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Flash desorption and thermalization
484
Surface Science 133 (1983) 484-498 North-Holland Publishing Company
FLASH DESORPTION Z.W. GORTEL
AND THERMALIZATION
and H.J. KREUZER
Department of Physics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada Received
15 February
1983
Starting from a master equation for phonon-mediated physisorption in a gas-solid system in a finite volume, we calculate the nonequilibrium time evolution of adsorbate density and pressure for flash desorption and subsequent thermalization of the gas due to wall collisions. The transient pressure overshoots its final equilibrium value by as much as a factor two. The asymmetry between flash desorption and adsorption caused by the reverse temperature flash is demonstrated.
1. Introduction Kinetic studies in physisorption and chemisorption probe the dynamics of the gas-solid interaction. The desorption kinetics is investigated in basically three different experimental arrangements, all, of course, starting from a gas-solid system in equilibrium at some initial temperature T and initial pressure Pi. In an isothermal desorption experiment the gas pressure is reduced suddenly to a much smaller final value Pf -GZPi that is maintained subsequently by continuous pumping also keeping the temperature q constant. The time evolution of the flux of desorbing particles is monitored and characterized by experiment the an isothermal desorption time I,(i) [ 1,2]. In a flash desorption nonequilibrium situation triggering the desorption process is created by a sudden increase in the temperature of the solid to T,, being maintained thereafter. Particles will desorb from the solid surface in a characteristic flash desorption time t$,‘) [2-51. Last1 y, temperature programmed desorption experiment (TPD) is basically a sequence of flash desorption experiments induced by a programmed rise in surface temperature as a function of time. From the resulting desorption signal one can, under favourable conditions, extract heats of adsorption and desorption times [1,6,7]. The difficulties in analyzing TPD spectra, e.g. caused by readsorption from the gas phase and the model dependence of the analysis, have been highlighted by King [8]. A microscopic theory of adsorption-desorption kinetics conveniently starts from a master equation. In the following we will restrict ourselves to physisorbed gas-solid systems for which the net interaction between a gas particle 0039-6028/83/0000-0000/$03.00
0 1983 North-Holland
Z. W. Gortel, H.J. Kreuzer / Flash desorption and thermaliration
and the solid can be described by a surface potential states of a gas particle in V, by I, the master equation dn,/dt
= c R,,n, I
- c R,,n,,
VS. Labelling reads
485
the energy (1)
I
where n, is the time dependent occupation probability of the states I which include both bound states (i.e. the adsorbate) and continuum states (i.e. the gas phase). R,, are transition rates per unit time for a gas particle to go from state I’ to 1. They must be calculated from microscopic theory. The three experimental arrangements described above must be modelled in the theory by imposing proper initial and boundary conditions on the solutions of the master equation (1). To describe an isothermal desorption experiment, one sets for time t > 0 the occupation functions of the continuum states equal to zero, thus ensuring that the pressure of the gas phase is zero, P = 0. For phonon-mediated physiso~tion kinetics, the procedure to calculate isotherms desorption times from (1) has been explained in detail in ref. [9]. To describe the flash desorption experiment, readsorption out of the gas phase must be included in the theory which has been done in refs. [ 10,l l] by imposing the appropriate final state reached by the gas-solid system after a long time without specifying the physical mechanism responsible for thermalization. Indeed, there are two processes which can equilibrate the gas phase: (if collisions between particles in the gas phase and (ii) collisions of gas particles with the walls. If the mean free path 1 for gas particle collisions is large compared to the size L of the experimental chamber, then only collisions with the walls provide an effective means of changing the energy distribution of the gas phase to a thermal one at the temperature of the walls. If I < L thermalization is speeded up by collisions between gas particles. For temperatures below room temperatures and pressure lower than 10m2 Torr and L - IO-’ m, one always gets I B L. In this paper we will set up the appropriate boundary conditions for the master equation (1) to describe a flash desorption experiment at low coverage and the subsequent thermalization due to wall collisions. We assume that at times t < 0 the gas-solid system is in equilibrium at Ti and Pi, so that the gas particle occupation probabilities are
n,(t < 0) = e-(E/-P)/k6Tt, _
(2)
where
efi/k8Tl =
pi(2am)-3’2(kg~)-s’2,
(3)
with p being the chemical potential of the gas phase. Within the framework of mean field theory [12] the present ideas can be extended to finite coverage. Having raised the temperature of the solid to Tf for times t 2 0 we calculated the phonon-mediated transition rates R,, at Tf. The experimental situation we
486
Z. W. Gortel, H.J. Kreuzer / Flash desorption and thermalization
thus describe is one in which all walls surrounding the gas phase are at T,. From (1) we can then calculate the time evolution of the relative bound state occupation
where the sum in the numerator runs over all bound states of the surface potential V, and N, is the total number of gas particles in the system. Likewise we get the time dependent pressure W/W)
= C~J&&) k
C&%(O)1 I
k
where both sums go over all continuum states of energy E, = h2k2/2m with m the mass of a gas particle. In the next section we will write down the master equation (1), explicitly reducing the problem to a one-dimensional one for a highly mobile adsorbate. In section 3 we then calculate the time dependent coverage and pressure for weakly bound physisorbed systems like He-graphite, He-LiF, and He-A. Our main results are as follows: for small incremental flashes, i.e. when T, is not much larger than Ti the adsorbate density decreases little with the pressure rising monotonically on the same scale of the flash desorption time to its final value Pf which at low temperature and in a small chamber, however, can be orders of magnitude larger than Pi. If, however, Tf is si~ificantly larger than Ti (by a factor two or so), then the adsorbate desorbs almost completely in a much shorter time with a simultaneous pressure rise through orders of magnitude which overshoots the final equilibrium pressure substantially; the latter is attained only after a much longer equilibration time. Such non-monotonic pressure evoiution must be accounted for in the analysis of experimental data and should be directly observable under favourable conditions.
