1431
J. Phys. Chem. 1980, 84,1431-1445
Thermal Accommodation Coefficients Frank 0. Goodman* Department of Physics, CJniversity of Queensland, St. Lucia, Queensland 4067, Australia; and Department of Applied Mathematics, University Of Waterloo, Waterloo, Ontario, Canada (Received June 2 1, 1979)
The subject of energy, that is thermal, accommodation of gases at surfaces is reviewed in both its experimental and theoretical aspects. The intention of this review is to give participants in this symposium a sufficient background to allow them to more fully appreciate the outstanding contributions of Professor Lloyd B. Thomas to this field.
1. Introduction
This paper is not intended to be a major review of the entire field of the energy accommodation coefficient (EAC or a), Such a review would be untimely in any case, as there have appeared only a few papers in the field since my 1974 review;l also, the EAC has a whole chapter devoted to it in the 1976 monograph2 by Wachman and myself. Rather, it is intended to give readers a sufficient background in order that they may more fully appreciate the outstanding contributions which Professor Lloyd B. Thomas has made to his field during the past 4 decades. Lloyd Thomas has never been a prolific publisher of papers, but the papers which he and his colleagues have published contain work which is still essentially state-ofthe-art (this is true even of some of his oldest papers, for example, his classic papers with Brown3in 1950 and with Schofield4 in 1955), There are many gems of information still hidden in Lloyd’s desks in Columbia, and some of us have encouraged him for up to 20 years now to publish more of them: I think our greatest success was at the most enjoyable Oxford Symposium in 1966, at which he (together with Dr. D. Vincent Roach) presented516us with most valuable collections of data on the EAC. Since then, he has presented papers on the EAC at severa17-11subsequent symposia, as well as in the ordinary literature.12 His other, much earlier, work13J4on the EAC should also be mentioned. Lloyd‘s interest in the EAC has naturally led to work on other, related topics, for example, momentum ACs9J5and topics associated with the experimental setup and difficulties in AC measurements,16-20although the present review is restricted to the EAC. Over a century ago, Kundt and Warburg21discovered that the viscosity 7 of a gas is not independent of the gas pressure P, at low P , in contradiction with the now wellknown prediction22of the classical kinetic (dynamical) theorg of gases. They found that d7ldP > 0 at low P, and attributed this fact to gas molecular “slip” at a solid surface; that is, they attributed it to a nonzero average tangential velocity of gas molecules at the surface. MaxwelP then showed that a modification of his existing theory of gases could account for this slip phenomenon; he assumed that a fraction a of the surface scatters gas molecules diffusely, while the remaining fraction (1 - a)of the surface scatters gas molecules specularly. It soon became clear that Maxwell’s assumption implies not only velocity slip, that is a “velocity jump”, at the surface, but also a temperature slip, that is a “temperature jump”, if the gas and surface are not in thermal equilibrium. It should be noted that there are nonzero velocity and temperature jumps at a surface even with complete accommodation, a = 1(see text just after (2.7)). The existence of the temperature jump *Address correspondence to the University of Waterloo address. 0022-3654/80/2084143 1$01 .OO/O
was verified by Smoluchowski.24~25 Maxwell’s a used above is in fact equivalent to the EAC, although the term “accommodation coefficient” is due to Knudsen,26who defined a from where T , and Tg are the temperatures of the solid and incident gas, and T f is the effective temperature of the scattered gas (subscript f final), defined from
( E ) = CUT (1.2) where ( E ) is the average of E , the gas molecular energy, and C, is the streaming heat capacity of a gas molecule;lI2 for monatomic gases, for example
( E ) = 2bT
(1.3)
where b is the Boltzmann constant. The dependence of a on both Tgand T, is written explicitly in (Ll), although an “equilibrium EAC”, which depends on only one temperature, is defined by a(T) =
T.--T, lim= T
(”)
T, - T g
(1.4)
A third type of EAC may be defined in terms of average energies from
(E,- Ei) = { E , - E i ) a
(1.5) where Eiand Ef are the incident (or initial) and scattered (or final) gas molecular energies, and where (E,) is defined by (1.2) with T = T,.When the incident gas is associated with a thermodynamic temperature Tg, then (1.5) is equivalent to (1.1); (1.5) may also be used to define a when the incident gas is not associated with a thermodynamic temperature, for example, in a molecular beam experiment, although in my opinion the concept of an EAC is misleading in this case, and discussion is restricted to situations in which a thermodynamic temperature Tgexists. As is evident from (1.2), the above energies E are total energies of a gas molecule, and the EAC is the total energy AC. For monatomic gases, all energies are translational kinetic energies and a is the translational kinetic energy AC as well. as the total energy AC. For nonmonatomic gases, the different energies (translational, rotational, and vibrational) will accommodate differently in general during scattering by a surface, and sometimes different “partial EACs” are defined,l one for each degree of freedom, by analogy with (1.11, (1.4), or (1.5); that is, one defines atran,, arOt, and (Yvib, and it is important to bear in mind that none of these is in general simply related to a. Discussion is restricted to the total energy AC, although this will usually be also the translational kinetic energy AC because the gases will usually be monatomic. 0 1980 American Chemical Society
