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Kinetic lattice gas model in one dimension
Surface Science 291 (1993) 242-260 North-Holland
surface science
Kinetic lattice gas model in one dimension I. C a n o n i c a l a p p r o a c h S.H. P a y n e , A. W i e r z b i c k i a n d H.J. K r e u z e r Department of Physics, Dalhousie University, Halifax, NS, Canada B3H 3J5
Received 17 December 1992; accepted for publication 1 March 1993
A closure approximation for the one-dimensional kinetic lattice gas model is formulated to truncate the hierarchy of equations of motion for correlation functions. Essentially exact results are obtained both for the equilibrium properties and the time evolution of the adsorbate when up to five-site correlation functions are included. Implications for thermal desorption in systems with slow and fast diffusion are studied for nearest-neighbor interactions.
I. Introduction The kinetic lattice gas model for surfaces was originally set up in close analogy to the kinetic Ising model for magnetic systems [1]. It is based on a master equation that treats adsorption, desorption and surface diffusion as Markovian processes. To obtain the time evolution in an adsorbate, one can solve the master equation directly, either by matrix diagonalization [2] or by renormalization group techniques [3,4] or by Monte Carlo methods [5,6]. Recently, a perturbation approach has been suggested that constitutes a time-dependent generalization of Kikuchi's method in the equilibrium statistical mechanics of the lattice gas [7]. In another approach [8-13] one first derives a hierarchy of equations of motion for n-site correlation functions which must be truncated via a closure approximation to yield a finite set of coupled equations. The n-site correlation functions are of course site specific, i.e. they can vary across the adsorbate and thus can describe inhomogeneities. In practice, however, this approach is only useful as long as the adsorbate remains homogeneous in which case the coverage and a few site correlation functions are sufficient to describe the equilibrium and non-equilibrium properties of the adsorbate. If the lateral interactions between adparticles are attractive the adsorbate will develop two-phase regimes below critical temperatures and the above approaches become untractable. To understand this point we briefly look at the canonical and grand-canonical approaches to calculate the equilibrium properties of the adsorbate. As long as the partition function is evaluated exactly the two approaches yield identical results. However, when approximations are invoked, such as the mean field theory, the inherent assumption of homogeneity introduces unphysical features in the canonical ensemble such as van der Waals loops. This is discussed at length by Hill [14], who also points out that, to avoid such artifacts, it is better to work in the grand-canonical ensemble. When a system is taken out of equilibrium these difficulties are compounded even more. However, a solution has been put forward recently that is based on a time-dependent generalization of the grand-canonical ensemble [15]. As with any other new theory some intuition must be developed for the new concepts and this is best done in an exactly soluble model. We will therefore, in this and a following paper, revisit the one-dimensional kinetic lattice gas model, in the canonical and the grand-canonical approaches to develop approximation schemes that can later be generalized for two-dimensional systems. Some exact results and various closure approximations for the linear Ising lattice have been obtained to study finite 0039-6028/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
243
chain effects [16], reactions [17-20], k-mer desorption [20,21], etc. The one-dimensional kinetic lattice gas model without surface diffusion was treated by Evans et al. [22]. For systems with fast diffusion the desorption kinetics are known exactly, because the adsorbate will then remain in quasi-equilibrium throughout desorption and the desorption rate is given by [23,24] dOdt
des = - S ( O , T ) a s 2 ~ r mhak E T 2
exp[/ (0,
(1)
in terms of the chemical potential, /~(0, T), of the adsorbate to be taken at the instantaneous coverage, 0 = O(t). Here S(O, T ) is the coverage and temperature-dependent sticking coefficient and a s is the area of one adsorption site. Because the chemical potential can be calculated exactly, the desorption kinetics is known equally well according to eq. (1). However, in situations where diffusion is slow on the time scale of desorption, one must resort to kinetic equations, e.g. those obtained from non-equilibrium thermodynamics [24-26] or from the kinetic lattice gas model [8-13]. This paper is concerned with the canonical approach. The results will then form the basis for a grand-canonical description in a subsequent paper. In the next section we collect the relevant formulae for the kinetic lattice gas model. In section 3 we will describe a procedure to obtain the hierarchy of equations for the n-site correlation functions, look again at the equilibrium properties in section 4, and define a decoupling s c h e m e used for the closure of the hierarchy in section 5. Numerical examples for desorption are studied in section 6 for slow, intermediate and fast diffusion. For slow diffusion we demonstrate the strong dependence of the kinetics on the initial conditions including a discussion of the usefulness of the Arrhenius parametrization. In section 7 we briefly comment on the coverage dependence of the sticking coefficient. We conclude with a comment on the implications for the two-dimensional lattice gas. 2. The kinetic lattice gas model
To set up the lattice gas model, one assumes that the surface of a solid can be divided into cells, labelled i, for which one introduces microscopic variables n i 1 or 0, depending on whether cell i is occupied by an adsorbed gas particle or not. To introduce the dynamics of the system one writes down a model Hamiltonian [11-12] -----
1
(2)
H = E s ~ n i + ~ V 2 E ninj + " ' ' . i
(ij)
Here E s = - V o - k B T ln(q3qi,t)
(3)
is a single particle energy with V0 the depth of the surface potential, q3 is the single particle partition function accounting for the vibrations of the adsorbed particle in the surface potential and qint is the internal partition function of the adsorbed particle. V2 is the two-particle interaction between nearest neighbors (ij). The coupling to the gas phase is achieved via its chemical potential /xg = k a T In kB r (2~rrnkBT)3/2
-kaT
In Zi~t,
(4)
where P is the pressure and Zi, t is the internal partition function of the free particle accounting for its vibrational and rotational degrees of freedom. Interactions between next-nearest neighbors etc. and many-particle interactions can be added to eq. (2). We restrict ourselves to gas-solid systems in which all relevant processes, like diffusion, adsorption, desorption etc., are Markovian. We introduce a function P ( n ; t) which gives the probability that a given
244
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microscopic configuration n = ( n l , n 2 , • • • , n t) is realized at time t, where I = N s is the total number of adsorption sites on the surface. It satisfies a master equation [11,12,27] d P ( n ; t ) / d t = E [ W ( n ; n')P(n'; t) - W(n'; n ) e ( n ; t ) ] ,
(5)
nt
where W(n'; n) is the transition probability that the microstate n changes into n' per unit time. It satisfies detailed balance
W(n'; n)Po(n ) = W(n; n')Po(n' ),
(6)
where
Po(n)
= Z -1
exp( - H ( n ) / k s T )
(7)
is the equilibrium probability. In principle, W(n'; n) must be calculated from a Hamiltonian that includes, in addition to eq. (2), coupling terms to the gas phase and the solid that mediate mass and energy exchange. In this paper we rather follow the procedure initiated by Glauber [27] in setting up the kinetic Ising model and guess an appropriate form of W(n'; n). If the residence time in a given state is much longer than the time needed to complete a transition to another state, one can write the transition probabilities as a sum of adsorption-desorption and diffusion processes
W(n'; n) = Wa_d(n'; n) + Wdif(n'; n).
(8)
In a recent paper [11] we have examined various choices and studied their physical implications. For the most part of this paper we choose the "Langmuir kinetics" for which one argues that adsorption into a site i is independent of its local environment. One thus gets for a one-dimensional lattice [11] Wa(n'; n) = W0]~ ( 1 - n i ) 6 ( n ~ , i
1 - n i ) I - I 6 ( n ~ , nt).
(9)
l~i
For desorption we write Wd(n'; n) = WoCoEni[1 + Cl( ni+ 1 + ni_l) q- C2ni+lni_l]8(n ~, 1 - ni) l--I 6(n~, nl) i
=
WoCoEn i i
14=i
exp[(ni+ 1 + n i _ l ) V 2 / / k B
T]
(10)
l--I6(n~, nl). l~i
The second equality follows readily by using the identity n[exp(ot)- 1] = e x p ( n a ) - 1 for n = 0, 1. Detailed balance demands that
C O= exp(Es/kBT )
(11)
C, = [exp(V2/kaT) - 1] r
(12)
and
We note that eq. (10) is the standard form of the desorption transition probability used in Monte Carlo simulations. The adsorption-desorption dynamics in eqs. (9)-(12) are of course different from Glauber's spin flip dynamics when rewritten in lattice gas language. Indeed, the Glauber dynamics produces a trivial desorption kinetics, i.e. one in which the desorption rate is strictly first order with a rate constant independent of coverage [11]. Turning next to diffusion, we write
Wdif(n' ; n)=JoEni(1-ni+a)(lx i,a
d-C 1 E ni+b)t~(n~, b~a
1-ni)t~(n:+a,
1
1-I a(n , n,).
l~i,i+a
(13)
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
245
T o m a k e c o n n e c t i o n with experimental observables we define average occupation n u m b e r s ~i(t)
=
~,nie(n, t),
(14)
n
w h e r e the sum runs over all microscopic configurations n with each ni = 0 and 1. T h e coverage is then given by
O(t) = N s l ~ _ , n i ( t ) = ( n i ) .
