100 (1980) 381. [S] R.P. Tnorman, D. Anderson and S.L. Bemasek. Phys. Rev. Letters 44 (1980) 743.
393
[6] A. Schutte, D. Bassi, F. Tommasini and A. Turelli, G. Stoles and L.J.F. Hermans, 3. Chem. Phyx. 64 (1976) 4135. [7] D. Hollenbach, M-W. Werner and E.E. Saltpeter, Astrophys. J. 163 (1971) 165. [i3] K.S. Soslick, US Patent 4,010,100 (1977). [9] K.S. Kochelashvili, N.V. Karlov, A.N. Orlov, R.P. Petrov, Y&N. Petrov and A.M. Prokhorov, JEPT Letters 21 (1975) 302. [lo] T.E. Gangwer, Opt. Laser Technol. 10 (1978) 81. [ll] C.A. Co&on and G.N. Robertson, Proc. Roy. Sot. (London) A337 (1974) 167. [12] J.A. Beswick and J. Jortner, 3. Chem. Phys. 68 (1978) 2277. [13] G. Ewing, Chem. Phys. 29 (1978) 253. [14] J.E. Lennard-Jones and A.F. Devonshire, Proc. Roy. Sot. (London) A156 (1936) 6. This paper contains references to earlier papers of the series. [Xl G. Ew5ng. J. Chem. Phys. 72 (1980) 2096. [16] K. Hcrzfeld and T. Litovitz, Absorption end dispenion of ultrasonic waves (Academic, New York, 1959). 1171 Z.W. Gortef, H.J. Kreuzer and R. Teshiia, Phys. Rev. B221 (1980) 5655. [18] E.H. Carter Jr. and H.A. Smith, I. Phys. Chem. 67 (1963) I512; D.L. West and A.L. Marston, J. Am. Chem. Sot. 86 (1964) 4731. [19] J. KingJr. and S.W. Benson, J. Chem. Phys. 44 11966) 1007. [ZO] N. Sheppard and D.J.C. Yates, Proc. Roy. Sot. 238A (1956) 69. 1211 G. Ewing, 5. Chem. Phys. 71 (1979) 3143. [22] G. Herzberg, Infrared and Raman spectra of polyatomic molecules (Van Nostrand. Princeton, 1945). [23] H. Bialas and H.J. Stolz, Z. Physik B21 (1975) 319. [24] R.A. Smith, FE. Jones and R.P. Chasmar, The detection and measurement of infrared radiation (Cfarendon, Oxford, 1968). [25] LX. Bras, J. Chem. Phys. 74 (1981) 737. [26] L.E. Bms, I. Chem. Phys. 73 (1980) 940. [27] H.J. Kreuzer and D.N. Lowy, Chen. Phys. Letters 79 (1980) 50. [28] D. Morales and G. Ewing, Chem. Phys. 53 (1980) 141. [29] MS. Child and C-l. Ashton, Faraday Discussions Chem. Sot. 62 (1976) 307. [30] J. Beswick and J. Jortner, J. Chem. Phys. 71 (1979) 4737. [31] J. Beswick and J. Jortner, J. Chem. Phys. 68 (1978) 2217.
SPONTANEOUS PHYSPCAECY Donald
LUCAS$
DESORPTION ADSORBED and George
Depnrzment ofChemistry, Indiana
OF VIRRATIONALLY
EXCITED
MOLECULES
ON SURFACES” E. EWING
Unir;ersity,
Bbomingron,
IN
474M.
USA
Received 12 December 1980
The energy within 2 vibrationally excited phy-sisorbed molecu!e often exceeds that needed to break its bond to the surface. Energy transfer from the vibrating chemical bond. to the surface bond causes the surface bond to ruptore and the vibrationdly relaxed adsorbate is released from the surface. We present a theoretical model which allows an estimation of the residence time of a vibrationally excited adsorbate on a surface. Because of uncertainties in the nature of the surface bond. the lifetimes obtained from the analytical expressions presented have only qualitative significance. The results are interpreted in terms of Franck-Condon overlaps between the wavefunctions which describe the adsorbatesu’bstrate complex and the released adsorbate. Lifetimes are calculated for hydrogen isotopes adsorbed on sapphire surfaces. Guide-lines are given for estimating lifetimes of other systems in terms of a few easily calculated parameters.