1432
The Journal of Physical Chemistry, Vol. 84, No. 12, 1980
Goodman
Perhaps Lloyd Thomas’ most useful achievements may + i o vacuum + to test gas ( a ) be summarized as follows: (a) he has made us fully aware of the extreme importance, first recognized by R0berts,2’-~l of controlling the surface conditions during EAC measurements; (b) he has taught us how to control the surface gloss tube filament at Ts conditions, how to monitor these conditions using the EAC as an indicator, and in particular how to obtain clean surfaces; and (c) he has presented us with clean-surface EAC measurements, the results of which are now regarded as “standard” clean-surface values. As an example, let us tube w a l l s a t Tw consider his standard clean-surface 4He-W(filament) result6Ss a(303 K) = 0.0166 f 0.0004; it is generally accepted that any surface contamination would increase a above this value. If any laboratory in the world were to repeat Thomas’ experiment on this system today and obtain, say, 4303 K) = 0.0176, then this would be regarded as an excellent result, but not quite as good as the standard result, most probably because of some residual surface contamination, perhaps by 0 atoms. If time and/or other constraints allowed, there is little doubt that the laboratory in question would strive harder to more carefully control the experiment and to get closer to the standard result. The EAC is such a sensitive indicator of surface condition, however, that even surface contamination as low as about 0.1% may be sufficient to yield the laboratory’s “dirtysurface” value. Indeed, there is the possibility that the present standard value is not the true clean-surface value, because of some contamination which even Lloyd Thomas’ meticulous methods have failed to eliminate. It is interesting to speculate, but not too seriously, on what may Figure 1. (a) Schematic diagram of the Knudsen cell used for measuring the EAC. (b) Dlagram of an experimental cell. (From Thomas’ and transpire if our laboratory above obtains, say, a(303 K) = Goodman and Wachman’.) 0.0156 instead! Apart from Lloyd Thomas and c o - w ~ r k e r sanother ,~~~~~~ Schofield4and Thomas: for example. The filament temgroup which has clearly mastered the art of controlledand C, is accurately perature T, is accurately mea~urable?~ surface EAC work is that of M e n ~ e l . Wachman, ~ ~ ? ~ ~ one known, so it remains only to specify I and (Ei),and the of Lloyd‘s students, has also presented data in this class.35n36 method of specifying these quantities depends, essentially, 2. Experimental Methods for Measuring the on P. There follows a brief description of the two main Energy Accommodation Coefficient experimental methods, called for convenience the lowpressure (LP) method and the high-pressure (HP) method: 2.1. The Knudsen Cell. Attention is confined here in the LP method, P = 0.01-0.1 torr, whereas in the HP mainly to the two experimental methods which incorporate method, P = 10-100 torr. (The name “HP method” has the Knudsen ~ e l l (Figure ~ ~ 3 ~l ) ,~as essentially all reliable not been used in the literature before now.) data have been obtained therefrom. Other methods are 2.2. The LP Method. The pressure P is chosen so that, mentioned briefly (section 2.5), but are not described in after interaction with the filament, a gas molecule will detail. generally have many encounters with the cell walls before It follows from the definition (eq 1.5) of a that one must returning to the filament. In the LP method, then, Tgand measure ( E , - Ei) and ( E , - Ei) in order to determine a. (Ei)are essentially equal to T , and CUTw, respectively, and In the Knudsen cell, the test gas is held in a cylindrical it follows from (2.2) and (2.3) that container, close to the axis of which lies a thin electrically conducting filament whose surface is the test surface. The a = J(~TM,~T,)‘/~/PC,AT (2.4) walls of the container (cell) are held at temperature T,. An electric current is passed through the filament, thereby where raising the filament temperature T, above the gas temAT = T , - T , (2.5) perature. The power loss per unit filament area, J , to the When measuring the equilibrium EAC (1,4), AT is chosen gas is m e a ~ u r a b l ein~terms ~ of the filament current and to be as small as possible, bearing in mind that J must be resistance, and we may write sufficiently large to be accurately measurable; a common (2.1) (Ef- Ei) = J / I p r ~ c e d u r e is ~ to ~ ~plot ~ ’ a~ vs. ~ ~AT and extrapolate to AT where I is the gas molecular intensity = 0. 2.3. The HP Method. The pressure P is sufficiently I =P/(~TM,~T,)~/~ (2.2) large to allow a temperature gradient to build up between It follows from (1.2),(1.5), and (2.1) that the filament and the cell walls, and temperature jumps (section 1)appear at the filament surface and at the cell walls (Figure 2). Two different approximations have been used in conjunction with the HP method, called2the temperature-jump (TJ) and the mean-free-path (MFP) apThe tedious corrections (for example, power losses from proximations. the filament by conduction through its ends and by ra2.3.1 The TJ Approximation. A temperature T(x)is diation) necessary to get accurate values of J from meaassumed to be meaningful for all r C x C R (Figure 2 ) , and sured power losses are not discussed here: see Thomas and I
The Journal of Physical Chemistry, Vol. 84, No. 12, 1980
Thermal Accsmmodation Coefficients T
1433
't i
Tr \
I
\
'\
\ \
gradient = T ' [ R )
---_ ;
dw I
0
R
0
Figure 2. The temperalure Tin the Kndsen cell vs. the radial distance x from the filament axis, under conditions relevant to the HP method. (From Goodman' and Goodman and Wachman*.)