(15)
i
3. Equations of m o t i o n
W e p r o c e e d to derive the hierarchy of coupled equations of m o t i o n for multi-site correlation functions. T o this end we multiply eq. (5), e.g. by n i, and sum over all configurations according to eq. (14) to obtain d ( n i ) / d t . W e point out that for an adsorbate with n e a r e s t - n e i g h b o r interactions only, the e q u a t i o n of m o t i o n of the n-site correlation function involves m-site correlation functions w h e r e m -- n, n + 1, • • •, n + c, with c = 2 being the site coordination n u m b e r for a one-dimensional chain of atoms. N o t e that in the n-site cluster at most one site is a d d e d in each direction. T h u s to get the equations up to the n-site correlation functions, one needs to consider a finite chain of (n + 2) sites. In an adsorbate that remains h o m o g e n e o u s t h r o u g h o u t its time evolution the correlation functions are not site specific. T h u s we can use a symbolic notation (')
= (hi) =0,
(" ") = (nini+l) ,
(o) = 1 - ('),
('o')
(''")
= (nini+lni_l)
= (ni(1 -ni+l)ni+2),
,
etc.,
(16)
were the new configuration averages are defined similarly to eq. (14), e.g. (.-)
=N~-I E Enini+lP(n, i
t).
(17)
n
Relations between correlators follow, e.g. (o- -) = (. • ) - (---), (o'o)
= ('o)
(18)
- (" " o ) = ( " ) -
2(" - ) + ( - - ' ) .
(19)
F o r the lowest four correlation functions we get d ( - ) / d t = W o ( o ) - WoCo[( • ) + 2 C 1 ( . d(- •)/dt
(20)
= 21410(0- ) - 2WoCo(C ~ + 1 ) [ ( . • ) + C1( • • • ) ] -- 2 J o [ ( O • • ) - ( . o . )
+C1((o..) d(--.
• ) "]- C 2 ( " " • ) ] ,
- ( ..o. ))],
(21)
) / d t = W012(o • • ) - ( . o . )] - WoCo(1 + C1) [(3 + C1)( • • • ) + 2 C l ( . . . . )] - 2Jo[(O -.. ) - (..o.)
d( . o . ) / d t
= W o [ 2 ( o . ) - 3( . o . ) ]
+ C~((o
...
) -
- WoCo[2(.o. )-
- C 2 ( • • • )] + 2 J 0 [ ( "oo" ) - 2 ( ' o - )
(..o..))]. (---)
(22)
+ 2C1((.o..)
+ (1 - C 1 ) ( " o ' '
- (...))
) "b (1 + C 1 ) ( - " o )
+ C 1 ( " OO" " ) ] .
T h e equations for the correlation functions on four sites are listed in the appendix.
(23)
S.H. Payne et al. / Kinetic lattice gas model in one dimension. 1
246
Comparison of eq. (20) with the phenomenological rate equation [11,23] identifies W0 to be P W o = Soa s ( 2 . n . m k B T ) l / 2 ,
(24)
where S o is the sticking coefficient at vanishing coverage. Note that eq. (20) implies a sticking coefficient S(O) = S 0 ( 1 - 0). On this account we have termed the underlying kinetics the "Langmuir kinetics" because Langmuir invoked site exclusion, and thus the above sticking coefficient, in his kinetic derivation of the Langmuir isotherm [28].
4. Equilibrium properties Before we discuss the closure approximation for the equations of motion, we want to examine the structure of the equilibrium correlation functions. We note that in equilibrium not only must the equations of motion be set equal to zero, but in addition the adsorption-desorption and diffusion terms in each equation must vanish individually. The coverage is then given exactly via exp(lx/kBT)
e x p [ ( V 2 - V o ) / k B T ] a - 1 + 20 = q3qint a + 1 - 20'
(25)
where the chemical potential must be equated to that of the gas phase, given in eq. (4). The two-site nearest-neighbor correlation function is ( " " )0 = 0(1 - 2 1----~a 1 - 0 ),
(26)
a 2 = 1 - 40(1 - 0)[1 - exp( - V 2 / k B T ) ] .
(27)
with
More generally one finds for the correlations between two sites a distance k apart, irrespective of whether the intermediate sites are occupied or not, that [29] (ni+kni)o=02+
(t °~-~-i1 - - 1 ]i k 0 ( 1 - 0 ) .
(28)
From eq. (20) and the adsorption-desorption term in eq. (21) we then find ("")o
= (" • ) 2 / ( . ) .
(29)
We also have (-o')0 = (('o)0)2/(o)
= ((")-
(" • ) 0 ) 2 / ( 1 - ( " ) ) ,
(30)
or
(.) ( . o . )0 = - ~ S ( ( • ) - 2 ( . • )o + ( " " ) o ) ,
(31)
where eqs. (28) and (29) have been used. Higher order correlation functions also factorize exactly and in numerous ways, e.g. ( . . . . ) 0 = ( "'" )o(" " ) o / ( ' )
= (" " ) 3 / ( ' ) 2 =
( . . . ) 2 / ( " ")o,
(" " o ' ) 0 = (" "o)0( . o . ) o / ( ' O ) o = (" • )0( . o . ) 0 / ( .
),
(32) (33)
247
S.H. Payne et aL / Kinetic lattice gas model in one dimension. I
( ....
o.)o=(
....
o)o('"o')o/(""
o)o=(
....
3,
(34) (.)2 ( "oo- )o = 7 , ~ ( (
....
>o- 3(""
)o + 3 ( " • ) o -
("))
+ ( "o" )o = ("o>2o(OO>o/(O) 2,
(35)
and for m holes
('oo...oo')o=
~_~ ( - - ( ' O ) o ) k + l / ( o ) k=l
~,
(36)
and so on. N o t e that these factorizations are such that the sub-clusters on the right can have one or more occupied or empty sites in common. In an earlier paper on the two-dimensional kinetic lattice gas model [13] correlation functions of the kind ( - x . ) = ( . - - ) + ( • o . ) were factored instead of ( . o . ). For example, one finds that ( - x . ) (.x)/(x) = ( . ) 2 and unnecessary difficulties arise in the truncation scheme for the hierarchy of equations of motion that can be overcome easily with the present set of functions. In fig. 1 we present some numerical examples of correlation functions for a strong repulsive nearest-neighbor interaction. N o t e in particular the alternating structure in the two-site correlation functions as k in eq. (28) increases, see fig. la. Indeed, in the low-temperature limit one finds that at 0 = 1 / 2 these functions are zero for k odd and equal to 1 / 2 for k even, reflecting the ordered structure of every second site being occupied. The correlations with h o l e s / u n o c c u p i e d sites present, display this feature explicitly, see fig. lb. Those with two or more adjacent holes decay rapidly to zero above 0 = 0.5, and ( - o - o . ) has a maximum there. For attractive nearest-neighbor interactions the functions ( n i n i + k ) simply increase monotonically as a function of coverage. It is also interesting that for strong attractive interactions all correlation functions are equal to the coverage. T o understand this we note that the probability to find a given site occupied is 0. With strong attractions the probability to find a site nearby occupied is one due to clustering. 1.0
. . . .
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. . . .
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,~'
.5
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...,
....
, ....
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.4
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.6 L-
L*_
0 /
//// // ,, / I//
.2
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2
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--~/i
0
//
,,/,
_ _ > 0 (_)
/ - -
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(3
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5
6
Coveroge (ML)
7
8
e
.1
.2
.3
.4
.5
.6
.7
.8
.9
Coveroge (ML)
Fig. 1. Equilibrium correlation functions for a one-dimensional adsorbate with repulsive nearest-neighbor interactions (V 2 / k B T = 5): (a) n-particle (solid lines) and two-particle (dashed lines) correlations (left to right at arrow position) (. x" ), (- xxx. ), (. xxxxx • ). (.xxxx-), (-xx.), (--), (...), ( . . . . ), ( . . . . . ), (b) one-hole (solid), two-hole (dash) and three-hole (long/short dash) correlations (top to bottom at 0 = 0.6) (- o. ), (. o- o. ), (- o - . ), ( . . o-- ), (- oo. ), (. ooo. ).
248
S . H . P a y n e et al. / K i n e t i c lattice g a s m o d e l in o n e d i m e n s i o n . I
4 4
. . . .
,
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,
. . . .
i
. . . .
i
. . . .
i
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i
. . . .
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E
~ ~,!
40
......
E = =_.= ~ = =_.=_.~ =- ~- ~
"T-
34
O
32 0 Go m
30 2 8
. . . .
0
i
.1
. . . .
i
.2
. . . .
i
.3
. . . .
i
. . . .
.4 Coveroge
,
.5
. . . .
i
. . . .
.6
,
.7
. . . .
I
.8
. . . .
I
. . . .
.9
(ML)
Fig. 2. Coverage and temperature dependence of the isosteric heat of adsorption for repulsive and attractive interactions. Solid lines: V2 = 400 K for T = 90, 120, 150 K (top to bottom at 0 = 0.3). Dashed lines: I"2 = - 4 0 0 K with T = 120, 140, 160 K. Temperatures span the desorption range (cf. figs. 3 and 4) for substrate/adsorbate parameters (cf. eqs. (3) and (4)), V0 = 4500 K, v x = vy = v z = 1012 s -1 in q3; qint = Zint = 1. These parameters are assumed throughout.
