Kinetic equations for physisorption; With some remarks on the compensation effect

Applications of Surface Science 11 / 12 (1982) 793—802 North-Holland Publishing Company

793

KINETIC EQUATIONS FOR PHYSISORPTION; WITH SOME REMARKS ON THE COMPENSATION EFFECT H.J.KREUZER Theoretical Physics Institute and Department of Physics, University of AIberia, Edmonton, Alberta, Canada T6G 2J1 Received 8 June 1981

Starting from a set of rate equations for the bound state occupation function for gas—solid systems in which the surface potential has many physisorbed bound states, we derive a master equation; its kernel is explicitly calculated for phonon-mediated adsorption and desorption in a Morse potential. We give the equivalent Smoluchowski—Chapman—Kolomogorov equation for which we find the Kramers—Moyal expansion. Identifying Van Kampen’s large parameter ~2for such gas—solid systems, we establish explicit criteria for the validity of a Fokker—Planck equation. The various kinetic equations are then used to calculate desorption times. The exact time evolution of the adsorbate as calculated from the set of rate equations shows that quasi-equilibrium is only maintained at low temperatures where perturbation theory of the master equation yields a simple analytic expression for the desorption time in weakly coupled gas—solid systems. At intermediate temperatures we derive another simple expression from the Fokker—Planck equation. Lower limits for the pre-exponential factor in the desorption time of the order 10 IS s, proportional to the inverse of the heat of adsorption, are derived. We conclude with some remarks on the compensation effect.

In a series of papers [1—3]we have developed a quantum statistical theory of desorption of a gas from the surface of a solid in systems which show physisorption at low coverage. The latest paper was, in particular, devoted to a study of physisorption in gas—solid systems in which the surface potential, i.e. the net static interaction between the particles of the gas and solid phases, develops many bound states, say at energies E0 EN, into which gas particles can get trapped. Typical examples are the He—LiF system with four bound states, the He—graphite system with N 4, the Xe—W system with N 200. To calculate the isothermal desorption time for such systems, we have argued that the occupation numbers n1 of gas particles in the lth bound or continuum state of the surface potential are, at low coverage, subject to a set of rate equations dn,(t)/dt

~ ~

~ R,.,n,,

(1)

i,~I

where R,,~is the probability for a transition of a gas particle from the /th into the /‘th state of the surface potential, including all bound state—bound state, 0378-5963/82/0000-0000/$02.75

©

1982 North-Holland

794

Hf. Kreuzer

/

Kinetic equations for phvsisorption

bound state—continuum, and continuum—bound state transitions. In ref. [3] we have calculated these transition probabilities in second order perturbation theory (Fermi’s Golden Rule) for phonon-mediated adsorption and desorption in a one-dimensional model assuming that the surface potential is adequately represented by a Morse potential 2)~xo)—2 e~x0)). (2) = U 0( e_ To find the connection between the isothermal desorption time and the transition probabilities we have in ref. [3] solved (1), with appropriate boundary conditions namely that gas particles once in the continuum are pumped out of the system, by straightforward matrix diagonalization and found that the total (relative) adsorbate occupancy is given for a gas—solid system with (N + 1) surface bound states by td

N

N(t)/N(0)

N

~ n 1(t)/N(0)= ~

S~e’,

(3)

1=0

where the X,’s are the eigenvalues of the matrix R11. Under most experimental conditions one finds for the lowest eigenvalue A0 <
exp(Q/kBT),

(4)

where the prefactors t~°typically vary for physisorbed gases from 10 ~ s for helium desorbing from constantan [4] to 10—14_10—15 s for xenon desorbing from tungsten [5].The heat of adsorption Q/kB varies from 30 K for the He— constantan to 4662 K for the Xe—W system. In the microscopic theory of physisorption kinetics it turns out that the depth U 0 of the surface potential (2) is slightly larger that Q, and that its range y determines the prefactor t~. A smaller range implies a stronger coupling of the gas particle trapped in a bound state of the surface potential to the phonon bath of the solid increasing the probability for adsorption of a phonon thus decreasing t~. However, reducing the range ~‘ of the surface potential keeping its depth U0 fixed also implies that the number of bound states is reduced so that the number of channels through which the adsorbed particle can cascade up and down the bound states of the surface potential is reduced leading to a decrease in the desorption rate compensating the increase caused by the stronger coupling to the phonons. One thus finds for the desorption kinetics in gas—solid systems with many surface bound states that details of the surface potential are less important than they are in systems with only a few surface bound states [3]. As the number of bound states in the surface potential becomes large, it seems plausible to approximate the system of many discrete bound states by a quasi-continuum ranging from the bottom (— (Jo) of the surface potential well to zero [6]. Whereas in the original system the adatom cascades through a ‘