2. Theory For a highly mobile adsorbate we can assume that the surface potential is a function of the distance z of the gas particle above the surface only, i.e. that V,(r) = V,(z). Its wave functions then factorize into a plane wave with two-dimensional wave vector q along the surface and into an eigenfunction of V-,(z). Assuming that the surface kinetics is dominated by one-phonon processes conserving the parallel momentum q in a transition, the transition rates can be evaluated in second order perturbation theory (Fermi’s Golden Rule). Specifying a Morse potential V,( 2) = I/,(e-2Y’
- 2eCYr),
(6)
2. W. Gorrel, H.J. Kreurer / Flash desorprion and therntalization
and choosing a bulk Debye model for the phonons (the surface Debye has been analyzed in refs. [ 13,14]), we get for bound state-bound transitions (see ref. [9])
X (20, -j
-j’
(j - j’)(2uo
-j-j’
)
(j-j')(2uo-j-j'- 1) r
+@( j’-j) n [i
8 Ii
2mAoD
j (j’-j)(2u,-j-j’-
1)
Cjf-jH2%-j-j.-I; r
where we introduced
r=g’
- 1)
r
i
x
model state
- 1)3 lj -j’i3
+(j-j’+ Xn
487
+* )
the dimensionless
2mV, uo’ = (tzr>2’
v, U*=i&=19
Ii9
i
(7)
variables 4
t+=>,
(8)
B f
where wr, is the Debye frequency of the solid, m and &f, are the masses of a gas particle and a unit cell of the solid, respectively, and Tt is the temperature of the solid. Note that the Morse potential develops bound states at energies cj = E,/ho,
forj=O,... n(x)
= - (a,, -j
- $)‘/r,
(9)
, N with a, - 1 < N < uo. The phonon
= (es+ - l)-‘,
occupation
function, 00)
is assumed to remain unchanged throughout the course of the kinetic surface processes. Also in (7) we have j, = max(j, j’) and j, = min( j, j’>. The first term in the curly brackets in (7) thus accounts for a transition into a higher bound state mediated by the absorption of a thermal phonon, whereas the second term describes a transition into a lower bound state accompanied by the spontaneous (the factor 1 in the square brackets) and stimulated (the factor n) emission of a phonon. Likewise the transition rate from a bound state j into a continuum state of (dimensionless) wave number cP and energy cp, describing the desorption process, 2
cp =q,
p = I,
YL
2,...;
cp=--,
eP r
(11)
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2. W. Gortel, H.J. Kreurer / Flash desorprion and ihermnlization
is given by cp sinh(2rc,) cos2(&,)
+ sinh*( rrc,)
Ir( f + oO f icP >I’ x(20,-2j-
1)
J! F(2u,
-j)
(12) For continuum-bound state transitions, describing the adsorption process, the phonon occupation function n(x) in (12) must be replaced by (n(x) + 1). Note already that R, and R, are proportional to the inverse size of the box, L- ‘. Finally we list the rate for continuum-continuum transitions [ 151
R
2
cpcp, sinh(2nc,) sinh(2rc,.) --377 m an 2r-4 MS sinh2( T( cP + c,,)) sinh*( r( cP - c,,))
x{“(c~_c~,,e(l_~)~(~) +8($.-c;
)
e( 1 -
G..I+)[~(~)+,I~
‘.
(13)
Before rewriting the master equation (1) explicitly, we note that the delta functions in the transition rates conserving the momentum q of the gas particles parallel to the surface imply that (1) is, indeed, independent of q because with the energy of a gas particle given by E, = h2q2/2m
+ ejAuD,
E, = #i2q2/2m -I-c,tZo,,
(14)
2. W. Gortel, HJ. Kreuzer / Flash desorption and thermaiization
489
for bound states and continuum states, respectively, the initial conditions (2) factorize. Thus the relevant part of the master equation (1) reads for a highly mobile adsorbate 80, -=at
N c RJJ.‘njfy=o
$
Rj,pj-+
j'=O
g
R,p,
- pfJ, Rpini*
(154
p=l N
f$=
f
R,,,n,,.
-
p'-1
e
pr=
R
ppp
+
1
N C
R,jn,-
j=O
C
R,,n,.
(15b)
j=O
One usually argues that the size L of the box into which the gas is confined is so much larger than all microscopic length scales in the problem, that one can safely take the large volume or thermodynamic limit L -+ 00. This implies from (11) that one can replace summations over p by integrations
(16) where cP = (s~/Ly)p is the dimensionless wave vector (11). With the transition rates given explicitly in (7), (12) and ( 13), we thus see that in (15)
2
Rjpnp
=
O(l),
E R,nj=
p=l
O(l),
(17)
p=l
so that in the large volume limit the equations for the bound state occupation probabilities nj remain unchanged, whereas for the continuum states we find
an,/at
= o( I/~L),
(18)
implying that for L + 00 the continuum occupation is not affected by the surface processes; for a discussion see,, e.g., ref. [ 161. To describe the isothermal desorption process we must put the continuum occupation equal to zero to mimick the removal of the gas phase by fast pumping. Thus for isothermal desorption, eqs. (15) reduce to ani/&
= c Rjjaj.
- c Rjpjnj -
_i’
j’
5
Rpinj,
(19)
p=l
with initial conditions n, (t = 0) = e@pe-Q,_
(20)
These equations have been solved in refs. [9] and [17] by matrix diagonalization which allows one to identify the (inverse) desorption time as the smallest eigenvalue of the matrix R in (19). In this paper we want to study the processes induced in the gas-solid system by raising the temperature of the solid at time t = 0 to a value Tf > Ti where Ti was the temperature for the gas-solid system in equilibrium for t < 0.
490
2. W. Gortel, H.J. Kreuzer / Flash desorption and thermalization
The initial states
conditions
are still (20) with a similar
equation
for the continuum
nP (t = 0) = eP$P e-*,‘,.
(21)
The transition rates in (15) must be calculated at T, or 6, = Aw,/k,T,. To solve eqs. (15) numerically by matrix diagonalization, we must first truncate the wave number p at some upper limit p, to get a finite number of equations. To find an estimate for p, we note that the one-phonon processes can connect energy levels that are at most one Debye energy apart. Restricting ourselves for physisorption to temperatures k,T, < Aw, (otherwise multiphonon processes become important), we know that in the final equilibrium only continuum states with cP = 8;’ -C 1 are appreciably occupied. This suggests that p, can be chosen such that eP is at most 2 or 3 to allow for the possibility of some transient occupation of higher energy states due to continuum-continuum transitions during the desorption process and the thermalization towards the final equilibrium. From 177* CP = - p2
(22)
we get Pu = $a/~.