1. Inrroduction A physically adsorbed molecule may be driven off a surface in a variety of ways. If the system (adsorbate and substrate) is heated the surface bond wiU eventually break in the process called thermal desorption [l-3]. A more specific release of the adsorbate occurs if light from a laser source drives the vibration of the adsorbate against its surface bond until rupture occurs [4]. In thii paper we will discuss a third type of desorption. Here a chemical bond within the adsorbate is vibrationally excited. The energy in the vibrating chemical bond usually exceeds that needed to break a van der Waals surface bond. After a time, energy transfer from the excited chemical bond to the surface bond causes the surface bond to rupture and the adsorbed molecule is flung off_ This is then an
* Contribution No. 3549 from the Chemical Laboratories of Indiana University. Supported by the National Science Foundation. i Present address: Energy and Environment Division, Lawrence Berkeley Laboratory, University of California at EerkeIey, Berkeley. CA 94120, USA.
0301-0104/81/0000-0000/$02.50
@ North-I-Iolland
example of vibrational prediisociation of an adsorbed mo!ecule. We may describe this spontaneous vibrational predissodation process by a simple equation: A-D*
. . . SL_4-D+S+AE_
(I)
The adsorbed molecule is called A-D* and the asterisk tells us it is vibrationally excited. The substrate, S, holds the excited molecule by a van der Waals bond represented by the dots.. After time r the surface bond breaks and the adsorbate, A-D, now vibrationally relaxed, flies away with translational kinetic energy AE. It is our task in this paper to offer a theoretical model which aUows an estimation of the residence time 7 of the vibrationally excited adsorbate. There are at least two ways in which au adsorbed molecule can become vibrationaily
excited. The adsorbate may be the product of a chemical reaction on the surface which has left the adsorbate in a vibrational@ excited state. A few surface reactions have been studied where the vibrational state of the product has been speciiied. One example [S] is the nitrogen atom
386
D. Lucas, GE. Ewing / Desorprion of aibrationally excitedmo!ecules
recombination on iron, wit!? the subsequent release of vibrationally excited Nz_ Another example is the recombination of hydrogen atoms on surfaces prepared in the laboratory [6] or on grains in interstellar space [7]. The product, H$ -. - S, could be released by a process such as that in eq. (1). Alternatively, the adsorbate may absorb an infrared photon to become vlbrationally excited. This IR induced release of an adsoibed molecule is an example of photodesorption. Radiation, frequency tuned to match vibrational levels of specific adsorbates, could be used to induce the selective release of different chemical species or even different isotopes. Because of its potential for separating molecules this application of the photodesorption process has been named photochromatography by Suslick [8]. He has performed preliminary experiments for separating hydrogen isotopes adsorbed on AlzOj by photochromatography and has written a patent [S] which describes the practical aspects of the process. Others have also considered the feasibility of isotopic separation by IR induced photodesorption [9, lo]. Most of the mathematical machinery for discussing spontaneous desorption of vibrationally excited mo!ecules has recently beenset down in a slightly disguised form. The systems that have been discussed are vibrationally excited van der WaaIs molecules such as HCl* - - - O(CHJz, r?j . . . He, or Hz - - - Hz, and their vibrational predissociation lifetimes have been theoreticaIIy described [11-13-J. The vibrational prediisociation in these excited van der WaaIs molecules is formally similar to spontaneous desorption of -4-V -. . S. For this reason we will be drawing freely on the theory of relaxation of excited van der Waals moIecules in setting up our theoretical model for vibrational predissociation of molecules on surfaces. A series of papers 1143 on rhe thermal desorption of atoms on surfaces wri?ten over 40 years ago and largely ignored by -ihose working on van der -Waals molecules also contains essentially the same mathematical methods. In section 2, to follow next, the theoretical model will be presented. A numerical example
for calculation of desorption rates of HT - - - A1203 will be given in section 3. A general guide to desorption rates of other systems will be offered in section 4 which we
hope will be useful to those designing experiments. Finaily a summary is gathered section 5.