)‘ —

—‘

H.J. Kreuzer / Kinetic equations for physisorplion

795

series of discrete bound states, it now performs a random walk through the quasi-continuum of bound state energies. In such a situation it seems appropriate to replace sums over I in (1) by integrals over a dimensionless variable which is conveniently chosen to be a = E/h w D such that for E = E1 being one of the bound state r=2mtl~/hy2 energies in a Morse potential (3), one Here gets 2/r where and a~2mU~/(ky)2. he=a,=(—a0—i—-~) w D is the Debye energy of the solid which serves as the energy scale for the system. The bound state occupation functions n.(t) then go over into n(a, t) such that n(a 1, t) = n.(t). Eq. (1) can now be written as a continuous master equation On(et

f_~0da’ p(a’)W(a, a’) n(a’,

~‘

t)

—f

da’ p(e’) W(a’, a) n(a,

t),

(5)

where u~= Uo/hwD

a~/r

()

(~++)2,

(kIY~

Rather than give the explicit, rather lengthly analytical expressions for the transition probabilities W(a, a’), we display a typical example in fig. 1 plotted as a perspective view over a section of the (a + a’, a — a’) -plane. (The choise of this coordinate system is dictated by aesthetical considerations and has no physical significance.) The pictures are dominated by the bound state-bound state transition matrix elements for a + a’ < 0. In fig. 1 the transitions up to higher bounds states produce the lower peak on the left hand side. To see the bound state—continuum and continuum—bound state transitions more clearly, we display in fig. 2 two enlarged sections around the origin a = a’ = 0. Their matrix elements are, indeed, much smaller than those for bound state—bound state transitions, implying that during the desorption process the bound state occupation is reshuffled into a thermal distribution much faster than particles are actually desorbing. This suggests the perturbation theory advanced in ref. [7] which leads for low temperatures such that 8 s a0 /2u0 to a simple expression for the desorption time, namely 2 M r 3/2 eôuo 6 d =~~1 D m~ 18iru~” ~

Going from the discrete rate equations (1) to the continuous master equation (5) only involves a justifiable mathematical approximation. To go further and derive a simpler kinetic equation for physisorption, we have in ref. [6] rewritten the master equation as a Smoluchowski—Chapman—Kolmogorov equation which in turn we developed into a Kramers—Moyal—Van Kampen moment expansion. Truncating the latter after two terms yields a Fokker— Planck equation for physisorption kinetics 3p(a)n(a,t) 8t

a

I

~—(aI(a) p(a) n(e, t)) + ~

a2 ~

p(e) n(a, t)),

(7)

Hf. Kreuzer / Kinetic equations for pkvsisorption

796

W(e,

e)

r

MOIilfI~Illhf~llJ~IASU~

4969.0 11.56 1.0

= =

10 ~1.Q

C

C’

\.

1.0

Transitions Into Higher

Transitions Into —.

Bound States

.—

Lower

Bound States

—1.0 —23.0

—1.0 —23.0

CO

+ CO

1.0

1.0 ~

—

r

Transitions Into Continuum

Transitions From Continuum

—1.0

Fig. I. Perspective views of the kernel e, e’) in the master equation (5) plotted over the (e + e’, e — e’) -plane. Note the different scales along the two axes. The highest peaks are W(e —7.385, e’ —7.315) w~M,/m =3.4 and W(e= —7.315, e’= —7.385)3.17. In all numerical examples we choose in/M, =0.714, u~ 5.302X 1013 ~-i

/

Hf. Kreuzer

Kinetic equations for physisorplion

797

4969.0

W(e, r’)

U

0

8 “~0

=

11.56

=

1.0

_\P

2

~j~

—1.0_~--’ —1.0

—1.0

CO

+

1.0

1.0

— CC

—1.0

Bound States —Continuum

— Bound States

Fig. 2. Section of fig. I around the origin s = e’ = 0. The maximum in the continuum —‘ bound state transition is W(~ = —0.07155, e’ —0.07245)w~M,/mrr0.054. The maximum in the bound state-continuum transition is W(s0.07245, s’ —0.07155) w~M~/mzo0.0466.