the functional dependence of T ( x )in the bulk of the gas is assumed to hold throughout this region. A crucial assumption which is made is ( E f Ei) = 2CuT, (2.6) which says that, of the gas molecules at the filament surface which are supposed to be at temperature TI,half have average energy (E,) and half have (Ei).It follows from (2,1), (2.2) with T g= T I ,(2.3), and (2.6) that
+
(2 - a ) (2.lrM&T1)~"J T, - T, = --___ (2.7) 2a cu p which is the fundamental TJ relationship. It is from (2.7) that we see clearly the nonvanishing of the temperature jump ( T , - T,) if LY = 1. One procedure39is to plot T, VB. P1 with the hope of obtaining a straight line, when a may be found in the standard way from the slope and intercept. Another common p r o c e d ~ r e ~is~to j ~introduce ~ , ~ ~ the thermal conductivity K of the gas and to write the heat conduction equation (for our case of cylindrical symmetry) as follows: Jr/x = -KT'(x) (2.8)
It folllows from the geometry of Figure 2 that T , - T , = -d8T'(r) (2.9a) T R - T , = -d,T'(R) (2.9b) and from (2.7), (2.8) with x = r, and (2.9a) that (2- a) - CZd,/K (2.10) 2ff (27~M,bT,)'/~ Assuming K is constant, (2.8) is integrated from x = r to x = R, and it may be shown from (2.5), (2.81, and (2.9) that A T / J = ( s / K ) In ( R / r ) -t (Pd,/K)P1 (2.11) where it has been recognized that rd,/Rd, is small. The product Pd, is nearly independent of P, and hence plotting AT/J vs. P Lshould yield approximately a straight line of slope m, say, and it follows from (2.10) that (2- a) = '~CY
mC" ~-
(2~M,bT,)~1~
(2.12)
The TJ theory is not convincing, particularly the assumption that T(x)is meaningful in the entire region from the filament surface to the cell wall, which gives specification of (Ei)through (2.6). The MFP approximation bypasses this assumption, giving a different specification of (Ei).
r
R- L
r t l
R
X
Figure 3. As in Figure 2. Definitions relevant to the MFP approximation are used. (From Goodman' and Goodman and Wachman'.)