(23)
To get a feeling for numbers we look at the helium-graphite system for which -1 chamber of, say, 0.30 m Y = 1 A, r t: 25, so that for a typical experimental diameter, pu is of the order 10 lo . Obviously one cannot diagonalize a matrix of that size, so that a coarse graining of the energy spectrum is mandatory. We therefore group the wave numbers p into NCs blocks each containing nCg of them and label these blocks by P = 1,. . . , If_,, where NCp can be chosen of the order of 100 or 200 for a reasonably sized computer. We next define an average occupation tip=
5
np,
p,=P+fn
(24)
Csr
P=PI
which are subject
J’=o
kP
=
5 P=l
Kg
N
N
hj = c
to the rate equations
R,j.nj, -
c
Rlrjnjt
ncgRppiPf -
c
R,ii,,
P=l
j’=O
z P’=l
n,.,R pfpff, + Cn,,R,jnji
CRjpn’,e, i
where p and p’ are respresentative wave numbers within the blocks P and P’, respectively; e.g., p = ncg( P + f) and p’ = ncg( P’ + i). These choices and also
Z. W. Gortel, H.J. Kreuzer / Flash desorption and thermalization
491
the values for nCa, NCp and z, must, of course, be made so that the features the solution of (25) that are important for the kinetics are independent them. To solve (25) we write it as iI=
-Rn,
of of
(26)
where n is a column vector made up of the ni’s and Ap’s. Next we symmetrize the transition matrix R by a transformation [9,17] S,,, = e&c,/z R,,, ,-+1~/2, x,(t)
= n,(t)
e8r’o’2,
A=S*x,
(27)
and diagonalize
S
S. p(” = X,p”‘,
(28)
where all eigenvalues are real and non-negative and are assumed to be ordered x,
=C[c(“‘r.x(O)] I
e-X~‘$“,
(29)
which gives PI,(~) = c A,,, e-‘l,‘, A,,.
=
e-&4
p,(1’)
(3W
C p\!? rip(O)) e8fc~/2.
(3Ob)
I”
The n,(O) are the initial occupation functions (20) and (21) and pc1(1’) is the Ith component of the f’th eigenvector. We note that S and thus R has a lowest eigenvalue zero whose corresponding eigenvector describes the final equilibrium distribution at S, or Tf. With n,(t) known, the time evolution of the relative bound state occupation and the gas pressure can be calculated from (4) and (5), respectively as NA(t)/NT=CS,ePh/‘, I”
(31)
S,=
(32)
; A,, c A,,,.., I=0 I P.1”
P(r)/P(O)
= C T( e-‘l’,
T, = c P’A,,
c
P
P’,I’
I
P’2A,,..
(33) (34)
492
Z. W. Gortel, H.J. Kreuzer / Flash desorption and thermalization
3. Results We will now present numerical results on the time evolution of the relative bound state occupation (4) and the pressure (5) for flash desorption and subsequent thermalization of helium desorbing from a graphite, a solid argon, and a LiF surface. We start with the helium-graphite system which has a surface potential that develops 5 bound states into which helium atoms can get trapped. A fit to scattering data suggests [9] a well depth of V, = 171.34ka and a range y -’ = 0.95 A so that r = 27.464 and a, = 5.041 with AU, = 185k, for graphite. In a first series of calculations we start from an initial temperature of q = 6.166 K (Si = 30) and flash the solid at time t = 0 to q = 7.4 K (8, = 25).
30
Fig. 1. Flash desorption in the He-graphite system of size Ly/n after a flash of the temperature of the solid from T, to Tr. ty)(T,) is the isothermal desorption time at Tr. The time evolution of pressure P( t)/P(O) and relative bound state occupation N,(t)/N, is calculated from (31) and (33) respectively. The dashed line is calculated from (36). Debye temperature of graphite To = 185 K. Morse potential parameters Vo/k,=171.34 K and y -’ = 0.95 A from ref. [9]. Mass ratio m/M, = 0.333.
2. W. Gortel, H.J. Kreurer / Flash desorption and thermalization
493
The resulting time evolution of pressure and coverage is plotted in fig. 1 against log t to highlight the short time transient behaviour. For the top panel we have chosen a small experimental chamber of only 3 cm linear dimension. Note first that initially at q = 6.166 K more or less all particles are actually trapped in the bound states so that NA(t = 0)/N, = 0.6. Within a time of order 10P3 s it decreases to NA(t = co)N, = 0.605; however, not in an exponential fashion because the transition matrix R has about 30 eigenvalues of the order lo3 s-‘, which contribute with comparable weights to the sum (31). The time evolution for isothermal desorption at T, = 7.4 K, NA(t 10% = ( NA(O)/%-)
expf - t/Q’)
(35)
I
describes complete desorption to NA(t = co) = 0 with tii) = 10e3 s. Accounting for the residual gas phase at t = co, a phenomenological ansatz, employed in ref. [lo], &(t)
= N,(O) + [&two)
-N,(O)]
exp( -f/Q),
(36)
produces a fair fit to the flash desorption “data” (dashed lines). To calculate the equilibrium values of pressure and relative bound state occupation for t = 0 and t + 00, we first determine the chemical potential through normalization, [2:eWpEi+ L/h,,,
N,=zn,=ePp(L/X,,)2 1
h.s.
1,
(37)
where 1 includes also the summation over the momentum q parallel to the surface and At,, = ~/(2~~k~T) IL2is the thermal wave length. We then get N
N,(t=
co)
%(‘=
‘)
N
C edarr~ C, e-Si’l + L/Ath( 7:) I
j-o =
5 j=O
(38) e--si<, ‘ioe-&E;
+ L,h,,(T,)
’
j=O
3/2
,goe-
“‘1 + L/X,,( Ti) (39)
5 e-&f’f + L/A,,(T,) j-0
We note that only in the large volume limit L -3 00 will the pressures be in the ratio of the temperatures. For the present model system all terms in (38) and (39) contribute, resulting in a pressure ratio Pt/Pi = 34.4, whereas in an infinite box it would only be Pf/Pi = Tf/Ti = 1.2; the difference being due to a substantial shift in the chemical potential. Note from the top panel of fig. 1 that the final pressure is not reached monotonically, but that a slight overshoot occurs during the major part of the desorption process followed by a much
494
Z. W. Gortel, H.J. Kreuzer / Flash desorption and thermalizntion
1 0
t(s)
t(r)
Fig. 2. Flash desorption after a sudden temperature drop and flash desorption after another flash back to the initial high temperature in the He-graphite system. For other details see fig. 1
slower thermalization process during which wall collisions slowly adjust the occupation functions to the final equilibrium one. The desorption process thus puts the particles into continuum states that have too high an energy for T,. This implies that in a very large box the flux of desorbing particles, e.g. detected in a mass spectrometer, will not have a Maxwell velocity distribution, but rather one that is determined by the bound-state-continuum transition rates’ in the master equation. Indeed, deviations from a Maxwell velocity distribution have been seen for helium desorbing from a helium film [4] and in laser-induced thermal desorption of CO from an Fe surface [5]. For the lower panel of fig. 1 we have enlarged the experimental chamber to L = 30 cm. Fewer of the total number NT of particles are initially in the adsorbate (N,( t = 0)/N, = 0.877) and more of them desorb (NA( t = oo)/NT = 0.133) the flash desorption process being closer to an isothermal one. Also note that the pressure overshoot is relatively larger though the final pressure rise is lower by about a factor 3 compared to the smaller box in the top panel. In fig. 2 we look at the same system as in fig. 1 but enclosed in a yet larger box of length about 1 m. In the left hand side panel we first follow a flash adsorption experiment decreasing the solid temperature at time t = 0 from q = 7.4 K to T, = 6.17 K. The adsorption and thermalization process being over after a few seconds we then increase, in the right hand side panel, the temperature back to 7.4 K to monitor a flash desorption experiment. As the kinetic processes at the surface are mainly determined by the temperature of the phonon bath of the solid, the flash adsorption from 7.4 to 6.166 K is orders of magnitude slower than the flash desorption from 6.166 to 7.4 K. Also note from the dashed curve in the left hand panel of fig. 2 that the ansatz (36) with thi) the isothermal desorption time at Tf = 6.166 K is a rather poor description of the flash adsorption process indicating the importance of continuum-con-
2. W, Gortel, H.J. Kreuzer / Flash desorption and thermaiization
495
He-C 1, = 6.17KeTs = 13.3K - 1.0 300 - 0.8
Fig. 3. Flash desorption in the He-grapbite rise.