We next look at the isosteric heat which can be calculated from eqs. (4) and (25) according to 20In P Qi~o(0, T) --kBT
~
o"
(37)
Numerical examples are given for repulsive and attractive nearest-neighbor interactions in fig. 2. We note that the coupled substrate-adsorbate system has two dominant energy scales, namely the single-partide binding energy, V0, introduced in eq. (3) and the nearest-neighbor interaction energy, V2, in eq. (2). The ratios we choose are typical of many systems. The isosteric heat (eq. (37)) will also be useful later on to understand thermal desorption in systems with fast diffusion.
5. Closure approximations
To make the set of coupled equations of motion for the correlation functions the basis of an analytic theory of surface processes, the hierarchy must be truncated. The simplest such closure approximation is the Kirkwood approximation in which one expresses all higher correlation functions in terms of two-body correlation functions. To develop a systematic closure scheme we are guided by similar procedures in equilibrium statistical mechanics. We recall that in the quasi-chemical approximation [14] one calculates the partition function for two sites exactly and then distributes these pairs randomly over the lattice, normalizing the result to account for double counting of overlapping sites. Similarly in Hill's approximation [30] one calculates the partition function for a square of four sites (on the square lattice) exactly. Kikuchi [31] and Morita [32] generalized this approach also taking account of the environment of such clusters in a self-consistent way. Any scheme to truncate the infinite hierarchy of equations of motion for the correlation functions by factorizing higher order correlation functions in terms of lower order ones must reduce to the factorization properties in equilibrium contained, e.g., in eqs. (32)-(36).
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
249
5.1. Two-site closure
In the two-site (or Kirkwood) approximation one retains the equations of motion for the coverage, ( • ), and the two-site nearest-neighbor correlation function, ( • • ). In eqs. (20) and (21) we factorize all higher order correlation functions with single-site overlap so that only single-site functions ( . ) and ( o ) = 1 - ( • ) appear in the denominator, e.g. (...)
= (..)2/(.),
(38)
(" " o ' ) = (" " ) ( ' o ) 2 / ( " ) ( o ) .
(39)
The truncated set of equations thus reads in the two-site approximation d ( . ) / d t = W o ( o ) - WoCo(" )(1 + C1(. • ) / ( .
))2,
d ( . . ) ~ d r = 2W0(o • ) - 2WoCo(Cl + 1 ) ( ( . . ) + G ( ' " _ (..)2/(.
(40) ) 2 / ( . )) _ 2Jo[( • • )
) _ (. 0)2/(0)
+CI((. ")_ (..)2/(.
)_ ( . . ) ( O " ) 2 / ( . ) ( O ) ) ] .
(41)
repeat that eqs. (40) and (41) yield the exact equilibrium solution. On two-dimensional lattices the two-site closure yields the quasi-chemical approximation which is exact on a one-dimensional lattice. Note, however, that eqs. (40) and (41) are approximate away from equilibrium because the factorizations no longer hold exactly.
We
5. 2. Three-site closure and beyond
To go beyond the two-site closure we include all correlation functions on three sites, i.e. (.),(--),(--.),(.o.).
(42)
All higher order correlation functions appearing on the right-hand sides of eqs. (20)-(23) will be factorized in terms of three-site correlation functions in the numerator with two-site overlap. These overlapping sites appear as two-site functions in the denominator to compensate for overcounting, e.g. ( .... ) = (-.-)(...
)/(.
• ),
(" "o" ) = (" " o ) ( ' o " ) / ( ' o ) , ('oo"
• ) = ('oo)(oo")(o"
(43) (44)
• )/(oo)(o').
(45)
In the absence of diffusion this scheme produces an exact solution for desorption [22]. This arises because the equations of motion for the full n-site particle correlator, Pn, reduce to
de. dt
= - W ° C ° { [ n + 2C1(n - 1) + C2(n - 2 ) ] P , + 2(C 1 + C2)P~+I},
(46)
for n > 2. One then verifies that the time evolution of P~ for n > 4 is correctly given by that of Pn_ 1P3/P2. The ratio P3/P2 evolves independently of any correlator, i.e. d -~-f(P,,+I/P,,) = -- WoC0(1
+ C1) 2,
(47)
for n >_ 2. Hence, the equations for ( • ), ( • • ) and ( • • • ) suffice. Note that correlation funetions with holes, such as ( -o • ) cannot enter the desorption kinetics.
250
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
This factorization scheme is unique which is not obviously the case for any other scheme. For example, if we factorize with single-site functions in the denominator, we have the following possibilities (- - o . )---- (- - ) ( . o . ) / ( ' )
(48)
(..o.)
(49)
and = (..o)(o.)/(o).
Either choice is exact in equilibrium. However, away from equilibrium, eqs. (43)-(45), (48) or (49) yield different time evolutions. The "best" convergence as one includes more correlation functions in the basis set is achieved by the first choice above, namely two-site overlap in the three-site approximation. The generalization to larger basis sets is guided by the principle of maximum overlap. E.g., in the five-site approximation we factorize all higher correlation functions in terms of all possible five-site functions in the numerator with four-site overlap (and thus four-site functions in the denominator), e.g., < . o . - o - "> = < . o . . o > < o . . o . > < .
-o. ->/<.-o->.
(50)
If we were to choose less than maximum overlap, there would be a multiplicity of choices with no clear indication of which is the optimum in some sense.
6. Thermal desorption
6.1. Mobile adsorbate We begin our study of thermal desorption kinetics by looking at mobile adsorbates, i.e. at systems in which surface diffusion is much faster than adsorption and desorption. As a consequence, such adsorbates will remain in quasi-equilibrium throughout desorption, i.e. all correlation functions will attain their equilibrium values at the instantaneous coverage. The rate of desorption is then given by eqs. (1) and (25). In fig. 3 we present numerical results for temperature-programmed desorption from a one-dimensional chain of mobile atoms with repulsive nearest-neighbor interactions (solid lines). We note that for initial coverages 00 > 1/2, a second, low-temperature peak develops reflecting the drop in isosteric heat (solid lines in fig. 2). To make this connection we parametrize the desorption rate as
~t des= -S(O, T)v(O, T)
exp(--Ed(0,
r)/kBr)o
(51)
as a quasi-first-order process with a coverage-dependent rate constant, where Ed(0,
T) -kBT2-~0
In ( - l d 0dt0
des]l
(52)
is the differential desorption energy. We get, via eqs. (1), (4) and (37),
Ed( O, T)
0
= Qiso(0,
T) -kBT/2 + kBT2~-~ In S( O, T).
(53)
Because in most systems the sticking coefficient varies little with temperature (and not at all for Langmuir kinetics), we see that the desorption energy is essentially equal to the isosteric heat for mobile adsorbates. As explicitly displayed in eq. (51), the desorption rate is proportional to the sticking coefficient which for Langmuir kinetics is S(O)= S 0 ( 1 - 0). If we were to use eq. (51) as the starting point for a
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I 3.5
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140
150
Temperoture Fig. 3. T e m p e r a t u r e - p r o g r a m m e d d e s o r p t i o n rate (in units o f m o n o l a y e r s per s e c o n d ) for a m o b i l e a d s o r b a t e with repulsive i n t e r a c t i o n V 2 = 400 K. Initial c o v e r a g e s (left to right) 00 = 0.9,
0 .... !30
=r,~ 135
140
145
150
155
Temperoture Fig. 4. T e m p e r a t u r e - p r o g r a m m e d d e s o r p t i o n rate for a m o b i l e adsorbate with attractive i n t e r a c t i o n V 2 = - 4 0 0 K. O t h e r w i s e as fig. 3.
0.7, 0.5, 0.3, 0.1. Parameters: V0 = 4500 K, 1,I2= 400 K; other p a r a m e t e r s : h e a t i n g rate 1 K s -1, a s = 10 ~2, m = 28 amu. Solid lines: sticking S(O) = S0(1 - 0). D a s h e d lines: S = S O = 1.
phenomenological description, we could take any (measured) coverage dependence of the sticking coefficient, e.g., setting it equal to a constant S O up to a coverage close to a monolayer. Such a behavior is, e.g., to be expected in the presence of precursors. Temperature-programmed desorption is then given by the dashed lines of fig. 3, demonstrating the marked change of the low-temperature peak caused by the decreasing sticking coefficient. In fig. 4 we give an example of temperature-programmed desorption from a chain of atoms with attractive nearest-neighbor interactions and S(O) = S0(1 - 0), remaining in quasi-equilibrium throughout desorption (solid lines). For higher initial coverages the peaks shift to higher temperatures reflecting the fact that the isosteric heat, and thus the desorption energy, rises (dashed lines in fig. 2). The dashed lines in fig. 4 give equivalent results for a constant sticking coefficient, demonstrating again the radical effect of the sticking coefficient on desorption. In closing this section we note that in the limit of fast diffusion we necessarily get identical results to those obtained from eq. (1) in figs. 3 and 4 by using the equations of motion with large hopping rate J0 (in any approximation).