Hf. Kreuzer / Kinetic equations for physisorption

798

where the moments are given by a~(a) da (—e)’2 (a—a”)” W(a,a”),

f~

(8)

in terms of the kernel W(a, a”) of the master equation (5). Eq. (7) should be compared to the Fokker—Planck equation for a particle diffusing over a potential barrier as derived by Kramers [8], and recently revisited by Caroli, Roulet and Saint-James [9]. For weakly coupled gas—solid systems, i.e. for large r, the first two moments (8) can be approximated by simple expressions [6] which allow one to solve the Fokker— Planck equation (7) in the temperature region 5/3u 0<<6<<~I2r/5u0,

(9)

yielding another approximation to the isothermal desorption time namely FPt...l

w~

~

e~°.

_~_~

(10)

This equation allows one to identify [6,7] a posteriori Van Kampen’s large parameter [10] as = a~with which one can delineate the range of validity of the Fokker—Planck equation (7). Both expressions (6) and (10) for the isothermal desorption time are of the Frenkel—Arrhenius form (4). They have been obtained for weakly coupled gas—solid systems in their respective temperature regimes from a well-defined quantum statistical model by means of mathemati-

:

1.0 0

..

ID

04

T

,,,~u0

=

11.56

02

0

r= 17O~~= 42.5

1.0

2.0

3.0 4(u0/r)~

Fig. 3. Test for the approximate desorption times td from (6) (dashed line) and FPtd from (10) (dotted line) for three different systems. Larger r implies weaker coupling. The curves with u0 = 11.56 have høD/k =405 K and m/M, =0.714 appropriate for the Xe—W system. The lowest curve with u0 = 11.56 has hw0/k =450 K and m/M0 =0.277 for the CO—Ru system.

Hf. Kreuzer / Kinetic equations for physisorption

799

cally acceptable approximations. It is important to notice their nontrivial dependence on the parameters U0 and y of the surface potential and on the Debye frequency aD of the solid as a consequence of the phonon-mediated dynamic coupling between solid and gas. This is in contrast to certain “equilibrium theories” [11] where the pre-exponential factor t~ is simply identified as proportional to the inverse of the frequency Wclass with which an adparticle of mass m oscillates classically at the bottom of the surface potential. In fig. 3 we plot the approximate desorption times (6) and (10) together with the exact desorption calculated for rlow = 550.0 and r = 4969.0 111’~d istimes a lower bound from on td (1) in the temperature region We see clearly thata bound in the temperature region (9). Indeed, these bounds whereas FPtd gives are better for larger a 0 or r and can therefore be used with confidence to estimate desorption times for such gas—solid systems. Note that the pre-exponentials in (6) and (10) depend quite strongly on the heat of adsorption, i.e. cum grano salis on u0. The r-dependence of the pre-exponentials can be used to estimate the range of the surface potential. The transition probabilities R,,. in (1) are calculated in second order time dependent perturbation theory (Fermi’s Golden Rule) and thus take account of one-phonon process only. This obviously puts restrictions on the kind of gas—solid systems that can be described. In particular, we must demand that any two neighbouring bound states in the surface potential can be linked by a one-phonon process. As the deepest two bound states are separated the farthest, this implies for a Morse potential that htaD I E0 — E1 or in terms of our dimensionless parameters, ~‘

2(a0—l)~
,

(11)

r~4u0,

where we used the fact that for most gas—solid systems a0 ~ 1. This inequality implies a lower limit on the pre-exponential factor in (4) which2 we can get from either (6) or (10). From the former we find for 6>> ~(r/u0)~’ (12 d D m u 0 m 9s U0’ 91T

and the latter yields for 5/3u0>>8~i’ii/2r ~ y .)U0 FP

M~ 2 m

911-

1 u08

M5 2 m

911-

kT U0

(13)