system as in fig. 1 and 2 but for a larger temperature
tinuum transitions to thermalize the gas phase during the course of the flash adsorption. Also note the slight undershoot of the pressure. Fig. 3 demonstrates the rather dramatic nonequilibrium effects in the He-graphite system after a substantial temperature flash of the solid from q = 6.166 to 13.33 K. In this situation the adsorbate evolution is very well described by the ansatz (35). Interesting is the fact that in the bigger boxes the pressure after rising substantially during a time of the order of the isothermal desorption time ty)( T = 13.33 K) = 3.2 x lo-’ s, remains constant for about a millisecond before it gradually decreases over about one second to its final equilibrium value.
496
2. W. Gortel, H.J. Kreurer / Flash desorption and thermalization
I
He-A T, = 4K+T,
0.6
= 6K
0.4
I
0.2
D
t(s) Fig. 4. Flash desorption of helium from solid argon. Debye temperature To = 92 K; Morse potential parameters Va/k, = 160.07 K and y - ’ = 1.98 A from ref. [9]. Mass ratio m/M, = 0.111.
We next turn to a system with more surface bound states, namely helium adsorbed on solid argon with Morse potential parameters such that 10 bound states develop. Fig. 4 shows a flash desorption in a box of 30 cm length from T, = 4 K to T, = 6 K. Although not many particles desorb, the pressures rises again substantially but in a monotonic fashion. Finally in fig. 5 we look at the He/LiF system with four surface bound
He-LIF T, = 3.65K
- o.4
Fig. 5. Flash desorption of helium from LiF; temperature flash from I; = 3.65 K to Tr indicated at the curves. Debye temperature Tn= 730 K; Morse potential parameters Vo/kB = 81.75 K and y-’ = 1.09 A from ref. [9]. Mass ratio m/M, = 0.152.
Z. W. Gortel, H.J. Kreuzer / Flash desorption and thermalization
497
states. Whereas a small flash from Ti = 3.65 K to Tf= 4.87 K produces a slight overshoot in the nonequilibrium pressure as we have seen in the He-graphite system, new features emerge at larger flashes. For a flash from Ti= 3.65 K to T,= 7.3 K, the pressure decreases slightly after about 10-l s to rise again to its final value. For a flash from T,= 3.65 K to Tf= 9.73 K, finally the pressure rises within roughly an isothermal desorption time t$‘)( T = 9.73 K) = 5 X lop6 s to a plateau from which another rise starts after some 0.1 s to the final equilibrium pressure.
4. Summary In this paper we have developed a theory to study transient effects in a flash desorption experiment. In situations where the mean free path for gas particle collisions is much larger than the size of the experimental chamber particles desorbing after the adsorbent temperature has been raised to its final value will thermalize due to wall collisions. Whereas the number of adsorbed particles decreases in a more or less exponential fashion within a typical flash desorption time, the gas pressure will rise with the same time scale but overshoots its final value by as much as a factor two depending on the temperature difference in the flash and the size of the vacuum chamber. The theoretical model assumes that all walls of the vacuum chamber are covered by the adsorbent (as is the case, e.g., with films deposited on the inside of glass systems) and are subject to the temperature flash. If only a small single crystal surface or a filament is the adsorbent, then the desorbing ‘particles will most likely thermalize with the other walls in the vacuum chamber. If the latter adsorb the gas readily, then a pressure device may be arranged to record the transient pressure only. It is important to note that it would be wrong to measure the number of desorbed particles and calculate a pressure using the ideal gas law. Such a procedure is only admissible for the final but not for the transient pressure. Our model is one-dimensional. This is appropriate for highly mobile adsorbates. It would appear that in a one-dimensional model only the momentum component perpendicular to the surface from which the particle desorbed will be thermalized. That this is not the case can be best seen in an idealized experiment for which we choose a perfectly cubic vacuum chamber with all surfaces covered by the adsorbent and uniformly flashed to a higher temperature. A particle leaving surface A with momentum pI perpendicular to A and momentum p,, parallel to it will either hit the surface B opposite to A thus thermalizing p I or hit a surface C perpendicular to A thermalizing the component of p,, perpendicular to C. In a one-dimensional model it is assumed that the thermalization of p,, proceeds at the same rate as that of pl . It is feasible to set up a three-dimensional model in which the angular distribution
498
2. W. Gortel, H.J. Kreuzer / Flash desorption and thermalization
of the desorbing particles is taken into account as well. We feel, however, that nothing qualitatively new can be learnt about the thermalization process that is not contained in the present model.
Acknowledgements This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. We would like to thank R. Teshima for developing the computer codes used in this paper.