6.2. Immobile adsorbate If surface diffusion is slow in the temperature range of desorption we must make use of the equations of motion to describe the desorption kinetics. The largest deviations from the desorption kinetics of a mobile adsorbate obviously occur for an immobile adsorbate where surface diffusion is negligible throughout desorption. If the time scale of surface diffusion is of the order of that of desorption we expect results intermediate between those of fast and negligible diffusion. In fig. 5a we present temperature-programmed desorption spectra for an immobile adsorbate with nearest-neighbor repulsion for varying initial coverages. The dashed lines are obtained in the two-site closure, while the solid lines result from the three-site closure (which, according to section 5.2 is exact). Clearly the two-site approximation is inadequate. Evans et al. [22] have interpreted the three peaks in the
S.H. Payne et aL /
252 4.,5
0 0
. . . .
,
....
, ....
, ....
, ....
, ....
Kinetic lattice gas model in one dimension. ,
4.0
(D c)
3.5
x
x
c
, ....
i/ . . 1 , .... t i
2.0
o
1.5
C3
1.0
i ....
t ....
,
b
iI /II
6
iI
2.5
0
i ....
7D
3.0 o r~
....
. . . .
I
II
J iJ
I~lll I
3
i
tI
7D ~ 2
]D I
.5
0
0 90
100
110
120
Tempereture
130
140
150
90
, , i
100
110
120
1.30
140
. . . .
i
. . . .
150
Temperoture
Fig. 5. ( a ) T e m p e r a t u r e - p r o g r a m m e d d e s o r p t i o n r a t e f o r a n i m m o b i l e a d s o r b a t e w i t h r e p u l s i o n V 2 = 4 0 0 K. S o l i d lines: t h r e e - s i t e c l o s u r e . I n i t i a l c o v e r a g e s ( l e f t t o r i g h t ) O0 = 0.9, 0.7, 0.5, 0.3, 0.1. D a s h e d lines: t w o - s i t e c l o s u r e ( f o r initial c o v e r a g e s 00 = 0.9, 0.5; f o r initial c o v e r a g e s l e s s t h a n 0.3 t w o - a n d t h r e e - s i t e c l o s u r e s c o i n c i d e ) . (b) C o r r e s p o n d i n g e v o l u t i o n r a t e s d ( . - ) / d t (solid lines) a n d d ( • - • ) / d t ( d a s h e d l i n e s ) in t h r e e - s i t e c l o s u r e f o r 00 = 0.9, 0.7, 0.5.
desorption spectra as "staged" desorption, with the low-temperature peak reflecting desorption of particles from an environment of two occupied neighboring sites and the smaller middle peak as desorption from the ends of chains. This picture is nicely confirmed by looking at the time evolution of ( . • ) and ( . . . ), fig. 5b. We have performed an Arrhenius analysis of the desorption spectra in fig. 5a by plotting isosteric rates versus T-1. For an immobile adsorbate these curves are not expected to be straight lines. However, instead of using eq. (52) to define a coverage- and temperature-dependent desorption energy we do a linear fit for simplicity. The resulting (temperature-averaged) desorption energy and prefactor are plotted in figs. 6a and 6b for the two-site closure and the exact immobile and mobile cases. Several connections can be made to preceding graphs. First, as we know from eq. (54), for a mobile adsorbate the desorption energy equals the heat of adsorption, to within a factor kBT/2. With this correction, the long/short dashed curve of fig. 6a is just the temperature average of the (solid) curves in fig. 2. Second, the two peaks in fig. 3 correspond to the two plateaus in the desorption energy, fig. 6a, at low and high coverage. Third, although we get a two-peak structure in the desorption spectrum for an immobile adsorbate calculated in the two-site approximation, similar to the mobile case, the desorption energy looks quite different with the appearance of a peak at half coverage. The exact (three-site) calculation is even more intriguing with the appearance of two peaks in the desorption energy. In the analysis of experimental desorption data, such structures might be interpreted as arising from attractive interactions. This is clearly not the case here. Rather, because the minima in the desorption rate are much lower for an immobile adsorbate than for a mobile one, the effective desorption energy must be much higher. The corresponding prefactors also show sharp peaks, which would be interpreted as an ordering above that for equilibrium. Clearly these structures are an artifact of the Arrhenius analysis. Indeed, their form depends on the strength of the interaction, the initial conditions (initial coverage and adsorption temperature) and the range of desorption temperatures over which the (linear) analysis is performed. Such effects have already been discussed for models of desorption of two-dimensional systems [24]. As examples we display, in fig. 6c, the differential desorption energy for a different interaction value and initial coverage (prepared in equilibrium) obtained by isothermal desorption. The variation of the peak structure between the lowest and the highest temperature traces (corresponding to
S.H. Payne et aL / Kinetic lattice gas model in one dimension. I 40
. . . .
]
. . . .
,
. . . .
,
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,
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i
+oo 13.5 o o
"\ i ~
33
, . . . .
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_
35
c
. . . .
,~
36
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,
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C
, . . . .
II
. . . .
~
. . . .
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~
. . . ,
1 4 . 5
253
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i-
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/
'/ ~ ' / /
E~
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¢\
G)
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29
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,
.1
. . . .
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. . . .
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.5
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. . . .
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. . . .
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.7
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, . . .
. . . .
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. . . .
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(ML)
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. . . .
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Coverege
. . . .
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.6
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.7
. . . .
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.8
i . . . .
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(ML)
42 41
40 39
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~
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g 35 ~
54 0 m
33
C3 32
31
. . . .
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p . . . .
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i . . . .
.4
i ,
.5
Coveroge
.+
. . . .
.6
, . . . .
.7
, , , , , j . , .
.8
.9
(ML)
Fig. 6. (a) Temperature-averaged desorption energy, Ed(8) , for adsorbate with Vz = 400 K for cases: immobile, three-site closure (solid); immobile, two-site closure (dash); mobile (long/short dash). (b) Corresponding prefactors, cf. eq. (51). (c) Differential desorption energy, Ea(O, T), f o r V 2 = 400 K, 00 = 0.55 (solid curves) and V 2 = 200 K, 00 = 0.99 (dash) and four temperatures spanning the desorption range (top to bottom at O = 0), T = 90, 110, 130, 150 K.
0 < 0 0 and 0 --* 0 in the T P D of fig. 5a, respectively) is significant on the scale of the interaction. The temperature average of both sets of traces in fig. 6c will yield peaks of Ed(O) at 0 = 0.4 noticeably different from that in fig. 6a. The Arrhenius analysis should be reserved for systems that remain in quasi-equilibrium where the interpretation of the desorption energy as the heat of adsorption and the prefactor as a measure for the entropy is rigorous. In fig. 7a we show desorption from an immobile adsorbate with attractive interactions, with the curves of the mobile case included for comparison. We note that the peak desorption from an immobile adsorbate occurs at a higher temperature than from a mobile adsorbate. This can be traced to the fact that during desorption there are more nearest-neighbor pairs and fewer holes (e.g. ( . o . >) in the immobile system, compare figs. 7b and 7c, respectively. Details of this evolution also depend on the initial conditions, as for repulsive interactions, but in a more marked manner. For instance, at low initial coverages a low-temperature s h o u l d e r / p e a k develops in the rate at a position corresponding to the
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
254
20
1.0 C]
o o ~ x
l\
b .8
15
Q)
¢ .6
c? rY I t
10
c
Ill~ l
0 o ~q ~ C2~
_o
.4
o 0
5
.2 ~.,'1 /
/
L
i
0 130
~-,-.-?-,-.-,_.',.
0 140
150
160
140
130
Temper0ture
. 150
.
.
. 160
Tempercture 3.5
. . . .
,
. . . .
::i
3.0 o
x
2.5
."
,
~-
. . . .
C
~
~ ~
1.5
1.0
(.9
. . . .
~-~
0
c
,
-
-
.......
_
.5
~
~ . . . . . .
0 L_ . . . . 130
_ , 140
~
~ \\
LL ~ L
~
.
. 150
.
.
160
Temperoture
Solid lines: t e m p e r a t u r e - p r o g r a m m e d d e s o r p t i o n rate for an i m m o b i l e adsorbate with attraction V 2 = - 4 0 0 K , in three-site closure for initial c o v e r a g e s (top to b o t t o m ) O0 = 0 . 9 , 0 . 7 , 0 . 5 , 0 . 3 , 0 . 1 . D a s h e d lines: m o b i l e adsorbate f r o m solid lines o f f i g . 4 . ( b ) C o r r e s p o n d i n g e v o l u t i o n o f < - • > for i m m o b i l e (solid) and m o b i l e (dash) systems. (c) E v o l u t i o n o f < . o . > for both cases. Fig.