Note that these two estimates, resulting from two quite different approximation schemes and valid in two different temperature regimes, are identical apart from the temperature factor ~kT/hwD in FP1~• Also recall that both are valid for large r and u0. Let us apply (12) to the Xe—W system for which u0 = 11.56. We get Id°~ 3.2 X 10— 16 s, whereas experiment says that t~° = l0 15 s. We would, however, caution the reader that our estimate above is not too stringent and out at 6= 1 by about a factor 15 because r = 4u0 = 46.24

800

1ff. Kreuzer

/

Kinetic equations for physisorpzion

is not a large number for (12) or (13) to be good approximations. Such an r value implies that the Xe—W surface potential would have a range = 0.15 A which, it seems to us, is too small to be acceptable without reservation [12]. It might actually reflect the fact that Xe gets weakly chemisorbed on W. Let us then look at gas—solid systems with large heats of adsorption. They typically show chemisorption, i.e. the gas particle undergoes some structural rearrangement as it gets adsorbed. We feel that such a process cannot adequately be described by a model, appropriate for physisorption, in which the gas particle otherwise unchanged gets trapped into the bound states of the surface potential. With these misgivings in mind, that seem to be of amazingly little or no concern in some of the literature, let us look at the CO—Ni system. Ibach et al. [13] measure in the low coverage regime Q~ 150 kJ/mole and td°~iO~~ s. With hwD/kB = 440 K for Ni, we get from (12) that t~~ 10~ s, i.e. not quite as fast as the experiment requires. For the CO—Ru(00l) system measured by Pfnür et al. [14] we get with u0 = 42.5 Id0 ~ l0 16 s overlapping with their experimental value t~° = l0 16 s. Similar conclusions can be reached for the systems Cu—W, Ag—W and Au—W measured by Bauer et al. [15]. Our microscopic model is thus quite capable of producing very small pre-exponential factors t~. Let us, however, stress once more that the ability of out theory to yield very small pre-exponential factors t~°, does not explain fast desorption in chemisorption systems but should only be taken as an encouragement to extend our line of attack, so far restricted to physisorption, to phenomena in chemisorption. An analysis of desorption data with eqs. (6) and (10) or (12) and (13) should at this stage be viewed as complementing the phenomenology based on equilibrium or transition state theories to get a fuller understanding of desorption kinetics. Let us finally present some “computer experiments” on the compensation effect. The latter is supposed to show up in most thermally activated processes and implies that in (4) one has [16] Q-’1n(t~)1.

(14)

Table 1 contains our “data”. For a fixed gas—phonon coupling r = 2mwD/hy2, we vary a 2y2 to get different heats of adsorption or u 0 = 2mU0/h 0 = Uo/hwD. For different 8 we calculate the isothermal desorption rate rd = t~ from (1). because the prefactor ~d° in (4) is in the relevant 6uo)/8 6-regimeinroughly the lastproportional column of to 8~’,see (10) or fig. 3, we calculate rdexp( table 1. This quantity varies little with 6 and, indeed, increases with increasing or Q for fixed r. However, the dependence is not logarithmic as (14) postulates. Indeed, in the weak coupling limit it obeys the simple power laws (6) and (10). Let us lastly point out that an attempt to fit the rates rd in table 1 to a Frenkel—Arrhenius formula (4) with a temperature independent prefactor Id° would produce a Q value smaller than the isosteric heat of adsorption of the

Hf. Kreuzer / Kinetic equations for physisorplion

801

Table I “Data” on the compensation effect; m/M, 2m~~ oo(~~)I,~~2

rz’r

100

400

0.1,

WD

5 X 1013 ~—I

u 0-~—

~

rd—td

10.2

1.04

20.2

4.08

30.0

9.0

6 156 2 5 0.5

4 X 10~ 6 5.4 X l0~ 8 x10 l.l9X109 9.9 x103 7.13X lOb

3.4 X1010 X 1010 2.6 2.16X 1010 2.08X10’2 1.4 Xl0’2 1.3 X l0’~

35.502

12.6

45.0

20.25

20.2

1.02

60.0

9.0

70.5

12.4

100.2

25.1

125

39.06

135

45.56

system. To see this, take

td

(s~)