2.3.2. The MFP Approximation. In this approximation, due to W a ~ h m a n (Ei) , ~ ~ is specified by (Ei) = CuTl (2.13)
where Tl is supposed to be the gas temperature a radial distance 1 (about one mean free path) from the filament surface (Figure 3); now (2.4) applies with T , = Tl and AT = (T, - T J ,that is (2.14) CY = J ( ~ T M , ~ T J ~ / ~ / P C Ti) ,(T, and it remains to specify Tl. The heat conduction equation (eq 2.8) is again integrated, this time from x = (r 4- 1) to x = ( R - L ) , and assuming that
K = KIT"-l to obtain T l - TL = (aJr/KJ In ( ( R - L ) / ( r + 1))
(2.15) (2.16)
where L (about one mean free path) and TL are defined near the cell wall, by analogy with 1 an Tl near the filament surface (Figure 3). TL is calculated by defining a, as the EAC of the gas at the cell wall, when (2.4) again applies but with T , = TL,AT = TL - T,, J = J r / R , and a = a,, that is a, = ( J ~ / R ) ( ~ ~ M , ~ T L ) ~ ' -~ T,) / P C (2.17) ~(TL
this equation is solved for TL, and a is calculated from (2.14) and (2.16). We note that five quantities (a,, a, K1, 1, and L ) must be specified before a is calculable. 2.4. Comparison of the Methods and Approximations. There is no doubt that data from the L P method are strongly preferred in general by workers in the field to data from any other method. I am not convinced that the TJ approximation in the HP method has a sound theoretical basis, and others agree,42v43 although it seems that the theory has been verified in the sence that the TJ approximation gives reproducible values of a in agreement with corresponding values from the LP m e t h ~ d . " ~ J ~ . ~ ~ Although Wachman's MFP approximation in the H P method avoids the major uncertainty in the TJ approximation, the fact that the five quantities a,, a, K1, 1, and L must be specified before a is calculable must surely detract from the value of the approximation, particularly as two of the five are mean free paths, notoriously misleading quantities. 2.5. Other Methods for Measuring the EAC. Other methods for measuring a have been used to a far smaller extent than have the LP and H P methods. The Carpenter-Humphries-Mair (CHM),44molecular beam (MB),46g46 and moment47methods were discussed either in (CHM and MB) or before (mornent)l2ref 1, and are not discussed
1434
The Journal of Physical Chemlstry, Vol. 84, No. 12, 1980
Goodman
here. Two interesting methods have appeared since ref t 1, the acoustical method of Shields,48based on earlier work of Shields and Faughn,” and the vibrating-surface method of Lemons and R o ~ e n b l a t and t ~ ~later ~~ D r a ~ e r . ~ ~ ~ ~ ~ Acoustical Method. In deriving his equations for the speed and absorption of sound in cylindrical tubes, K i r ~ h h o f f assumed 6 ~ ~ ~ ~ that the gas particle velocity was zero, and that the temperature was continuous, at the tube walls. These assumptions are valid if the gas pressure is sufficiently large but, as we have seen above, they fail at lower pressures due to the appearance of the velocity and o 0 He temperature jumps (even with complete accommodation). Shields and F a ~ g h corrected n~~ Kirchhoff s equation for t the tangential velocity and temperature jumps, relating I I I I 200 400 600 T ( K ) 800 them to the tangential momentum and energy ACs, and it became clear that measurements of the speed and abFigure 4. Experimental data on a(T ) for He and Ne on W. (From sorption of sound in tubes could be used to infer values Kouptsidis and M e n ~ e I Goodman,’ ,~~ and Goodman and Wachman.*) for the ACs. The results48are clearly not yet good enough to be clean-surface values, but the method has the advantage over the LP and HP methods that it is not restricted to surfaces in the form of electrically conducting Xe filaments. I ** No correction has been made yet for the normal velocity jump, which is related to the normal momentum AC. (As a matter of fact, I have not seen the normal velocity jump mentioned before now.) Vibrating-Surface Method. When a solid surface is vibrated at high frequency in a gas at low pressure, it c I I I I 0 200 400 800 experiences a temperature increase due to the nonequi6oo T ( K ) librium nature of the gas relative to the surface. The Figure 5. As in Figure 4, except that Ar, Kr, and Xe on W are conresulting steady-state unequal gas and surface temperasidered. tures may be related50directly to the quantity a / € ,where = 0) values of (Y and the highest values of da 88, He and is the thermal emissivity of the surface. Again, the reNe being the most popular candidate^.^^^^-^^^ Bes ~ l t s are ~ not ~ ~clean-surface ~ ~ - ~ ~values of a , but the mecause of the relative difficulty in quantitatively interpreting thod has the same advantage as does the acoustical method type 3 data,l discussion is restricted hereafter to data of of being not restricted to electrically conducting filament types 1 and 2. surfaces. Further, both translational and internal EACs Data of both types 1 and 2 are referred to as “cleanare ~ b t a i n a b l e ,whereas ~ ~ ! ~ ~ only total EACs may be insurface” data in the literature because the contamination ferred from the LP, HP, and acoustical methods. A disby the test gas with type 2 data is unavoidable. Whether advantage is that t must be known before a is calculable, data from a particular clean-surface experiment are of type although it seems that the quantity a / €is itself important 1 or 2 depends to a large extent on the quantities Q/ bT in some contexts.50 and Q/bT,, where Q is the heat of adsorption. If both of 3. Experimental Data these quantities are small, then the equilibrium coverage 8 will be small but P dependent in all experimental ranges 3.1. Type 1 , Type 2, and Type 3 Data. Experimental