7. (a)
desorption peaks for a non-interacting system. This is the result of the initial desorption of isolated particles in the adsorbate (there are relatively fewer in number at higher coverage) which are not replenished by diffusion as in the mobile case, see fig. 8a [33]. Such a feature clearly alters the interpretation of the spectra, and the Arrhenius analysis in particular. As another demonstration we show in fig. 8b isothermal rates for three initial coverages, prepared in equilibrium. If we now (against better judgement) perform an Arrhenius analysis of isothermal rates we obtain vastly different results for the desorption energy depending on the initial conditions, fig. 8c. For the case 00 --- 1, Ea(O) is essentially independent of 0 for the value of V 2 assumed. ( U p o n increasing 11.I2 I we find Ea(O) - Ed(0) ~ -- 1.5V2, the limit deduced by Evans et al. [22].) This behaviour can only persist for lower initial coverages, if the initial distributions < • • >, < . . . > have those (non-equilibrium) values which would have evolved from 0 o = 1. Again, one can associate the peaks of Ed(0) with the temporary reduction of the desorption rate, in this case for the low 00 traces of fig. 8a. Yet another, and more complicated picture would emerge if we were to do an isosteric analysis of T P D data, obtained either by the method of variation of initial
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
1.6
.
.
.
.
i
.
.
.
.
I
0
,
1.4 0 0
A 0D
255
•
//"
-1
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1.2
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~-
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o D9 © r'm
.4
0 Go (1) d:3
~
6
0 120
~
-4
130
140
150
-2.0
160
i
i
i
-1.5
-1.0
-.5
Log (Coveroge)
TemperQture 44 ,
%
4.3
/ /'
, /
C
t~
E 42
/
/
1
if ~,
v
0) c LJ
4o
~
',
/
/
C 0
0 o3
37 36 -2.0
-1.5
'
'
1.0
.5
Log (Coverage) Fig. 8. (a) Temperature-programmed desorption for the immobile system of fig. 7a but initial coverages 00 = 0.2, 0.1, 0.05, 0.01. (b) Isothermal desorption rates for this system for temperatures spanning the TPD range for initial coverages 0.9 (solid), 0.2 (dash), 0.05 (long/short dash). Temperatures (top to bottom) T = 165, 155, 145, 135, 125 K. (c) Desorption energy, Ed(O), obtained from linearized Arrhenius analysis of isothermal rates for immobile adsorbate for 00 = 0.99 (solid line), 0.2 (dash), 0.05 (long/short dash), and mobile adsorbate (00 = 0.99, long dash) - cf. fig. 2 (dashed curves). coverage or variation o f h e a t i n g rate. A s n o n e o f these effective desorption energies, and, h e n c e , derived c o n c e p t s o f prefactor and d e s o r p t i o n order [22], are a m e n a b l e to a consistent physical interpretation, this m e t h o d o f analyzing T P D spectra must be a b a n d o n e d . 6.3. D i f f u s i o n
It r e m a i n s to l o o k at the role o f diffusion. Firstly, in the a b s e n c e o f desorption, fig. 9 shows the time d e p e n d e n c e o f a n u m b e r o f correlation functions calculated at the three-, four- and five-site truncation levels in the approach to equilibrium, starting initially from an ideal gas. W e n o t e that the e v o l u t i o n is slower the larger the basis. For the two- and three-particle correlators o f fig. 9a c o n v e r g e n c e to the
256
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I .4
0 x
O
.3
x
.2
._o
.1
.4
0 CI 0 c..J .2
\
";
x -.1
E 0
©
0
'~
d:b
C3
"~, .1
2 -.3
.4
.01
.02
.03
.04
.05
.06
.07
0
.01
.02
Time (sec)
.03
.04
Time
.05
.06
.07
(sec)
Fig. 9. I s o t h e r m a l e v o l u t i o n o f adsorbate c o r r e l a t i o n functions towards e q u i l i b r i u m f r o m a r a n d o m distribution in the a b s e n c e o f desorption, c a l c u l a t e d in three-site ( . . . . ), four-site (- - -), and five-site ( ) closure approximations. O r d i n a t e is n o r m a l i z e d deviation f r o m e q u i l i b r i u m value. 0 = 0.9, V 2 = 400 K, and h o p p i n g rate J0 = 1 s -1. (a) n-particle correlations (top to b o t t o m at t = 0): ( . . . . . ), ( . . . . ), < . . . >, < . . >, (b) c o r r e l a t i o n s with holes: ( . o o - > x 10 - 2 , < . . o - . >, < - o . >, < -o . o . >.
"exact" evolution has already been achieved at the four- and five-site levels, respectively. This is confirmed for the hole functions of fig. 9b: those containing 2 particles, e.g. < • o . > and ( • o o . >, are accurately calculated at the four-site closure level. Obviously multi-site correlators require a truncation within a larger basis, but out results for diffusion alone suggest a rule of thumb for repulsive interactions: correlations on m sites with n < rn particles require closure at the (n + 2)-site level for "exact" evolution. Qualitatively similar results are obtained for attractive interactions, but on a longer time scale. Next we look at desorption from adsorbates with finite hopping rates, J0, larger and smaller than the desorption rate constant, WoC o, in the midrange of desorption temperatures. We display, in fig. 10, TPD spectra for a system with repulsive interactions. The variation of J0 between the immobile and mobile 4
O O
.
.
.
.
,
. . . .
,
.
.
.
.
,
.
.
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,
. . . .
,
.
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.
,
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3
x
.
!
E~ r~
E 2 .£ O if)
81
90
100
110
120
130
140
150
TemperQture
Fig. 10. T e m p e r a t u r e - p r o g r a m m e d d e s o r p t i o n rate for adsorbate with repulsion, V 2 = 400 K, for 00 = 0.9 and varying hopping rates H o p p i n g barrier Q = 0, a n d v o = 10 . 4 s - 1 (solid, t h r e e - p e a k e d ) (effectively immobile), 10 . 3 s - 1 ( - - - ) , 10 - z s - t (__ _ _ __), 10 - 1 s - 1 (__ _ _ __), 1 s - 1 ( . . . . ), and 10 s - ] ( ) (effectively mobile).
Jo = vo e n d ( - Q / k B T ) .
S.H. Payne et a L / Kinetic lattice gas model in one dimension. I
257
limits is approximately five orders of magnitude. These results for the desorption rate are obtained in the three-site approximation, there is negligible improvement in the four-site approximation. This does not contradict our earlier statement that isothermal and isosteric diffusion towards equilibrium can only be described accurately within the four- or even five-site approximation: the exponential decay of the coverage in desorption eliminates the need for a detailed knowledge of the long-time evolution of the multi-site correlators due to diffusion, thus making the factorization scheme almost exact.
7. Sticking coefficient
Everything done so far in this paper is based on the "Langmuir kinetics" contained in the transition probabilities eqs. (9)-(12). They imply in particular that the sticking coefficient is given trivially by S(O) = S0(1- 0). For most systems this simple form, based on site exclusion only, does not apply. In particular, if precursors mediate adsorption one expects a sticking coefficient that is initially coverage independent. One might also argue that the adsorbing particle experiences the local environment in a way analogous to the desorbing particle. In such cases lateral interactions will affect the adsorption probability, e.g., repulsive nearest-neighbor interactions will aid desorption and inhibit adsorption. For such a scenario, which we term interaction kinetics, we can choose the transition probabilities for adsorption and desorption to read [11] Wa(n'; n) = W o E ( 1 - n / ) [ 1 - Cl(ni+l-I-ni_x) -C2ni+lni_l]~(n~, i
Wd(n';
n) = WoCo~ni[1
+ Cl(ni+l
--1-n i _ l )
d- C 2 n i + l n i _ l ] t ~ ( n ~ ,
i
1-ni)l-Is(n~, l~i
1 - ni) I"Is(n~, l~i
nl).
nt),
(54) (55)
Detailed balance gives C1 = tanh(V2/2kBT ) ,
(56)
C2 = tanh( V2/kBT
(57)
) - 2C 1. 1.1
....
"0
.7
('3
.5
"~
.3
, ....
, ....
, ....
, ....
, ....
, ....
, ....
, ....
~',xxx
.2
, ....
x
~,?..,,,
.1
o 0
.1
.2
.3
.4
.5
.6
.7
.8
.e
Coverage (ML) Fig. 11. Normalized sticking coefficient, S(O)/So, in the interaction kinetics for attractive interactions (dashed lines) and repulsive interactions (solid lines). Values of V2 / k n T (top to bottom at 0 = 0.4): - 2.0, - 0.5, - 5.0, - 10.0, 0.0, 0.5, 2.0, 5.0, 10.0.
258
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
From eq. (54) we get the effective coverage-dependent sticking coefficient as
S(O, T)=S0((o)-2C1<'o)-C2<'o')).
(58)
For adsorption, the correlation functions on the right of eq. (58) are those in equilibrium if adsorption occurs at temperatures where the adsorbate is mobile. On the other hand, at very low temperature adsorption is a r a n d o m process so that the correlation functions are ( ' o ) = ( - ) ( o ) and ( ' o ' ) =
<. >2(o>.
Numerical examples of sticking coefficients for a mobile adsorbate are given in fig. 11. For strong repulsive nearest-neighbor interactions the sticking coefficient falls initially as S(O)= S0(1 - 30) as both neighboring sites must be empty as well. Above half a monolayer sticking is essentially zero. For attractive interactions the sticking coefficient is always larger than 1 - 0. It rises above its value at zero coverage, as was also observed for a square lattice in the quasi-chemical approximation [11]. We also note that in the limit of very strong attraction the sticking coefficient approaches the limit 1 - 0 again due to the fact that all particle correlators are then equal to 0, cf. eqs. (26)-(29). Due to the fact that the transition probabilities must satisfy detailed balance in equilibrium, the rather complicated coverage dependence of the sticking coefficient will reflect itself in the desorption kinetics as well. Thus an attempt to fit desorption data only without considering the adsorption characteristics and also the equilibrium properties of the system might lead to a wrong model.