rdexp(ôuo)/S

l.54X109

l.25X10’3

2 1.5 2 0.5

3.34X l0~ 2 X iO~ 4.45X102 9.82x 108 7.64x l0~

1.1 X 1013 2.2 x 1013 2 XlO’3 49 X 10I~ 4.8 X10’3

3 4 1 2 1.25 2 0.5 I 0.4 0.75 0.4 0.2

5.22x 108 2.29X 108 3.2 X 108 7.26X l0~ I.33X 106 l.69X 102 5.6 x l0~

3.7 x l0~ 3.33)< iO~ 2.38>< 1012 5.9 x io~ 5.24X 1012 3.16X 1013

3.73X 102

2.97X 1013

4.87X l0~ 9.54 4.75X iO~ 2.05X l0~

7.4 X 10~ 6.7 X 10’s 9.75X l0’~ 9.3 x 10I~

2.6 >< lOb

= (~

16Y‘exp(8u0) and fit it with (4) at two temperatures 61 and 62. One, indeed, gets kT1

kT2

-~ U0—kT2. T2 r-.r,

This is not new physics but simply the result of carelessness.

References [1] Z.W. Gortel, H.J. Kreuzer and D. Spaner, J. Chem. Phys. 72 (1980) 234; Z.W. Gortel and H.J. Kreuzer, Chem. Phys. Letters 67 (1979) 197; Z.W. Gortel, H.J. Kreuzer and R. Teshima, Can. J. Phys. 58 (1980) 376; H.J. Kreuzer, Surface Sci. 100 (1980) 178; Z.W. Gortel and H.J. Kreuzer, J. Quantum Chem.: Quantum Chem. Symp. 14 (1980) 617. [21 Z.W. Gortel, H.J. Kreuzer and R. Teshima, Phys. Rev. B22 (1980) 612;

Chem. Phys. Letters 73 (1980) 365.

(15)

802

H.f. Kreuzer / Kinetic equations for physisorption

[3] Z.W. Gortel, H.J. Kreuzer and R. Teshima, Phys. Rev. B22 (1980) 5655. [4] S.A. Cohen and J.G. King, Phys. Rev. Letters 31(1973) 703. [5] H.J. Dresser, T.E. Madey and J.T. Yates Jr., J. Chem. Phys. 55 (1971) 3236; Surface Sci. 42 (1974) 533. [6] Z.W. Gortel, HJ. Kreuzer, R. Teshima and L.A. Turski, Phys. Rev. B24 (1981) 4456. [7] Hi. Kreuzer and R.Teshima, Phys. Rev. B24 (1981) 4470. [8] HA. Kramers, Physica 7 (1940) 284. [9] C. Caroli, B. Roulet and D. Saint-James, Phys. Rev. B18 (1978) 545. [10] N.G. van Kampen, Can. J. Phys. 39 (1961) 551; also see N.G. van Kampen, in: Fluctuations in Solids, Ed. R.E. Burgess (Academic Press, New York, 1965). [11] See, e.g., P. Jewsbury and J.L. Beeby, J. Phys. C (Solid State Phys.) 8 (1975)3541; S. Holloway and P. Jewsbury, J. Phys. C (Solid State Phys.) 9 (1975)1907; or G. Armand, Surface Sci. 66 (1977) 321, for a recent crop of such classical equilibrium theories of desorption and for reference to earlier work. [12] The object of desorption theories is to understand and reproduce desorption times that are given experimentally as a two parameter Frenkel—Arrhenius fit (4). Physisorption theories all have as one of their ingredients a surface potential that is also characterized by two parameters, namely depth U 0 and range ~‘ Fixing U0 = Q then forces one to fit y~’ to reproduce the pre-exponential factor t~. This is not much of a feat. One must therefore be very aware that the resulting numbers, in particular ~‘ ~, are reasonable. If, e.g., Holloway and Jewsbury in their theory (see ref. [II]) must choose ~ = 0.09 A for the Ga—GaAs system, more caution should be exercised. [13] H. Ibach, W. Erley and H. Wagner, Surface Sci. 92 (1980) 29. This article gives a detailed discussion of desorption data in terms of the phenomenological equilibrium theory. [14] H. Pfnur, P. Feulner, H.A. Engelhoodt and D. Menzel, Chem. Phys. Letters 59 (1978) 481. [15] E. Bauer, F. Bonczek, H. Poppa and G. Todd, Surface Sci. 53 (1975) 87. [16] A recent discussion has been presented by E. Peacock-Lopez and H. SuhI, UCSD preprint.