in
2. The theoretical model 2.1.
The golden rule
An energy level diagram for the desorption process of eq. (1) may be viewed in fig. 1. We begin, on the left, with the vibrationally excited system A-D* . . . S in one of the discrete levels labeled by the u, quantum number. These levels represent the stretching vibration of the adsorbed molecule against the van der Waals bond which holds it to the surface. Isoenergetic with A-D* - - - S is the continuum of A-D (now vibrationally relaxed) as it fries away with translational energy AE and velocity urn from the surface S. This picture, which represents energy transfer from a discrete level into a continuum, invites application of the golden rule or its equivalent [15]:
(2) In order to obtain r-l we need to specify the final and initial state wavefunctions, qgt and Yf’ and give a coupling term V,,. 2.2.
The wauefunchxzs
We will take the initial state wavefunction
to
be
WP’ = r-‘R,(r)&(x), which describes the state of A-D* . - - S. The geometry for the problem is shown in fig. 2. The distance between the center of mass.of A-D and the surface is given by r. We can provide R,(r), which describes the vibration bf the adsorbate against the surface, only after the surface bond potential function is specified. We
D_ Lucas, G.E. Ewing / Desorpfion of vibrationally excitedtmfecules
2IA OA
387
A-D---S
Fig. 1. Spontanwus desorption of a vibrationally excited moiecule adsorbed on a surface. The scale for the energy is that for Hz .__ ALO3 considered in the text.
will use the Morse function for this potentid, V(r)=D,{exp[-2a(r-rc)]-2
exp[-a(r-r,)l), (4)
with D, the depth at the bottom of the well at separation r,. The steepness of this potential energy function is given by the range parameter a. The analytical form of the Morse oscillator wavefunctions R,(r) for the discrete v, levels is given elsewhere [13,14]. An harmonic oscillator wavefunction &(x) describes the internal vibration of A-D with x the v&rational displacement coordinate and u the quantum level. The adsorbate may be generalized to a polyatomic molecule in which case A represents other atoms in the molecule. In this paper, however, we will imagine A-D as a diatomic moIecule with atom D directed toward the surface. The initial state for the harmonic oscillator is &I(X). The final state wavefunction, P, m = r-l&(r) defines A-D
&(X) , + S; The molecule A-D
(3 which has
been released from the surface is now vibrationally relaxed and &(x) describes this condition. As A-D flies far away from the surface with translational kinetic energy LIE its wavefunction resembles a plane wave. Near the surface the translational motion of A-D is moderated by its interaction with the surface. The wavefunction R,(r) describing the continuum behavior of A-D under the influence of a Morse potential is given elsewhere [13]. FortunateIy, we will not need the involved analytical forms of either R,(r) or R,(r) for this paper. 2.3.
The coupling term
The coupling term V,, must connect the coordinates of the vibrating chemical bond of A-D” with the surface bond of A-D’ .- - S. We will make a simple assumption to connect the x and r coordinates. We imagine that motion of the chemical bond of A-D moduiates the surface potential energy only because the distance between D and S changes. This coupling term
D. Lucas, G.E. Ewing j Desorption of uibrationallyexcited molecules
388
proposed many years ago for vibrational relaxation processes and discussed elsewhere [13, 16) is given by Vmn(x, r) = -2asaDCx{exp -exp
[-Za(r
C-a (r - rC)]L
where the fraction S is u = nlAliiI&D,
[13] yields
-I-,)] (61
of displacement
algebra given elsewhere
of D toward
(7)
with mA the mass of A and m~-o the adsorbate mass. The component of the vibrational displacement of A-D toward the surface is given
X { n$l ecn -w+q2_1]
x{[(d0-~~-~~~iq2m1/2d0~*exp
(8)
where 13is defined in fig. 2. 2.4.
Desorptior:
rates
Fortunately the matrix element (T~‘\V,,\Yr(no’) of the golden rule using our expressions for the wavefunctions and the coupling term has a simple analytical result. The
(9)
Several terms need defining in this rate expression. The harmonic oscilIator matrix element for the chemical bond of the adsorbate is [l6] (10)
by s =cos 8,
(-~q~).