of P; if these ratios are large, then 8 may be of the order details, such as (a) methods of preparing atomically clean of a monolayer and may approach saturation. surfaces, (b) procedures for minimizing the contamination 3.2. Equilibrium Data of Types 1and 2. Most people problems during an experiment, (c) procedures for obwould agree that the most reliable data are the L P data taining accurate values of P, T,, and so on, and (d) other of tho ma^^*^ and of Kouptsidis and MenzeP4 on the “nuts-and-bolts-type” details, developed by Thomas and equilibrium a(T) for He, Ne, Ar, Kr, and Xe on W, in the ~ o - w o r k e r sare , ~ perforce ~ ~ ~ ~ ~omitted ~~ here. temperature range 77-373 K (Figures 4 and 5). The other The recognition’ that there are three essentially different data outside this temperature range are LP or HP(TJ) types of data, resulting from three different types of exdata and are less reliable. The data shown in Figures 4 perimental conditions attainable with controlled surfaces, and 5 contain examples of what are believed to be all seems to be importanta2 possible types of behavior of the equilibrium a(T),and our Type 1 data are from experiments in which the surface discussion of interpretation of ( ~ ( 7 is ‘ ) almost entirely contamination is low enough to be negligible. confined to these data. The He and Ne data (Figure 4) Type 2 data are from experiments in which the surface are believed to be of type 1,and the Ar, Kr, and Xe data contamination by impurity gases is as in type 1, but in (Figure 5 ) of type 2.2 which there is some unavoidable contamination by the test Useful and reliable data on 3He and 4He on W have also gas itself, in dynamic equilibrium at constant coverage 8. been presented, the first set by Thomas5and later refined Type 3 data are from experiments as in type 1or 2, but sets by Kouptsidis and MenzeP3and Menzelsg(Figure 6). with a partial or complete surface coverage of an adsorbate These data are particularly useful in testing theories, begas which has been purposely introduced in a controlled cause 3He and 4He should behave essentially identically manner. except for effects of the difference in atomic mass. I quote Type 3 data are relevant to experiments on the effect from Lloyd Thomas’ Oxford Symposium paper:5 “We on a of adsorbed gases, experiments which may in fact use know of no properties of these species, other than atomic the value of a as an indicator for adsorption. The best weight, which should affect the EAC and hope that these indicator gases are those with the lowest clean-surface (0 r
.
.
i
5936v57v58
The Journal of Physical Chemistry, Vol. 84, No. 12, 1980
Thermal Accommodation Coefficients
1435
C 217,
1
L
C 016
k---
0
I
_7_
100
I
I
200
300 T ( K )
400
I
t
0 011
Flgure 0. Experimental data on a( r ) for 3He and 4He on W. Straight lines are drawn between neighboring pairs of data points. (From Koupteidis and M e n ~ e ! ,Menzel,” ~~ Goodman,’ and Goodman and Wachman.’)
TABLE I: Nonequilibrium EACs from Ref 60a
Ts,K 1073 1335 1568 1785 a
lOOa (He-W)
1.8 f 1.8 f 1.8 * 1.8 c
7.0 8.5 10.0 12.0
:3’2
1 OOa ( Ar-W) 25 i: 5.5 25 i: 6.0 22 i: 6.5 23 c 7.0
The gas temperature Tg is about 303 K in each case.
studies may effectively isolate [the] relationship of atomic weight of the gas to thermal accommodation”. Several other sets of a(T) data of types 1 and 2 have been presented, for several different gases on several different surfaces. These are not discussed further because the data in Figures 4-6 are sufficient for our limited purpose here. It if3important to note the following features of the data in Figures 4 and 5. (A) For Xe and Kr, da/dT < 0 and it is probable that a(0) = 1. (B) For He and Ne, the a(T) have minima with Tminx 40 K and 180-280 K, respectively. (C) For Ar, cr(T) may have a shallow minimum with Tminin the range 500-700 K. 3.3. Nonequilibrium Data of Types 1 and 2. The early clean-surface studies by Thomas and Schofield4 of the dependence of a(Ts,Tg)on AT (= T, - T ) using the LP method, indicated that, at least for He-8; with AT > 0, a is independent of AT, and hence of T,, at given Tg. The most comprehensivestudies of the dependence of a on AT are those of Kouptsidis and M e n ~ e 1in , ~which ~ a is measured for He, Ne, Ar, Kr, and Xe on W, by the LP method, over the range 77-318 K of Tgwith ranges of AT of about 20-80 K. The data for the extreme cases, He and Xe, are shown in Figures 7 and 8. Much larger values of AT were used by Watt, Moreton, and Carpentereofor He and Ar on W (Table I). For He at all experimental Tgand for Ne at all but the lowest Tg, a(Ts,TJ is independent of T,, confirming Thomas and Schofield‘s4earlier work. For Ne at the lowest Tgand for Ar, Kr, and Xe at all experimental Tg!a(T,,T,) decreases with increasing T,, the decrease being more prominent for lower T,and higher Mr The He and Ne data, except perhaps for Ne at the lowest T are probably of type 1,and the other data, including pergaps Ne at the lowest T9,are probably of type 2. A possible link between the dependence of a(Ts,Tg)on T, discussed1 in the previous paragraph and the type (1or 2) of data involved was suggested in ref 1. For type 2 data, the surface coverage 8 should decrease with increasing T,, and the T, dependence of the data may be due to a com-
0
20
LO
60
EO
AT(K)
Flgure 7. Data on the nonequilibrium a for He-W. The temperatures on the right are gas temperatures Tg;the surface temperatures T, are given by T, = Tg4- AT. The different symbols at the same Tg indicate measurements at different P. (From Kouptsidls and Menzel,” Goadman,‘ and Goodman and Wachman.’) 0,95
0.85
-**.*.*
r
,0-
T
T
l-
60
80
a
0.8
0,7
0.5 0
20
O I AT( K )
Flgure 8. As in Flgure 7 except that Xe-W is considered.