8. Conclusions
In this p a p e r we have studied the one-dimensional kinetic lattice gas model to examine the convergence of closure approximations for the hierarchy of equations of motion governing the n-site correlation functions. We have shown that a unique and in a sense " b e s t " factorization scheme involves maximum overlap in which an n-site correlator is factorized with ( n - 1)-site correlators in the numerator and (n - 2)-site correlators in the denominator. In a subsequent p a p e r we will see that the analogous closure scheme is also " b e s t " for the time-dependent generalization of the grand-canonical approach. For strongly interacting one-dimensional systems in which the time scale of diffusion is of the order of that of desorption, three-site closure is sufficient for an accurate description of the desorption kinetics, whereas isothermal and isosteric diffusion is only accurate in the four- to five-site closure. In systems with slow diffusion, the kinetics depend on initial conditions and non-equilibrium effects show up strongly in desorption. This implies, as we have shown in numerical examples, that an isosteric Arrhenius analysis of desorption data is inappropriate and more often than not, misleading. However, a differential desorption energy can be defined and, indeed, measured by the threshold method [34,35], as demonstrated for rare gases on metals where non-equilibrium effects due to slow diffusion are known to be present [36]. The lesson to be learnt is that one should model equilibrium and kinetic data directly. The implications for the two-dimensional kinetic lattice gas model remain to be explored. In a previous p a p e r [13] we have developed a systematic closure approximation for a square lattice including all correlation functions on a square of four sites, i.e.
<->, (- • >, <..>, <:.>, <::>,
(59)
yielding good results. However, it has remained a mystery so far why this scheme did not work well for the diffusional part of the equations of motion. The present study of the one-dimensional lattice gas suggests a solution, namely that correlation functions of the form ( • o • ) must be kept. We intend to do this in a systematic way by including all correlation functions on a four- and on a nine-site cluster.
S.H. Payne et aL / Kinetic lattice gas model in one dimension. I
259
Acknowledgement F u n d i n g for this r e s e a r c h h a s b e e n p r o v i d e d by t h e N e t w o r k o f C e n t r e s o f E x c e l l e n c e in M o l e c u l a r a n d I n t e r f a c i a l D y n a m i c s , o n e o f t h e f i f t e e n N e t w o r k s o f C e n t r e s o f E x c e l l e n c e s u p p o r t e d by t h e Government of Canada.
Appendix In addition to the equations of motion in eqs. (20)-(23) we have the following for the remaining c o r r e l a t i o n f u n c t i o n s o n f o u r sites.
d( ....
) / d t = 2W0(( • • • o) + (- -o" )) - 2WoCo(1 + C,) [(2 + Ct)( +C,(
.....
)] + 2J0[(.o
....
)
. - - > + C l < - "O " ' " ) - - (1 + C 1 ) < o . . . .
d(" "o- ) / d t = Wo((O'O" ) + ( "oo" ) + (" -oo) - (" "o" )) - WoCo[(3 + 2C1)(" -(1
+ C1)2(
+J0[("
-oo->
+ ( 1 -I..-C 1 ) ( < o
.
.
.
.
) +C1(''0""
) +C,(1
+ < "o " o " > + C l ( ( " '''
+C,)(
>],
(A.1)
"o" )
. o . . . )]
"oo" • ) + ( " " o ' o " >)
> .--I- < "o ' ' " >) - ( 3 + C 1 ) ( " o " • ) - ( C 1 - 1 ) ( -
"o'"
)], (A.2)
d( "oo" )/dt
= 2Wo((OOO" ) - ("oo"))
- 2WoC0[(
"oo" ) - ("o"
" ) + CI(( "oo" • )
-('o''>)] +2]o[<'ooo'>-2('oo'>-('o''>+('o')+('oo'') +C1( < "ooo""
> - < "oo""
>)].
(A.3)
References [1] For a review of the kinetic Ising model, see K. Kawasaki, in: Phase Transitions and Critical Phenomena, Vol. 2, Eds. C. Domb and M.S. Green (Academic Press, New York, 1972). [2] D.J.W. Geldart, H.J. Kreuzer and F.S. Rys, Surf. Sci. 176 (1986) 284. [3] J. Luscombe, Phys. Rev. B 29 (1984) 5128. [4] E. Oguz, O.T. Vails, G.F. Mazenko and J.H. Luscombe, Surf. Sci. 118 (1982) 572. [5] See, e.g., J.D. Gunton, in: Kinetics of Interface Reactions, Eds. M. Grunze and H.J. Kreuzer (Springer, Berlin, 1987). [6] K. Binder, Ed., Monte Carlo Methods (Springer, Berlin, 1986). [7] K. Wada, M. Kaburagi, T. Uchida and R. Kikuchi, J. Stat. Phys. 53 (1988) 1081. [8] A. Surda and I. Karasova, Surf. Sci. 109 (1981) 605. [9] V.P. Zhdanov and K.I. Zamaraev, Sov. Phys. Usp. 29 (1985) 755. [10] J.W. Evans, J. Chem. Phys. 87 (1986) 3038, and references therein. [11] H.J. Kreuzer and Zhang Jun, Appl. Phys. A 51 (1990) 183. [12] H.J. Kreuzer, Surf. Sci. 231 (1990) 213. [13] A. Wierzbicki and H.J. Kreuzer, Surf. Sci. 257 (1991) 417. [14] T.L. Hill, Statistical Mechanics (McGraw-Hill, New York, 1956). [15] H.J. Kreuzer, Phys. Rev. B 44 (1991) 1232. [16] R.H. Lacombe and R. Simha, J. Chem. Phys. 61 (1974) 1899. [17] A. Silberberg and R. Simha, Biopolymers 6 (1968) 479. [18] G. Schwarz, Ber. Bunsenges. Phys. Chem. 75 (1971) 40. [19] R. Simha and R.H. Lacombe, J. Chem. Phys. 55 (1971) 2936.
260
[20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
[34] [35] [36]
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
D. Ben-Avraham and J. K/Shier, Phys. Rev. A 45 (1992) 8358. J.J. Luque and A. Cordoba, J. Chem. Phys. 76 (1982) 6393. J.W. Evans, D.K. Hoffman and H. Pak, Surf. Sci. 192 (1987) 475. H.J. Kreuzer and S.H. Payne, in: Dynamics of Gas-Solid Collisions, Eds. C.T. Rettner and M.N.R. Ashfold (Roy. Soc. of Chemistry, Cambridge, UK, 1991). S.H. Payne and H.J. Kreuzer, Surf. Sci. 222 (1989) 404. H.J. Kreuzer and S.H. Payne, Surf. Sci. 198 (1988) 235. S.H. Payne and H.J. Kreuzer, Surf. Sci. 205 (1988) 153. R.J. Glauber, J. Math. Phys. 4 (1963) 294. I. Langmuir, J. Am. Chem. Soc. 38 (1916) 2221. B.M. McCoy and T.T. Wu, The Two-Dimensional Ising Model (Harvard University Press, Cambridge, MA, 1973). T.L. Hill, J. Chem. Phys. 18 (1950) 988. R. Kikuchi, Ann. Phys. 10 (1960) 127; Phys. Rev. 124 (1961) 1682, 1691; Prog. Theor. Phys. (Kyoto) Suppl. 35 (1966) 1. T. Morita, J. Phys. Soc. Jpn. 12 (1972) 753; J. Math. Phys. 13 (1972) 115. By analyzing the equations of motion for isothermal desorption kinetics in the low-coverage limit, Evans et al. [16] showed explicitly the competition between desorption of isolated particles and that from the ends of clusters. However, the evolution of clusters is such that, for (their) initial coverages of 00 = 1, the effect is practically unobservable. E. Habenschaden and J. Kiippers, Surf. Sci. 138 (1984) L147. J.C. Miller, H.R. Siddiqui, S.M. Gates, J.N. Russell, Jr., J.T. Yates, Jr., J.C. Tully and M.J. Cardillo, J. Chem. Phys. 87 (1987) 6725. H. Schlichting and D. Menzel, Surf. Sci. 272 (1992) 27; 285 (1993) 209.
surface science
Kinetic lattice gas model in one dimension I. C a n o n i c a l a p p r o a c h S.H. P a y n e , A. W i e r z b i c k i a n d H.J. K r e u z e r Department of Physics, Dalhousie University, Halifax, NS, Canada B3H 3J5
Received 17 December 1992; accepted for publication 1 March 1993
A closure approximation for the one-dimensional kinetic lattice gas model is formulated to truncate the hierarchy of equations of motion for correlation functions. Essentially exact results are obtained both for the equilibrium properties and the time evolution of the adsorbate when up to five-site correlation functions are included. Implications for thermal desorption in systems with slow and fast diffusion are studied for nearest-neighbor interactions.