References [l] G. Ehrlich, Advan. Catalysis 14 (1963) 255; J. Appl. Phys. 32 (1961) 4. [2] D. Menzel, in: Interactions on Metal Surfaces, Topics in Applied Physics 4, Ed. R. Comer (Springer, Berlin, 1975). [3] S.A. Cohen and J.G. King, Phys. Rev. Letters 31 (1973) 703. [4] F. Taborek, M. Sinvani and D. Goodstein, Phys. Letters 95A (1983) 59. [5] G. Wedler and H. Ruhmann, Surface Sci. 121 (!982) 464; the rapid flash is achieved here by laser heating. [6] L. Apker, Ind. Eng. Chem. 40 (1948) 846. [7] P.A. Redhead, Vacuum 12 (1962) 203. [8] D.A. King, Surface Sci. 47 (1975) 384. 191 Z.W. Gortel, H.J. Kreuzer and R. Teshima, Phys. Rev. B22 (1980) 5655. [lo] Z.W. Gortel, H.J. Rreuzer and D. Spaner, J. Chem. Phys. 72 (1980) 234. [ 1 l] Z.W. Gortel, H.J. Kreuzer, Chem. Phys. Letters 67 (1979) 197. [ 121 P. Summerside, E. Sommer, R. Teshima and H.J. Kreuzer, Phys. Rev. B25 (1982) 6235; E. Sommer and H.J. Kreuzer, Phys. Rev. B26 (1982) 658, 4094. [13] E. Goldys, Z.W. Gortel and H.J. Kreuzer, Solid State Commun. 40 (1981) 963; Surface Sci. 116 (1982) 33. [14] J. Stutzki and W. Brenig, Z. Physik B45 (1981) 49. [15] The integral in (13) involving the continuum wavefunctions in the Morse potential can be evaluated with a method used for the repulsive part only by J.E. Lennard-Jones and A.F. Devonshire, Proc. Poy. Sot. (London) Al58 (1937) 253. [16] Z.W. Gortel, H.J. Kreuzer, R. Teshima and L.A. Turski, Phys. Rev. B24 (1981) 4456. 1171 H.J. Kreuzer, R. Teshima, Phys. Rev. B24 (1981) 4470.
Surface Science 133 (1983) 484-498 North-Holland Publishing Company
FLASH DESORPTION Z.W. GORTEL
AND THERMALIZATION
and H.J. KREUZER
Department of Physics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada Received
15 February
1983
Starting from a master equation for phonon-mediated physisorption in a gas-solid system in a finite volume, we calculate the nonequilibrium time evolution of adsorbate density and pressure for flash desorption and subsequent thermalization of the gas due to wall collisions. The transient pressure overshoots its final equilibrium value by as much as a factor two. The asymmetry between flash desorption and adsorption caused by the reverse temperature flash is demonstrated.
1. Introduction Kinetic studies in physisorption and chemisorption probe the dynamics of the gas-solid interaction. The desorption kinetics is investigated in basically three different experimental arrangements, all, of course, starting from a gas-solid system in equilibrium at some initial temperature T and initial pressure Pi. In an isothermal desorption experiment the gas pressure is reduced suddenly to a much smaller final value Pf -GZPi that is maintained subsequently by continuous pumping also keeping the temperature q constant. The time evolution of the flux of desorbing particles is monitored and characterized by experiment the an isothermal desorption time I,(i) [ 1,2]. In a flash desorption nonequilibrium situation triggering the desorption process is created by a sudden increase in the temperature of the solid to T,, being maintained thereafter. Particles will desorb from the solid surface in a characteristic flash desorption time t$,‘) [2-51. Last1 y, temperature programmed desorption experiment (TPD) is basically a sequence of flash desorption experiments induced by a programmed rise in surface temperature as a function of time. From the resulting desorption signal one can, under favourable conditions, extract heats of adsorption and desorption times [1,6,7]. The difficulties in analyzing TPD spectra, e.g. caused by readsorption from the gas phase and the model dependence of the analysis, have been highlighted by King [8]. A microscopic theory of adsorption-desorption kinetics conveniently starts from a master equation. In the following we will restrict ourselves to physisorbed gas-solid systems for which the net interaction between a gas particle 0039-6028/83/0000-0000/$03.00
0 1983 North-Holland
Z. W. Gortel, H.J. Kreuzer / Flash desorption and thermaliration
and the solid can be described by a surface potential states of a gas particle in V, by I, the master equation dn,/dt
= c R,,n, I
- c R,,n,,
VS. Labelling reads
485
the energy (1)
I
where n, is the time dependent occupation probability of the states I which include both bound states (i.e. the adsorbate) and continuum states (i.e. the gas phase). R,, are transition rates per unit time for a gas particle to go from state I’ to 1. They must be calculated from microscopic theory. The three experimental arrangements described above must be modelled in the theory by imposing proper initial and boundary conditions on the solutions of the master equation (1). To describe an isothermal desorption experiment, one sets for time t > 0 the occupation functions of the continuum states equal to zero, thus ensuring that the pressure of the gas phase is zero, P = 0. For phonon-mediated physiso~tion kinetics, the procedure to calculate isotherms desorption times from (1) has been explained in detail in ref. [9]. To describe the flash desorption experiment, readsorption out of the gas phase must be included in the theory which has been done in refs. [ 10,l l] by imposing the appropriate final state reached by the gas-solid system after a long time without specifying the physical mechanism responsible for thermalization. Indeed, there are two processes which can equilibrate the gas phase: (if collisions between particles in the gas phase and (ii) collisions of gas particles with the walls. If the mean free path 1 for gas particle collisions is large compared to the size L of the experimental chamber, then only collisions with the walls provide an effective means of changing the energy distribution of the gas phase to a thermal one at the temperature of the walls. If I < L thermalization is speeded up by collisions between gas particles. For temperatures below room temperatures and pressure lower than 10m2 Torr and L - IO-’ m, one always gets I B L. In this paper we will set up the appropriate boundary conditions for the master equation (1) to describe a flash desorption experiment at low coverage and the subsequent thermalization due to wall collisions. We assume that at times t < 0 the gas-solid system is in equilibrium at Ti and Pi, so that the gas particle occupation probabilities are
n,(t < 0) = e-(E/-P)/k6Tt, _
(2)
where
efi/k8Tl =
pi(2am)-3’2(kg~)-s’2,
(3)
with p being the chemical potential of the gas phase. Within the framework of mean field theory [12] the present ideas can be extended to finite coverage. Having raised the temperature of the solid to Tf for times t 2 0 we calculated the phonon-mediated transition rates R,, at Tf. The experimental situation we
486
Z. W. Gortel, H.J. Kreuzer / Flash desorption and thermalization
thus describe is one in which all walls surrounding the gas phase are at T,. From (1) we can then calculate the time evolution of the relative bound state occupation
where the sum in the numerator runs over all bound states of the surface potential V, and N, is the total number of gas particles in the system. Likewise we get the time dependent pressure W/W)
= C~J&&) k
C&%(O)1 I
k
where both sums go over all continuum states of energy E, = h2k2/2m with m the mass of a gas particle. In the next section we will write down the master equation (1), explicitly reducing the problem to a one-dimensional one for a highly mobile adsorbate. In section 3 we then calculate the time dependent coverage and pressure for weakly bound physisorbed systems like He-graphite, He-LiF, and He-A. Our main results are as follows: for small incremental flashes, i.e. when T, is not much larger than Ti the adsorbate density decreases little with the pressure rising monotonically on the same scale of the flash desorption time to its final value Pf which at low temperature and in a small chamber, however, can be orders of magnitude larger than Pi. If, however, Tf is si~ificantly larger than Ti (by a factor two or so), then the adsorbate desorbs almost completely in a much shorter time with a simultaneous pressure rise through orders of magnitude which overshoots the final equilibrium pressure substantially; the latter is attained only after a much longer equilibration time. Such non-monotonic pressure evoiution must be accounted for in the analysis of experimental data and should be directly observable under favourable conditions.