bination of a dependence of a on 8 at constant T , and a dependence of a on T , at constant 8. See ref 1 or 2 for more detailed discussion, but it turns out that the simplest assumption which is consistent with all the experimental data is
(aa/aTs)Tg,e =0
(3.1)
although the only cases in which (3.1) is known to hold are the particular type 1 (8 = 0) cases mentioned in the previous paragraph. The assumption 3.1 is clearly attractive, particularly from the theoretical point of view, and will be referred to again in sections 4.2 and 4.3. 3.4. Other Experimental Data. Other experimental data are not discussed here. For further discussions and for references on the topics of (a) EAC data of type 3 and
1436
The Journal of Physical Chemistry, Vol. 84, No. 12, 1980
Goodman
E s = i M s ~ i = O
Figure 9. The Baule hard-spheres model of an encounter of a gas atom with a surface atom. (From Goodman’ and Goodman and Wachman.*)
(b) EAC data for non-monatomic gases (including data on the partial EACs), readers are referred to ref 1 and 2. 4. Classical Theory of the Energy Accommodation Coefficient 4.1. The Hard-Spheres Model. The first theory of the EAC was that of Baule,61who modeled the gas molecule (atom) and the surface atom by hard spheres (Figure 9). With the surface atom initially at rest, it follows from the definition (eq 1.5) of a, with (E,) = 0, that a = 1- (E,)/Ei (4.1) where (Ei)is set equal to Ei because a is independent of Ei in the hard-spheres model. For the encounter shown in Figure 9, it follows from (4.1) and the conservation of energy and momentum that, for fixed $ a = 4p cos2 $/(l+ p)2 (4.2) where
p
is the mass ratio, defined by P = Mg/M,
(4.3)
Two different “Baule results” are quoted in the literature, one of these being the “head-on” result for $ = 0 a = 4p/(1
+ p)’
(4.4)
and the other the more correct, $-averaged result61 a = 2p/(1
+ p)2
(4.5)
The “actual” hard-sphere value of a lies between (4.4) and (4.5) because of multiple-collision and shielding effects which occur at larger ; ) I more accurate calculations a = 2.4p/(1
+ p)2
Figure 10. The classical scattering of a gas atom by a lattice model. The gas atom interacts in general with each of the lattice atoms, this interaction being represented by a pairwise potential V(u), where u is the separation of a gas atom and a lattice atom. (From Goodman.‘)
by the surface of a three-dimensional lattice model (reasons why the model must be three dimensional are first given in ref 64), and the scattering of a gas molecule (atom) by this surface is determined by calculating the classical trajectory of the atom in the presence of the lattice (Figure 10). The general interaction implied in Figure 10 has not in fact been calculated in the context of the EAC, because the calculations involved are too complicated and time consuming.l,‘ The simplifications made are described in the following paragraph. Only the relatively large gas-surface repulsive forces are important as far as gas-surface energy transfer is concerned, and the gas atom is generally very close to at most one surface atom at any one time; so it is assumed that the gas atom interacts directly with only one surface atom, other solid atoms being involved indirectly via the lattice spring forces. It is then assumed that the three-dimensional nature of the gas atom velocity is unimportant when energy transfer is considered, and that the gas atom motion may be restricted to the one dimension normal to the surface. Thus the trajectory of the gas atom must be such that a “head-on” encounter with a surface atom occurs. Because of the difficulty of dealing with the thermal motion of the lattice for nonzero T,, it is assumed that (3.1) holds, that is, that the nonequilibrium a(T,,Tg)is independent of T,, allowing the following enormous simplification for the equilibrium a(T): a(T) = 4 , T )
(4.6)
The hard-spheres EACs (eq 4.4-4.6) are limiting values, as the incident speed u i becomes large, in more general models. Even the simple Baule model gives a reasonable qualitative interpretation of the dependence on p of a, that is, a is an increasing function of p (Figures 4-6). However, the experimental dependence of a on p is complicated by the fact that larger values of p are generally associated with larger gassolid interaction potential well depths, W ,which are also associated with larger a. Even the apparently straightforward isotopic dependence of a (Figure 6) is complicated, because atoms of different isotopes have different speed distributions at the same Tg,and a depends strongly on incident gas atom speed (a dependence which is absent from the Baule model). 4.2. The Lattice Model. In spite of some serious defects, the classical lattice (based on earlier work on one-dimensional lattices by Cabrera67and ZwanzigGs)was the first to give a quantitative interpretation@,@ of the EAC over a wide range of experimental conditions (essentially those in Figures 4 and 5). The solid surface is represented