I. Introduction The kinetic lattice gas model for surfaces was originally set up in close analogy to the kinetic Ising model for magnetic systems [1]. It is based on a master equation that treats adsorption, desorption and surface diffusion as Markovian processes. To obtain the time evolution in an adsorbate, one can solve the master equation directly, either by matrix diagonalization [2] or by renormalization group techniques [3,4] or by Monte Carlo methods [5,6]. Recently, a perturbation approach has been suggested that constitutes a time-dependent generalization of Kikuchi's method in the equilibrium statistical mechanics of the lattice gas [7]. In another approach [8-13] one first derives a hierarchy of equations of motion for n-site correlation functions which must be truncated via a closure approximation to yield a finite set of coupled equations. The n-site correlation functions are of course site specific, i.e. they can vary across the adsorbate and thus can describe inhomogeneities. In practice, however, this approach is only useful as long as the adsorbate remains homogeneous in which case the coverage and a few site correlation functions are sufficient to describe the equilibrium and non-equilibrium properties of the adsorbate. If the lateral interactions between adparticles are attractive the adsorbate will develop two-phase regimes below critical temperatures and the above approaches become untractable. To understand this point we briefly look at the canonical and grand-canonical approaches to calculate the equilibrium properties of the adsorbate. As long as the partition function is evaluated exactly the two approaches yield identical results. However, when approximations are invoked, such as the mean field theory, the inherent assumption of homogeneity introduces unphysical features in the canonical ensemble such as van der Waals loops. This is discussed at length by Hill [14], who also points out that, to avoid such artifacts, it is better to work in the grand-canonical ensemble. When a system is taken out of equilibrium these difficulties are compounded even more. However, a solution has been put forward recently that is based on a time-dependent generalization of the grand-canonical ensemble [15]. As with any other new theory some intuition must be developed for the new concepts and this is best done in an exactly soluble model. We will therefore, in this and a following paper, revisit the one-dimensional kinetic lattice gas model, in the canonical and the grand-canonical approaches to develop approximation schemes that can later be generalized for two-dimensional systems. Some exact results and various closure approximations for the linear Ising lattice have been obtained to study finite 0039-6028/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
243
chain effects [16], reactions [17-20], k-mer desorption [20,21], etc. The one-dimensional kinetic lattice gas model without surface diffusion was treated by Evans et al. [22]. For systems with fast diffusion the desorption kinetics are known exactly, because the adsorbate will then remain in quasi-equilibrium throughout desorption and the desorption rate is given by [23,24] dOdt
des = - S ( O , T ) a s 2 ~ r mhak E T 2
exp[/ (0,
(1)
in terms of the chemical potential, /~(0, T), of the adsorbate to be taken at the instantaneous coverage, 0 = O(t). Here S(O, T ) is the coverage and temperature-dependent sticking coefficient and a s is the area of one adsorption site. Because the chemical potential can be calculated exactly, the desorption kinetics is known equally well according to eq. (1). However, in situations where diffusion is slow on the time scale of desorption, one must resort to kinetic equations, e.g. those obtained from non-equilibrium thermodynamics [24-26] or from the kinetic lattice gas model [8-13]. This paper is concerned with the canonical approach. The results will then form the basis for a grand-canonical description in a subsequent paper. In the next section we collect the relevant formulae for the kinetic lattice gas model. In section 3 we will describe a procedure to obtain the hierarchy of equations for the n-site correlation functions, look again at the equilibrium properties in section 4, and define a decoupling s c h e m e used for the closure of the hierarchy in section 5. Numerical examples for desorption are studied in section 6 for slow, intermediate and fast diffusion. For slow diffusion we demonstrate the strong dependence of the kinetics on the initial conditions including a discussion of the usefulness of the Arrhenius parametrization. In section 7 we briefly comment on the coverage dependence of the sticking coefficient. We conclude with a comment on the implications for the two-dimensional lattice gas. 2. The kinetic lattice gas model
To set up the lattice gas model, one assumes that the surface of a solid can be divided into cells, labelled i, for which one introduces microscopic variables n i 1 or 0, depending on whether cell i is occupied by an adsorbed gas particle or not. To introduce the dynamics of the system one writes down a model Hamiltonian [11-12] -----
1
(2)
H = E s ~ n i + ~ V 2 E ninj + " ' ' . i
(ij)
Here E s = - V o - k B T ln(q3qi,t)
(3)
is a single particle energy with V0 the depth of the surface potential, q3 is the single particle partition function accounting for the vibrations of the adsorbed particle in the surface potential and qint is the internal partition function of the adsorbed particle. V2 is the two-particle interaction between nearest neighbors (ij). The coupling to the gas phase is achieved via its chemical potential /xg = k a T In kB r (2~rrnkBT)3/2
-kaT
In Zi~t,
(4)
where P is the pressure and Zi, t is the internal partition function of the free particle accounting for its vibrational and rotational degrees of freedom. Interactions between next-nearest neighbors etc. and many-particle interactions can be added to eq. (2). We restrict ourselves to gas-solid systems in which all relevant processes, like diffusion, adsorption, desorption etc., are Markovian. We introduce a function P ( n ; t) which gives the probability that a given
244
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
microscopic configuration n = ( n l , n 2 , • • • , n t) is realized at time t, where I = N s is the total number of adsorption sites on the surface. It satisfies a master equation [11,12,27] d P ( n ; t ) / d t = E [ W ( n ; n')P(n'; t) - W(n'; n ) e ( n ; t ) ] ,
(5)
nt
where W(n'; n) is the transition probability that the microstate n changes into n' per unit time. It satisfies detailed balance
W(n'; n)Po(n ) = W(n; n')Po(n' ),
(6)
where
Po(n)
= Z -1
exp( - H ( n ) / k s T )
(7)
is the equilibrium probability. In principle, W(n'; n) must be calculated from a Hamiltonian that includes, in addition to eq. (2), coupling terms to the gas phase and the solid that mediate mass and energy exchange. In this paper we rather follow the procedure initiated by Glauber [27] in setting up the kinetic Ising model and guess an appropriate form of W(n'; n). If the residence time in a given state is much longer than the time needed to complete a transition to another state, one can write the transition probabilities as a sum of adsorption-desorption and diffusion processes
W(n'; n) = Wa_d(n'; n) + Wdif(n'; n).
(8)
In a recent paper [11] we have examined various choices and studied their physical implications. For the most part of this paper we choose the "Langmuir kinetics" for which one argues that adsorption into a site i is independent of its local environment. One thus gets for a one-dimensional lattice [11] Wa(n'; n) = W0]~ ( 1 - n i ) 6 ( n ~ , i
1 - n i ) I - I 6 ( n ~ , nt).
(9)
l~i
For desorption we write Wd(n'; n) = WoCoEni[1 + Cl( ni+ 1 + ni_l) q- C2ni+lni_l]8(n ~, 1 - ni) l--I 6(n~, nl) i
=
WoCoEn i i
14=i
exp[(ni+ 1 + n i _ l ) V 2 / / k B
T]
(10)
l--I6(n~, nl). l~i
The second equality follows readily by using the identity n[exp(ot)- 1] = e x p ( n a ) - 1 for n = 0, 1. Detailed balance demands that
C O= exp(Es/kBT )
(11)
C, = [exp(V2/kaT) - 1] r
(12)
and
We note that eq. (10) is the standard form of the desorption transition probability used in Monte Carlo simulations. The adsorption-desorption dynamics in eqs. (9)-(12) are of course different from Glauber's spin flip dynamics when rewritten in lattice gas language. Indeed, the Glauber dynamics produces a trivial desorption kinetics, i.e. one in which the desorption rate is strictly first order with a rate constant independent of coverage [11]. Turning next to diffusion, we write
Wdif(n' ; n)=JoEni(1-ni+a)(lx i,a
d-C 1 E ni+b)t~(n~, b~a
1-ni)t~(n:+a,
1
1-I a(n , n,).
l~i,i+a
(13)
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
245
T o m a k e c o n n e c t i o n with experimental observables we define average occupation n u m b e r s ~i(t)
=
~,nie(n, t),
(14)
n
w h e r e the sum runs over all microscopic configurations n with each ni = 0 and 1. T h e coverage is then given by
O(t) = N s l ~ _ , n i ( t ) = ( n i ) .
(15)
i
3. Equations of m o t i o n
W e p r o c e e d to derive the hierarchy of coupled equations of m o t i o n for multi-site correlation functions. T o this end we multiply eq. (5), e.g. by n i, and sum over all configurations according to eq. (14) to obtain d ( n i ) / d t . W e point out that for an adsorbate with n e a r e s t - n e i g h b o r interactions only, the e q u a t i o n of m o t i o n of the n-site correlation function involves m-site correlation functions w h e r e m -- n, n + 1, • • •, n + c, with c = 2 being the site coordination n u m b e r for a one-dimensional chain of atoms. N o t e that in the n-site cluster at most one site is a d d e d in each direction. T h u s to get the equations up to the n-site correlation functions, one needs to consider a finite chain of (n + 2) sites. In an adsorbate that remains h o m o g e n e o u s t h r o u g h o u t its time evolution the correlation functions are not site specific. T h u s we can use a symbolic notation (')
= (hi) =0,
(" ") = (nini+l) ,
(o) = 1 - ('),
('o')
(''")
= (nini+lni_l)
= (ni(1 -ni+l)ni+2),
,
etc.,
(16)
were the new configuration averages are defined similarly to eq. (14), e.g. (.-)
=N~-I E Enini+lP(n, i
t).