2. Theory For a highly mobile adsorbate we can assume that the surface potential is a function of the distance z of the gas particle above the surface only, i.e. that V,(r) = V,(z). Its wave functions then factorize into a plane wave with two-dimensional wave vector q along the surface and into an eigenfunction of V-,(z). Assuming that the surface kinetics is dominated by one-phonon processes conserving the parallel momentum q in a transition, the transition rates can be evaluated in second order perturbation theory (Fermi’s Golden Rule). Specifying a Morse potential V,( 2) = I/,(e-2Y’
- 2eCYr),
(6)
2. W. Gorrel, H.J. Kreurer / Flash desorprion and therntalization
and choosing a bulk Debye model for the phonons (the surface Debye has been analyzed in refs. [ 13,14]), we get for bound state-bound transitions (see ref. [9])
X (20, -j
-j’
(j - j’)(2uo
-j-j’
)
(j-j')(2uo-j-j'- 1) r
+@( j’-j) n [i
8 Ii
2mAoD
j (j’-j)(2u,-j-j’-
1)
Cjf-jH2%-j-j.-I; r
where we introduced
r=g’
- 1)
r
i
x
model state
- 1)3 lj -j’i3
+(j-j’+ Xn
487
+* )
the dimensionless
2mV, uo’ = (tzr>2’
v, U*=i&=19
Ii9
i
(7)
variables 4
t+=>,
(8)
B f
where wr, is the Debye frequency of the solid, m and &f, are the masses of a gas particle and a unit cell of the solid, respectively, and Tt is the temperature of the solid. Note that the Morse potential develops bound states at energies cj = E,/ho,
forj=O,... n(x)
= - (a,, -j
- $)‘/r,
(9)
, N with a, - 1 < N < uo. The phonon
= (es+ - l)-‘,
occupation
function, 00)
is assumed to remain unchanged throughout the course of the kinetic surface processes. Also in (7) we have j, = max(j, j’) and j, = min( j, j’>. The first term in the curly brackets in (7) thus accounts for a transition into a higher bound state mediated by the absorption of a thermal phonon, whereas the second term describes a transition into a lower bound state accompanied by the spontaneous (the factor 1 in the square brackets) and stimulated (the factor n) emission of a phonon. Likewise the transition rate from a bound state j into a continuum state of (dimensionless) wave number cP and energy cp, describing the desorption process, 2
cp =q,
p = I,
YL
2,...;
cp=--,
eP r
(11)
488
2. W. Gortel, H.J. Kreurer / Flash desorprion and ihermnlization
is given by cp sinh(2rc,) cos2(&,)
+ sinh*( rrc,)
Ir( f + oO f icP >I’ x(20,-2j-
1)
J! F(2u,
-j)
(12) For continuum-bound state transitions, describing the adsorption process, the phonon occupation function n(x) in (12) must be replaced by (n(x) + 1). Note already that R, and R, are proportional to the inverse size of the box, L- ‘. Finally we list the rate for continuum-continuum transitions [ 151
R
2
cpcp, sinh(2nc,) sinh(2rc,.) --377 m an 2r-4 MS sinh2( T( cP + c,,)) sinh*( r( cP - c,,))
x{“(c~_c~,,e(l_~)~(~) +8($.-c;
)
e( 1 -
G..I+)[~(~)+,I~
‘.
(13)
Before rewriting the master equation (1) explicitly, we note that the delta functions in the transition rates conserving the momentum q of the gas particles parallel to the surface imply that (1) is, indeed, independent of q because with the energy of a gas particle given by E, = h2q2/2m
+ ejAuD,
E, = #i2q2/2m -I-c,tZo,,
(14)
2. W. Gortel, HJ. Kreuzer / Flash desorption and thermaiization
489
for bound states and continuum states, respectively, the initial conditions (2) factorize. Thus the relevant part of the master equation (1) reads for a highly mobile adsorbate 80, -=at
N c RJJ.‘njfy=o
$
Rj,pj-+
j'=O
g
R,p,
- pfJ, Rpini*
(154
p=l N
f$=
f
R,,,n,,.
-
p'-1
e
pr=
R
ppp
+
1
N C
R,jn,-
j=O
C
R,,n,.
(15b)
j=O
One usually argues that the size L of the box into which the gas is confined is so much larger than all microscopic length scales in the problem, that one can safely take the large volume or thermodynamic limit L -+ 00. This implies from (11) that one can replace summations over p by integrations
(16) where cP = (s~/Ly)p is the dimensionless wave vector (11). With the transition rates given explicitly in (7), (12) and ( 13), we thus see that in (15)
2
Rjpnp
=
O(l),
E R,nj=
p=l
O(l),
(17)
p=l
so that in the large volume limit the equations for the bound state occupation probabilities nj remain unchanged, whereas for the continuum states we find
an,/at
= o( I/~L),
(18)
implying that for L + 00 the continuum occupation is not affected by the surface processes; for a discussion see,, e.g., ref. [ 161. To describe the isothermal desorption process we must put the continuum occupation equal to zero to mimick the removal of the gas phase by fast pumping. Thus for isothermal desorption, eqs. (15) reduce to ani/&
= c Rjjaj.
- c Rjpjnj -
_i’
j’
5
Rpinj,
(19)
p=l
with initial conditions n, (t = 0) = e@pe-Q,_
(20)
These equations have been solved in refs. [9] and [17] by matrix diagonalization which allows one to identify the (inverse) desorption time as the smallest eigenvalue of the matrix R in (19). In this paper we want to study the processes induced in the gas-solid system by raising the temperature of the solid at time t = 0 to a value Tf > Ti where Ti was the temperature for the gas-solid system in equilibrium for t < 0.
490
2. W. Gortel, H.J. Kreuzer / Flash desorption and thermalization
The initial states
conditions
are still (20) with a similar
equation
for the continuum
nP (t = 0) = eP$P e-*,‘,.
(21)
The transition rates in (15) must be calculated at T, or 6, = Aw,/k,T,. To solve eqs. (15) numerically by matrix diagonalization, we must first truncate the wave number p at some upper limit p, to get a finite number of equations. To find an estimate for p, we note that the one-phonon processes can connect energy levels that are at most one Debye energy apart. Restricting ourselves for physisorption to temperatures k,T, < Aw, (otherwise multiphonon processes become important), we know that in the final equilibrium only continuum states with cP = 8;’ -C 1 are appreciably occupied. This suggests that p, can be chosen such that eP is at most 2 or 3 to allow for the possibility of some transient occupation of higher energy states due to continuum-continuum transitions during the desorption process and the thermalization towards the final equilibrium. From 177* CP = - p2
(22)
we get Pu = $a/~.