CL= m+_mJm+_-D,
(11)
is the reduced mass of A-D. The vibrational energy WA_,, is the difference between the o = 1 and o = 0 states of the adsorbate. The adsorbate trapped by the Morse potential [eq. (4)] can be represented by the dimensionless parameter d = (2mA_DD,)“‘/aii.
(12)
The vibrational energy levels of the adsorbate against its surface bond are
* -“,
ir
%
\ :, ;:
‘1, .D
:.: r : .
l3
W ,=-[ah(2d
-1-2r~Jj’/8m~_~+D,,
given relative to the bottom of the Morse potentia1 well. Since the Morse parameters for the surface bond are not well known we Iose little useful information by rounding off d to an integer and calling it do. This parameter which appears in the rate expression of eq. (9) may be interpreted as the number of bound states of A-D .a. S contained in the Morse potential well. The final translational energyhE of A-D now released from the surface is incorporated in the dimensionless term qm = (2mA-&E)“‘/a
Fig. 2. The coordinates for a physicallya&orbed molecule
on a surface.
(13)
h = p/a tr,
(14)
which is a reduced momentum scaled by the range parameter a. Since A = h/p the q,,, parameter is related to the de Broglie wavelength of A-D flying away from the surface
D. Lucas, G.E. Ewing f Desorption of L’ibmtima~ly by the expression A = 2sr/aq,.
(1%
The parameter q,,, appears in the rate expression of eq. (9). We obtain ti by finding the energy remaining after the adsorbate bond from the vv level has been broken, AE=
WA-D-D,+
W,.
(16)
We now have all the expressions needed to perform numerical calculations of 7:: by vibrational predissociation of the adsorbate on a surface. However, thermal desorption is a competitive mechanism for releasing molecules from a surface. In this case the adsorbate and substrate are in equilibrium at temperature T. The energy needed to dissociate the adsorbate (in its u,. = 0 level) from the substrate is Do = D,- W,. The probability of the adsorbate site on the surface having this energy is exp (-D,/kl). The adsorbate can pick up this energy at its collision frequency, v = We/h, with the surface so the resulting thermal desorption rate is (ro’)rr’
= Y exp (-Do/kT).
(17)
This crude estimate of the thermal desorption rate, Frenkel’s formula, has been known for decades Cl] and has been recast by others in more sophisticated forms [14, 171.
3. A numerical example One of the most thoroughly studied examples of physical adsorption is HZ on AlaOS. The properties of AlaO, for chromatographic separation of ortho- and para-Ha and its isotopes have been discussed [lS]. Moreover, the Hz a.. AIZOa system was selected for photochromatographic experiments [8]. It is particularly useful for our purposes that the surface bond for Hz - - - AlJ& has been described [19]. It is for these reasons that we select this system for our sample numerical calculation of desorption. King and Benson [19] view HZ polarized by the electric field from A13+ and O*- ions near
excited m&cute+
389
the A1203 surface. The resulting induced dipole of HZ orients the molecule so that it is held perpendicular to the surface from which the electric field emanates. {It is this induced dipole which enables HZ. to absorb infrared radiation easily [20] and becomes vibrationally excited in the photochromatographic experiments.) They offer a variety of potential functions for the adsorbate-surface bond. The one they prefer is HZ located over a vacancy on the surface and is of the form V(r) = &P(rJr)9
-9(relr151
(18)
with D, = 1.0 x lo-l3 erg and r, = 2.40 A. An ambiguity results in trying to fit the range parameter of the Morse potential of eq. (4) to the 9-5 King and Benson potential of eq. (18). The 9-5 potential energy goes to zero at r = u = 2.07 A and somewhat arbitrarily we adjust the Morse potential so that it goes to zero at this separation as well. The result is a = 2.09 X 10’ cm-‘. Using D, = 1.0 x lo-l3 erg and a = 2.09 x 10’ cm-’ together with rn*-n = rnsl = 3.32 x lo-” g we obtain d = 3.71 (da =4) by eq. (12). The zero point vibration of HZ against the surface is W, = 2.53 x 10-r” erg (128 cm-‘) from eq. (13). With the vibrational frequency of Hz, WA-u = W,, = 8.26~ lo-” erg (4130 cm-‘) [20], we obtain AE = 7.51 x 10-r’ erg (3795 cm-l) by eq. (16). We then find q,,, = 10.2 by eq. (14), which corresponds to a de Broglie wavelength of A =0.29 A by eq. (15). Finally, use of eqs. (10) and (11) gives (w) = 8.96 x lo-” cm. Application of eq. (9) reveals r, = 6.0 x lo-’ s as the lifetime of Hz - - - AlzOs from the v,=O level. The error limits that may be placed on this lifetime are large. An uncertainty in the range parameter of only LO% introduces a variation in -ru,by an order of magnitude. Variations in the range parameter by more than 20% result from different choices of potentials ofIered by King and Benson [19]. Reasonable values of a fall between 2 x 10’ cm-’ and 3 x 10’ cm-‘. We can then safely say only that the desorption lifetime is better described by a nanosecond time scale rather than microseconds or picoseconds.