(4.7)
that is, T , = 0 and the thermal motion is absent. With the simplifications described in the preceding paragraph, the equations of motion of the gas atom (position zg) and the surface atom (position z,) a r e l ~ ~ ~
z,(t) = M;’ltdT 0
X(7) f ( t - 7)
(4.9)
where zg and z, increase into the gas phase, u,, (< 0) is the initial velocity of the gas atom, subscripts 0 refer to time t = 0, X(t) is the “response function” of the lattice surface atom for motion in the normal direction, and f ( t )is the force on the surface atom, given by f ( t )= V’(U(t)) (4.10) where a@) = zg(t)-
(4.11)
V(u)is an empirical pairwise interaction potential, usually
Thermal Accommodation Coefficients
taken to be a Morse potential, defined from V(a)/W = exp(-2aa) - 2 exp(-aa)
The Journal of Physical Chemistry, Vol. 84, No.
where 2 ~ ish the I’lanck constant and where w, is the maximum of the modal frequencies of the lattice. It turns out tlhat,l~~~-@ for this interaction model, the two parameters a and 8, enter the theory only in the combination d,/a, the physical reason for this being that urnaxlais a characteristic interaction speed. At time t = -m, the gas atom approaches the lattice from z = a: with a (negative) velocity ui, and the important result of the calculation is uf, the (nonnegative) velocity of the gas atom at t = m. For a given gas-surface system, that is, given M,,M,,H,, V(a), or given Mg,M,,8,, a, W if V(a) is a Morse potential, the calculation proceeds as follows: (1) A velocity ui of the gas atom is chosen. (2) An initial gas atom position zgo (= uo) is chosen sufficiently far from the surface (that is, zgo is chosen sufficiently large) in order that no significant interaction occurs before t = 0, allowing us to write uo = ui (4.14) (3) The equations of motion (4.8) and (4.9) are integrated by using (4.10) until either (a) z,(t) reaches zgo again, in which case significant interaction has ceased, or (b) “trapping” of the gas atom occurs. On this model trapping may be defined to occur if the energy E(t)of the gas atom, given by
E(t3 = ‘/zMgu2(t)+ V(a(t))
(4.15)
becomes negative during the interaction. The gas atom velocity when z,(t) reaches zgo again is denoted by ul, and, by analogy with (4.14) uf = u1 (no trapping) (4.16a) uf = 0
(trapping)
(4.16b)
We note that uf = 0 for a trapped gas atom because it will be eventually “desorbed” at temperature T,= 0. The results are usually presented in the form of the dependence on Ei of an “effective EAC of a single gas atom”, denoted by y, and defined by analogy with (1.5) with ( E s ) =: 0: y(EJ = AE/E;
(4.17)
where A E is defined by
AE
.trapping region ( E ; c E i c )
(4.12)
Absence of thermal motion accounts for the simplicity of (4.9), in which the initial position zs0 of the surface atom is arbitrarily set equal to zero. The zero for the gas atom position z is chosen in order to eliminate the usual additional dorse length parameter am,which is arbitrary in one-dimensionalcalculations, from the exponents in (4.12). The response function X ( t ) depends on the particular lattice model chosen. Here, I consider my simplified m 0 d e l , ~ 9which ~ ~ ~incorporates ~ central and noncentral restoring springs with equal force constants, and for which X ( t ) depends on only the characteristic temperature 8, of the lattice model, defined from b8, = hum, (4.13)
12, 1980 1437
0
Eic
Ei
E i ;in
Flgure 11. Qualitative form of the y(EJ curve from the lattice theory. The high-energy limit y(m) is given by (4.4). The minimum value ymln occurs at an energy E,, of the order of W . The largest value of E, for which trapping occurs (y = 1) is the critical incident energy E, for trapping. (From Goodman‘ and Goodman and Wachman.’)
The two different methods of calculating AE form a useful check on the integration procedure. For small and large Ei, the forms of y(Ei) are well known:1,66@
?(Ei = small) = Ei,/Ei y(Ei = small) = 1
Ei 2 Ei,
Ei 5 Eic
(4.18)
and is calculated either (a) from the values of ui and uf or (b) from integration over time of the power supplied to the lattice atom during the interaction: (4.19)
(4.20b)
where Ei, is the critical value of Ei for trapping, and y(Ei = large) = y ( m ) ( l - cE;l
+ ...)