(23)
To get a feeling for numbers we look at the helium-graphite system for which -1 chamber of, say, 0.30 m Y = 1 A, r t: 25, so that for a typical experimental diameter, pu is of the order 10 lo . Obviously one cannot diagonalize a matrix of that size, so that a coarse graining of the energy spectrum is mandatory. We therefore group the wave numbers p into NCs blocks each containing nCg of them and label these blocks by P = 1,. . . , If_,, where NCp can be chosen of the order of 100 or 200 for a reasonably sized computer. We next define an average occupation tip=
5
np,
p,=P+fn
(24)
Csr
P=PI
which are subject
J’=o
kP
=
5 P=l
Kg
N
N
hj = c
to the rate equations
R,j.nj, -
c
Rlrjnjt
ncgRppiPf -
c
R,ii,,
P=l
j’=O
z P’=l
n,.,R pfpff, + Cn,,R,jnji
CRjpn’,e, i
where p and p’ are respresentative wave numbers within the blocks P and P’, respectively; e.g., p = ncg( P + f) and p’ = ncg( P’ + i). These choices and also
Z. W. Gortel, H.J. Kreuzer / Flash desorption and thermalization
491
the values for nCa, NCp and z, must, of course, be made so that the features the solution of (25) that are important for the kinetics are independent them. To solve (25) we write it as iI=
-Rn,
of of
(26)
where n is a column vector made up of the ni’s and Ap’s. Next we symmetrize the transition matrix R by a transformation [9,17] S,,, = e&c,/z R,,, ,-+1~/2, x,(t)
= n,(t)
e8r’o’2,
A=S*x,
(27)
and diagonalize
S
S. p(” = X,p”‘,
(28)
where all eigenvalues are real and non-negative and are assumed to be ordered x,
=C[c(“‘r.x(O)] I
e-X~‘$“,
(29)
which gives PI,(~) = c A,,, e-‘l,‘, A,,.
=
e-&4
p,(1’)
(3W
C p\!? rip(O)) e8fc~/2.
(3Ob)
I”
The n,(O) are the initial occupation functions (20) and (21) and pc1(1’) is the Ith component of the f’th eigenvector. We note that S and thus R has a lowest eigenvalue zero whose corresponding eigenvector describes the final equilibrium distribution at S, or Tf. With n,(t) known, the time evolution of the relative bound state occupation and the gas pressure can be calculated from (4) and (5), respectively as NA(t)/NT=CS,ePh/‘, I”
(31)
S,=
(32)
; A,, c A,,,.., I=0 I P.1”
P(r)/P(O)
= C T( e-‘l’,
T, = c P’A,,
c
P
P’,I’
I
P’2A,,..
(33) (34)
492
Z. W. Gortel, H.J. Kreuzer / Flash desorption and thermalization
3. Results We will now present numerical results on the time evolution of the relative bound state occupation (4) and the pressure (5) for flash desorption and subsequent thermalization of helium desorbing from a graphite, a solid argon, and a LiF surface. We start with the helium-graphite system which has a surface potential that develops 5 bound states into which helium atoms can get trapped. A fit to scattering data suggests [9] a well depth of V, = 171.34ka and a range y -’ = 0.95 A so that r = 27.464 and a, = 5.041 with AU, = 185k, for graphite. In a first series of calculations we start from an initial temperature of q = 6.166 K (Si = 30) and flash the solid at time t = 0 to q = 7.4 K (8, = 25).
30
Fig. 1. Flash desorption in the He-graphite system of size Ly/n after a flash of the temperature of the solid from T, to Tr. ty)(T,) is the isothermal desorption time at Tr. The time evolution of pressure P( t)/P(O) and relative bound state occupation N,(t)/N, is calculated from (31) and (33) respectively. The dashed line is calculated from (36). Debye temperature of graphite To = 185 K. Morse potential parameters Vo/k,=171.34 K and y -’ = 0.95 A from ref. [9]. Mass ratio m/M, = 0.333.
2. W. Gortel, H.J. Kreurer / Flash desorption and thermalization
493
The resulting time evolution of pressure and coverage is plotted in fig. 1 against log t to highlight the short time transient behaviour. For the top panel we have chosen a small experimental chamber of only 3 cm linear dimension. Note first that initially at q = 6.166 K more or less all particles are actually trapped in the bound states so that NA(t = 0)/N, = 0.6. Within a time of order 10P3 s it decreases to NA(t = co)N, = 0.605; however, not in an exponential fashion because the transition matrix R has about 30 eigenvalues of the order lo3 s-‘, which contribute with comparable weights to the sum (31). The time evolution for isothermal desorption at T, = 7.4 K, NA(t 10% = ( NA(O)/%-)
expf - t/Q’)
(35)
I
describes complete desorption to NA(t = co) = 0 with tii) = 10e3 s. Accounting for the residual gas phase at t = co, a phenomenological ansatz, employed in ref. [lo], &(t)
= N,(O) + [&two)
-N,(O)]
exp( -f/Q),
(36)
produces a fair fit to the flash desorption “data” (dashed lines). To calculate the equilibrium values of pressure and relative bound state occupation for t = 0 and t + 00, we first determine the chemical potential through normalization, [2:eWpEi+ L/h,,,
N,=zn,=ePp(L/X,,)2 1
h.s.