390
D. Lucas, G.E. Ewing / Desorptiqn of cibrationally excited molecules
In setting up experiments to detect molecules released from a surface by vibrational predissociation the thermal desorption channel must be considered. We will use eq. (17) for H; on Al& and our original set of parameters. We find v = We/h = 3.8 x IO” s-l and DO = Y&T W0 = 7.5 x lo-” erg and consequently = 3 x IO-to s at 77 I< and 7;“’ =0.2 s at 70 20 K. Thus the thermal and vibrational desorption rates may be competitive at 77 K but vibrational desorption clearly dominates at 20 K. A large isotope effect is predicted for op - - - AlzOs vibrational predissociation. Working through the same type of arithmetic as before with WA_,, = W,, = 5.92 x lo-r3 erg (2990 cm-‘) we find qm = 11.8 and r,= 1.2 x lo-’ s from the u,. = 0 level.
4. Other systems, other channels Rather than provide more detailed numerical calculations of other systems we will examine the form of ihe rate expression of eq. (9) to learn what factors are primarily responsible for the efficiency (or ineEiciency) of the desorption process. Two Franck-Condon factors appear in eq. (9). The first deals with changes in the vibrational wavefunctions when A-D* is relaxed to A-D. The overlap of the dr(x) and &(x) wavefunctions through the coupling term yields the vibrational Franck-Condon factor a’~~a’(x)~. The energy scale for the coupling term is D,, the we:1 depth of the surface bond. This accounts for all the system parameters to the Ieft of the bracket of eq. (9). The remaining terms reflect the FranckCondon factor involving changes in the motions of the adsorbate while on the surface and after it is released. The wavefunctions R,(r) and R,(r) through the ccupling term of eq. (6) yield this translational Franck-Condon factor. The result involves only the reduced parameters do and qm. While the vibrational Franck-Condon factor contains on!y quadratic terms whose numerical range is rather limited, the trans-
lational Franck-Condon factor contains a product expression, a factorial term and an exponential. It is the translational FranckCondon factor then which really dominates the outcome of the desorption rate. We have plotted the desorption lifetime from the u, = 0 level given by eq. (9) as a function of q,,, for several values of & in fig. 3. The vibrational Franck-Condon overlap has been given a typical value (r2~2u2{.x)2 = 2 x 10m3 and the adsorbate weI1 depth is taken to be 2 x lo-r3 erg, a representative value for physisorption [2]. The desorption lifetime follows a simple exponentiai dependence for qrn > do_ There is a simple physical explanation for this behavior. The departing molecule is approximately described by a plane wave of de Broglie wavelength A = 27r/aq,. The adsorbed molecule in its u, = 0 level is (almost) an harmonic oscillator against the surface and its wavefunction resembles a gaussian. It is the overlap of the near plane wave and the gaussian functions within the matrix element of the golden rule which determines the efficiency of the desorption process. The magnitude of this overlap is dominated by the de Broglie wavelength of the -4-D molecule flying away from the surface. Algebraically the exponential dependence on the de Broglie wavelength appears in the final term, exp (-mQ, of eq. (9). We have referred io the logarithmic lifetime dependence of the sort shown in fig. 3 as the momentum gap correlation [Zl]. The momentum gap can serve as a useful guide in estimating the lifetiie of the vibrational desorption process. Because q,,, = (2rn+,_oE)“*/a~ we see that three system parameters are involved: a, AE, and mA_D. A linear change in a results in an exponential change in r,,_ Thus the uncertainty in this parameter because of the poorly defined surface bond potential energy makes quantitative theoretical calculations unreliable. The energy gap AE is seen from eq. (16) to depend on three terms. An adsorbate which has both a low vibrational frequency and a strong surface bond favors a small AE and a short vibrational predissociation. time. Adsorbates in excited vibra-