(4.21)
where y(m) is the Baule head-on value (4.4) and where c is a positive constant. It follows that y ( E J has at least one minimum, which occurs at an energy Ei given approximately by the interaction potential depth W , the simplest possible form being that in Figure 11, and calculations from the lattice theory confirm this qualitative form for all gas-solid systems; the exact form of y(Ei) depends on the particular system considered, and particularly on the three parameters p, 8,/a, and W. A brief explanation of the physical basis for the shape of the r(Ei)curve in Figure 11 may be in order. The nonconservative energy transfer from the gas atom to the solid occurs mainly during the repulsive part of the int e r a c t i ~ n . l For * ~ ~small ~ ~ ~Ei such that Ei > Wand Ei >> b8, the conditions for a hard-spheres interaction are approached; slower interactions will be more conservative than faster ones, however, and (4.21) holds, giving the positive slope of y(Ei) for Ei, < Ei < a. It follows from (1.6) and (4.17) that the relation between a and y is
wa
a = (Eiy(Ei))/(Ei)
(4.22)
because the model is one dimensional as far as the gas is concerned, the averages in (4.22) are over the appropriate of Ei one-dimensional distribution fiD(Ei) fiD(Ei) = ( bTgF1exp(-Ei/ b T,)
= Ei - Ef
(4.20a)
(4.23)
for example, (Ei)= bTg for this model. It follows from elementary considerations that the theoretical a(Tg)curve (Figure 12) also has the same qualitative form for all gas-solid systems, with a minimum which occurs at a temperature Tg,, of the order of W/b. In fact, proponents of the flat-surface (“cubes”) mode l of gas-surface ~ ~ scattering ~ ~ (section ~ 4.3.2) may argue
1438
The Journal of Physical Chemistry, Vol. 84, No. 12, 1980
0‘
I ’
Tg min
Figure 12. Qualitative form of the a( T,) curve from the lattice theory. The high-temperature limit a ( m ) is equal to y(m) and is given by (4.4). The minimum value ami, occurs at a temperature T&, of the order of W l b . (From Goodman’ and Goodman and Wachman.*)
formula
A K
He Ne Ar Kr Xe
290 220 250 250 250
0.2 1.8 70 155 380
100
200
300
400
TI K)
Lattice theory
M a , (g gas K/mol)1’2 T o ,K 1.95 2.0 2.2 2.4 2.4
9
0
Flgure 13. Comparison of the lattice theory (- - -) and the formula (4.25) (-) with experimental data on a(r ) for He-W. The lattice theory and formula parameters are given in Table 11. (From Goodman and W a ~ h m a n ~ and , ’ ~ Goodman.’)
TABLE 11: Parameters from the Lattice Theory and Formula 4.25 Used in Ref 74 to Fit the Experimental Data on W
a
Goodman
W,
Q,a
kcal/mol kcal/mol 0.21 0.50 2.4 4.0 6.4
-1.9 1 at high T because of the mass-ratio effect as the hard-spheres limit (eq 4.6) is approached, whereas R < 1 at low T because the effect of the larger speeds of the less massive 3He dominates over the mass-ratio effect, and larger speeds lead generally to larger values of a. The data shown in Figure 16 are insufficient to either confirm or deny this prediction, although the more recent data in Figure 6 indicate that the prediction may be false. Effects of a nonzero surface temperature T, on the calculations have been considered by Trilling78J9using his continuum model of the surface, and by others7@72p80 using concluded various forms of the cube model. that the nonequilibrium a(T,,T,) should be essentially independent of T, if Tgand T, are both greater than W / b; he discussed also other cases, and obtained qualitative agreement with the data of Kouptsidis and MenzeP4 for the inert gases on W (Figures 7 and 8 show two examples of these data). 4.3.2. Flat-Surface (Cubes) Models. At relatively low incident gas atom energies, it is known that, for many
rigid wall Figure 17. The soft-cubes model of the interaction of a gas atom with a solid surface. (From Goodman‘ and Goodman and Wachman.*)
TABLE 111: Parameters from the Soft-Cubes Theory Used in Ref 7 2 to Fit the Equilibrium ( E ) and Nonequilibrium (NE) Experimental Data on Wa soft-cubes theory gas
type
He He Ne A r Ar K r Xe
E NE
E E NE E
E
ocla,
A K
140 140 140 110 120 110 130
*
W, Q, kcal/mol kcal/mol 0.1
2.1
0.1
0.7 1.9 1.9 4.5 9.0
1.6 -1.9 -1.9