1,
(37)
where 1 includes also the summation over the momentum q parallel to the surface and At,, = ~/(2~~k~T) IL2is the thermal wave length. We then get N
N,(t=
co)
%(‘=
‘)
N
C edarr~ C, e-Si’l + L/Ath( 7:) I
j-o =
5 j=O
(38) e--si<, ‘ioe-&E;
+ L,h,,(T,)
’
j=O
3/2
,goe-
“‘1 + L/X,,( Ti) (39)
5 e-&f’f + L/A,,(T,) j-0
We note that only in the large volume limit L -3 00 will the pressures be in the ratio of the temperatures. For the present model system all terms in (38) and (39) contribute, resulting in a pressure ratio Pt/Pi = 34.4, whereas in an infinite box it would only be Pf/Pi = Tf/Ti = 1.2; the difference being due to a substantial shift in the chemical potential. Note from the top panel of fig. 1 that the final pressure is not reached monotonically, but that a slight overshoot occurs during the major part of the desorption process followed by a much
494
Z. W. Gortel, H.J. Kreuzer / Flash desorption and thermalizntion
1 0
t(s)
t(r)
Fig. 2. Flash desorption after a sudden temperature drop and flash desorption after another flash back to the initial high temperature in the He-graphite system. For other details see fig. 1
slower thermalization process during which wall collisions slowly adjust the occupation functions to the final equilibrium one. The desorption process thus puts the particles into continuum states that have too high an energy for T,. This implies that in a very large box the flux of desorbing particles, e.g. detected in a mass spectrometer, will not have a Maxwell velocity distribution, but rather one that is determined by the bound-state-continuum transition rates’ in the master equation. Indeed, deviations from a Maxwell velocity distribution have been seen for helium desorbing from a helium film [4] and in laser-induced thermal desorption of CO from an Fe surface [5]. For the lower panel of fig. 1 we have enlarged the experimental chamber to L = 30 cm. Fewer of the total number NT of particles are initially in the adsorbate (N,( t = 0)/N, = 0.877) and more of them desorb (NA( t = oo)/NT = 0.133) the flash desorption process being closer to an isothermal one. Also note that the pressure overshoot is relatively larger though the final pressure rise is lower by about a factor 3 compared to the smaller box in the top panel. In fig. 2 we look at the same system as in fig. 1 but enclosed in a yet larger box of length about 1 m. In the left hand side panel we first follow a flash adsorption experiment decreasing the solid temperature at time t = 0 from q = 7.4 K to T, = 6.17 K. The adsorption and thermalization process being over after a few seconds we then increase, in the right hand side panel, the temperature back to 7.4 K to monitor a flash desorption experiment. As the kinetic processes at the surface are mainly determined by the temperature of the phonon bath of the solid, the flash adsorption from 7.4 to 6.166 K is orders of magnitude slower than the flash desorption from 6.166 to 7.4 K. Also note from the dashed curve in the left hand panel of fig. 2 that the ansatz (36) with thi) the isothermal desorption time at Tf = 6.166 K is a rather poor description of the flash adsorption process indicating the importance of continuum-con-
2. W, Gortel, H.J. Kreuzer / Flash desorption and thermaiization
495
He-C 1, = 6.17KeTs = 13.3K - 1.0 300 - 0.8
Fig. 3. Flash desorption in the He-grapbite rise.
system as in fig. 1 and 2 but for a larger temperature
tinuum transitions to thermalize the gas phase during the course of the flash adsorption. Also note the slight undershoot of the pressure. Fig. 3 demonstrates the rather dramatic nonequilibrium effects in the He-graphite system after a substantial temperature flash of the solid from q = 6.166 to 13.33 K. In this situation the adsorbate evolution is very well described by the ansatz (35). Interesting is the fact that in the bigger boxes the pressure after rising substantially during a time of the order of the isothermal desorption time ty)( T = 13.33 K) = 3.2 x lo-’ s, remains constant for about a millisecond before it gradually decreases over about one second to its final equilibrium value.
496
2. W. Gortel, H.J. Kreurer / Flash desorption and thermalization
I
He-A T, = 4K+T,
0.6
= 6K
0.4
I
0.2
D
t(s) Fig. 4. Flash desorption of helium from solid argon. Debye temperature To = 92 K; Morse potential parameters Va/k, = 160.07 K and y - ’ = 1.98 A from ref. [9]. Mass ratio m/M, = 0.111.
We next turn to a system with more surface bound states, namely helium adsorbed on solid argon with Morse potential parameters such that 10 bound states develop. Fig. 4 shows a flash desorption in a box of 30 cm length from T, = 4 K to T, = 6 K. Although not many particles desorb, the pressures rises again substantially but in a monotonic fashion. Finally in fig. 5 we look at the He/LiF system with four surface bound
He-LIF T, = 3.65K
- o.4
Fig. 5. Flash desorption of helium from LiF; temperature flash from I; = 3.65 K to Tr indicated at the curves. Debye temperature Tn= 730 K; Morse potential parameters Vo/kB = 81.75 K and y-’ = 1.09 A from ref. [9]. Mass ratio m/M, = 0.152.
Z. W. Gortel, H.J. Kreuzer / Flash desorption and thermalization
497
states. Whereas a small flash from Ti = 3.65 K to Tf= 4.87 K produces a slight overshoot in the nonequilibrium pressure as we have seen in the He-graphite system, new features emerge at larger flashes. For a flash from Ti= 3.65 K to T,= 7.3 K, the pressure decreases slightly after about 10-l s to rise again to its final value. For a flash from T,= 3.65 K to Tf= 9.73 K, finally the pressure rises within roughly an isothermal desorption time t$‘)( T = 9.73 K) = 5 X lop6 s to a plateau from which another rise starts after some 0.1 s to the final equilibrium pressure.
4. Summary In this paper we have developed a theory to study transient effects in a flash desorption experiment. In situations where the mean free path for gas particle collisions is much larger than the size of the experimental chamber particles desorbing after the adsorbent temperature has been raised to its final value will thermalize due to wall collisions. Whereas the number of adsorbed particles decreases in a more or less exponential fashion within a typical flash desorption time, the gas pressure will rise with the same time scale but overshoots its final value by as much as a factor two depending on the temperature difference in the flash and the size of the vacuum chamber. The theoretical model assumes that all walls of the vacuum chamber are covered by the adsorbent (as is the case, e.g., with films deposited on the inside of glass systems) and are subject to the temperature flash. If only a small single crystal surface or a filament is the adsorbent, then the desorbing ‘particles will most likely thermalize with the other walls in the vacuum chamber. If the latter adsorb the gas readily, then a pressure device may be arranged to record the transient pressure only. It is important to note that it would be wrong to measure the number of desorbed particles and calculate a pressure using the ideal gas law. Such a procedure is only admissible for the final but not for the transient pressure. Our model is one-dimensional. This is appropriate for highly mobile adsorbates. It would appear that in a one-dimensional model only the momentum component perpendicular to the surface from which the particle desorbed will be thermalized. That this is not the case can be best seen in an idealized experiment for which we choose a perfectly cubic vacuum chamber with all surfaces covered by the adsorbent and uniformly flashed to a higher temperature. A particle leaving surface A with momentum pI perpendicular to A and momentum p,, parallel to it will either hit the surface B opposite to A thus thermalizing p I or hit a surface C perpendicular to A thermalizing the component of p,, perpendicular to C. In a one-dimensional model it is assumed that the thermalization of p,, proceeds at the same rate as that of pl . It is feasible to set up a three-dimensional model in which the angular distribution
498
2. W. Gortel, H.J. Kreuzer / Flash desorption and thermalization
of the desorbing particles is taken into account as well. We feel, however, that nothing qualitatively new can be learnt about the thermalization process that is not contained in the present model.
Acknowledgements This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. We would like to thank R. Teshima for developing the computer codes used in this paper.
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