D. Lucas, GE. Ewing I DesorpTionof uibracionnllyexcited molecules
391
we now explore. We have ignored energy flow channels which may leave either the released adsorbate or the surface excited. We have treated the substrate, S, as a homogeneous mass. This bulk mass absorbs the momentum needed to balance that of the departing adsorbate but accepts negligible energy. However, let us now imagine S to be an atom or ion (or several atoms or ions) at the surface bound directly to A-D. This portion of the surface is in turn coupled to the rest of the lattice. The disposition of the energy and momenta now changes significantly. We can take care of this in the simplest way by replacing rnA_D [in eqs. (12)-(14) and the abscissa of fig. 33 by the reduced mass of A-D and S, 9,,,=(2n1~_~
AE) “2,(1x
Fig. 3. The lifetime of a vibrationally excited mo!ecule adsorbed on a surface.
tional energy states (IJ,= 1,2 . ..) against the surface with the energy W, will be released with a smaller energy gap and tend to have shorter residence times. Isotopic changes affect both m&u and dE (through WA-, and W,) and may result in either an increase or decrease in TO. In regions where do> qm, the released adsorbate has relatively low translational momentum and its effective de Broglie wavelength near the surface is dominated by the shape of the adsorbate-substrate potential well. The lifetime for desorption then depends exponentially on do with milder variations with q,,,. As an application of fig. 3 [or eq. (9)] let us estimate the lifetime of CH8 ... AlzOa. We take the molecular excitation to be [22] w&u = P& = 6 x lo-r3 erg (3010 cm-‘) and 0, = 2 x lo-” erg (1000 cm-l). We find do= 12 with Q ~2.5 X IO* cm-’ and appioximate CHZ by CH3-H* and take & before ~2s2~2(~)2 = 2 X lo-‘. Ignoring the zero point vibration against the surface we find AE = WA-D-D== 4 X 1013 erg (2000 cm-‘) which gives qm = 18. Fig. 3 then reveals r. = lo-’ s. The model we have presented leading to eq. (9) or fig. 3 contains critical limitations which
&A-D5 =
mA-_DmS/(mA-D+
mS.),
(19)
where ms is the effective mass of the surface beneath the adsorbate. The picture that emerges is that of A-D flying away from the surface and the rebounding vibrationally excited S* colliding with its neighbors which take up its translational ener-v and distribute it to the remainder of the solid. The substrate will retain energy WS into the localized vibrational modes of the surface which supports the adsorbate. This process is A-D*
..-S+A-D+S*+AEv-s
(20)
with AEv_s=
W..-D-D,+
W,-
WS.
(21)
We call this the V-S channel. Again referring to the CHZ -0 - Alr03 example, the substrate has fundamental vibrations up to 870 cm-’ [231 and strong optical absorption to ~1600 cm-’ [24]. The en:rgy gap may then be AEv-s = 400 to 1100 cm . If we imagine methane bound directly to A13+ then the effective reduced mass wil1 be J.Jc~.~ = 1.7 x 10eZ3 g. Using o = 2.3 x 10’ cm-l we have do = 10 and a resulting qEs = 6.3 to 10.4. Desorption leading to a vibrationally excited released adsorbate may be represented as A-D*
. ..S-+A+-D+S+~E~X..
(22)
The energy gap now is AEvv = W-j,_,-,- DC + W, - WA*_*
(23)
D. iiicar, G.E. Ewing / Desorpticn of oibrationally excited molecules
392
and has been reduced from eq. (16) by the energy WArD in the vibration of A*-D. This process is anaIogous to V-V transfer in gas phase relaxation [16]. In the CHT *.- Al103 system just considered the released adsorbate may be excited in the u2 vibration at 1525 cm-’ [22]. The energy gap originally A.E=2000 cm-’ is now lowered to &XE,-475 cm-’ and the reduced momentum becomes q,” - 8.6. While either the V-S or V-V channels lower the momentum gap they also introduce an additional Franck-Condon factor involving the excitation of a vibrational mode of the substrate or the adsorbate. As we saw a typical value for this Franck-Condon factor in a rate expression is =10e3 (or lo3 in a residence time). Thus whiie values of q,, ~6.3 to 10.4 for the V-S channel of CHZ -a- Al203 applied to fig. 3 would suggest times from rO= lo-’ to lo-’ s, the vibrational Franck-Condon factors to excite the substrate modes increase these times to To== 10e6 to lo-” s. Likewise qm 28.6 expected for the V-V channel when the v1 mode of the released methane is excited yield r0 2 lo-’ s rather than ro= 10e8 s read from fig. 3. In any event all these channels are more efficient than the process in which the released CH, deposited no energy into either itself or the substrate_ A final relaxation channel may leave the released adsorbate rotationally excited A-D*
--.S+A-D’+S-i&5,,_~.,
with the ener,7
gap
&v-X.,,
-D,
= w,,
+ w,-
(24)
W&D?.
(25)
The energy partitioned into the tumbling motion of the released adsorbate is given by WA-D+. In the case of CHZ k-. Alz03 we might expect to find the released methane in a high J state. It should be noted that there are channels in which the adsorbate is not released from the surface. Transfer of vibrational ener-9 from one adsorbed molecule to another adsorbed molecule nearby is possible through long-range
.
coupling terms. Another possibility is that the surface modes are in resonance with the excited adsorbate, resulting in energy transfer to the substrate with no energy remaining to break the surface-adsorbate bond. Brus [25] has treated this relaxation channel. He considers the energy transfer of a vibrationally excited molecule to the surface of a dielectric solid as a classical electromagnetic phenomenon. When there is a match in their vibrational frequencies and their oscillator strengths are large, the vibrating electric dipole of the adsorbate couples strongly with surface polarons. The lifetime associated with this channel of resonance energy transfer from the adsorbate to the substrate is 7” 10-l's. However, the energy transfer rate drops rapidly when the adsorbate vibrational frequency is even slightly off resonance from the characteristic surface polaron frequency. If these frequencies are mismatched by only 10% the lifetime increases two orders of magnitude to t = 10M1’s. If the mismatch widens, the surface mode adsorption coetiicient at the frequency of the vibrating adsorbate vanishes and this channel for energy transfer is closed. For the molecules we have been considering at VA_*-3000 to 4000 cm-’ on typical dielectric solids with 3;, L- 1000 cm-‘, we expect therefore that this non-desorptive channel will be closed. We have been talking about desorption from dielectric solids. What about metals? Here the free electrons are able to take up energy in a variety of ways. The surface plasmons can be driven over a wide frequency range by a vibrationally excited adsorbate as Brus shows {25,26]. With such an efiicient channel for vibrational relaxation without breaking the surface absorbate bond, metals may be poor subsirates for the desorption experiments we have been discussing. Calculations for such systems have, however, been performed [27J Exploration of the various V-V and V-T,R channels has been initiated for vibrationally excited van dar Waals molecules [i3,28-311. Further theoretica investigation of the V-S channel should probably await the first experimental results of vibratitinally induced desorption.
D. Lucas, G.E. Ewing / Desorption of vibrationally excited molecules
5. SMlmary Let us summarize this guide to spontaneous desorption of physically adsorbed vibrationally excited molecuies. The most efficient desorption processes will occur for adsorbates with a small number of bound states (do small) and when released the adsorbate has small translational momentum (small qm). This momentum gap correlation is most succinctly revealed by fig. 3. Smaller translational momentum will be achieved if the adsorbate can take up energy into its internal motions. Absorption of energy into lattice modes of the substrate will also serve to reduce the translational momentum and provide for more e9icient desorption. However, if the vibrational frequency of the adsorbate is in near resonance with surface polarons or plasmons of the substrate, energy transfer to the solid will be so efficient that desorption will be quenched. A test of these possible relaxation channels awaits the first experimental measurements of desorption of vibrationally exdited molecules.
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