(17)
n
Relations between correlators follow, e.g. (o- -) = (. • ) - (---), (o'o)
= ('o)
(18)
- (" " o ) = ( " ) -
2(" - ) + ( - - ' ) .
(19)
F o r the lowest four correlation functions we get d ( - ) / d t = W o ( o ) - WoCo[( • ) + 2 C 1 ( . d(- •)/dt
(20)
= 21410(0- ) - 2WoCo(C ~ + 1 ) [ ( . • ) + C1( • • • ) ] -- 2 J o [ ( O • • ) - ( . o . )
+C1((o..) d(--.
• ) "]- C 2 ( " " • ) ] ,
- ( ..o. ))],
(21)
) / d t = W012(o • • ) - ( . o . )] - WoCo(1 + C1) [(3 + C1)( • • • ) + 2 C l ( . . . . )] - 2Jo[(O -.. ) - (..o.)
d( . o . ) / d t
= W o [ 2 ( o . ) - 3( . o . ) ]
+ C~((o
...
) -
- WoCo[2(.o. )-
- C 2 ( • • • )] + 2 J 0 [ ( "oo" ) - 2 ( ' o - )
(..o..))]. (---)
(22)
+ 2C1((.o..)
+ (1 - C 1 ) ( " o ' '
- (...))
) "b (1 + C 1 ) ( - " o )
+ C 1 ( " OO" " ) ] .
T h e equations for the correlation functions on four sites are listed in the appendix.
(23)
S.H. Payne et al. / Kinetic lattice gas model in one dimension. 1
246
Comparison of eq. (20) with the phenomenological rate equation [11,23] identifies W0 to be P W o = Soa s ( 2 . n . m k B T ) l / 2 ,
(24)
where S o is the sticking coefficient at vanishing coverage. Note that eq. (20) implies a sticking coefficient S(O) = S 0 ( 1 - 0). On this account we have termed the underlying kinetics the "Langmuir kinetics" because Langmuir invoked site exclusion, and thus the above sticking coefficient, in his kinetic derivation of the Langmuir isotherm [28].
4. Equilibrium properties Before we discuss the closure approximation for the equations of motion, we want to examine the structure of the equilibrium correlation functions. We note that in equilibrium not only must the equations of motion be set equal to zero, but in addition the adsorption-desorption and diffusion terms in each equation must vanish individually. The coverage is then given exactly via exp(lx/kBT)
e x p [ ( V 2 - V o ) / k B T ] a - 1 + 20 = q3qint a + 1 - 20'
(25)
where the chemical potential must be equated to that of the gas phase, given in eq. (4). The two-site nearest-neighbor correlation function is ( " " )0 = 0(1 - 2 1----~a 1 - 0 ),
(26)
a 2 = 1 - 40(1 - 0)[1 - exp( - V 2 / k B T ) ] .
(27)
with
More generally one finds for the correlations between two sites a distance k apart, irrespective of whether the intermediate sites are occupied or not, that [29] (ni+kni)o=02+
(t °~-~-i1 - - 1 ]i k 0 ( 1 - 0 ) .
(28)
From eq. (20) and the adsorption-desorption term in eq. (21) we then find ("")o
= (" • ) 2 / ( . ) .
(29)
We also have (-o')0 = (('o)0)2/(o)
= ((")-
(" • ) 0 ) 2 / ( 1 - ( " ) ) ,
(30)
or
(.) ( . o . )0 = - ~ S ( ( • ) - 2 ( . • )o + ( " " ) o ) ,
(31)
where eqs. (28) and (29) have been used. Higher order correlation functions also factorize exactly and in numerous ways, e.g. ( . . . . ) 0 = ( "'" )o(" " ) o / ( ' )
= (" " ) 3 / ( ' ) 2 =
( . . . ) 2 / ( " ")o,
(" " o ' ) 0 = (" "o)0( . o . ) o / ( ' O ) o = (" • )0( . o . ) 0 / ( .
),
(32) (33)
247
S.H. Payne et aL / Kinetic lattice gas model in one dimension. I
( ....
o.)o=(
....
o)o('"o')o/(""
o)o=(
....
3,
(34) (.)2 ( "oo- )o = 7 , ~ ( (
....
>o- 3(""
)o + 3 ( " • ) o -
("))
+ ( "o" )o = ("o>2o(OO>o/(O) 2,
(35)
and for m holes
('oo...oo')o=
~_~ ( - - ( ' O ) o ) k + l / ( o ) k=l
~,
(36)
and so on. N o t e that these factorizations are such that the sub-clusters on the right can have one or more occupied or empty sites in common. In an earlier paper on the two-dimensional kinetic lattice gas model [13] correlation functions of the kind ( - x . ) = ( . - - ) + ( • o . ) were factored instead of ( . o . ). For example, one finds that ( - x . ) (.x)/(x) = ( . ) 2 and unnecessary difficulties arise in the truncation scheme for the hierarchy of equations of motion that can be overcome easily with the present set of functions. In fig. 1 we present some numerical examples of correlation functions for a strong repulsive nearest-neighbor interaction. N o t e in particular the alternating structure in the two-site correlation functions as k in eq. (28) increases, see fig. la. Indeed, in the low-temperature limit one finds that at 0 = 1 / 2 these functions are zero for k odd and equal to 1 / 2 for k even, reflecting the ordered structure of every second site being occupied. The correlations with h o l e s / u n o c c u p i e d sites present, display this feature explicitly, see fig. lb. Those with two or more adjacent holes decay rapidly to zero above 0 = 0.5, and ( - o - o . ) has a maximum there. For attractive nearest-neighbor interactions the functions ( n i n i + k ) simply increase monotonically as a function of coverage. It is also interesting that for strong attractive interactions all correlation functions are equal to the coverage. T o understand this we note that the probability to find a given site occupied is 0. With strong attractions the probability to find a site nearby occupied is one due to clustering. 1.0
. . . .
.8
,
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
,
Q
. . . .
i
. . . .
r
,~'
.5
. . . .
u) c-
...,
....
, ....
, ....
, ....
, ....
, ....
, ....
, ....
, ....
.4
O
0
~)
.6 L-
L*_
0 /
//// // ,, / I//
.2
o
.1
2
.3
.4
///
--~/i
0
//
,,/,
_ _ > 0 (_)
/ - -
/ / / //
(3
i/I /
/
0 C)
i i I /
.1
l/
5
6
Coveroge (ML)
7
8
e
.1
.2
.3
.4
.5
.6
.7
.8
.9
Coveroge (ML)
Fig. 1. Equilibrium correlation functions for a one-dimensional adsorbate with repulsive nearest-neighbor interactions (V 2 / k B T = 5): (a) n-particle (solid lines) and two-particle (dashed lines) correlations (left to right at arrow position) (. x" ), (- xxx. ), (. xxxxx • ). (.xxxx-), (-xx.), (--), (...), ( . . . . ), ( . . . . . ), (b) one-hole (solid), two-hole (dash) and three-hole (long/short dash) correlations (top to bottom at 0 = 0.6) (- o. ), (. o- o. ), (- o - . ), ( . . o-- ), (- oo. ), (. ooo. ).
248
S . H . P a y n e et al. / K i n e t i c lattice g a s m o d e l in o n e d i m e n s i o n . I
4 4
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
,
. . . .
i
. . . .
i
. . . .
i
. . . .
i
. . . .
42 B 0
E
~ ~,!
40
......
E = =_.= ~ = =_.=_.~ =- ~- ~
"T-
34
O
32 0 Go m
30 2 8
. . . .
0
i
.1
. . . .
i
.2
. . . .
i
.3
. . . .
i
. . . .
.4 Coveroge
,
.5
. . . .
i
. . . .
.6
,
.7
. . . .
I
.8
. . . .
I
. . . .
.9
(ML)
Fig. 2. Coverage and temperature dependence of the isosteric heat of adsorption for repulsive and attractive interactions. Solid lines: V2 = 400 K for T = 90, 120, 150 K (top to bottom at 0 = 0.3). Dashed lines: I"2 = - 4 0 0 K with T = 120, 140, 160 K. Temperatures span the desorption range (cf. figs. 3 and 4) for substrate/adsorbate parameters (cf. eqs. (3) and (4)), V0 = 4500 K, v x = vy = v z = 1012 s -1 in q3; qint = Zint = 1. These parameters are assumed throughout.
We next look at the isosteric heat which can be calculated from eqs. (4) and (25) according to 20In P Qi~o(0, T) --kBT
~
o"
(37)
Numerical examples are given for repulsive and attractive nearest-neighbor interactions in fig. 2. We note that the coupled substrate-adsorbate system has two dominant energy scales, namely the single-partide binding energy, V0, introduced in eq. (3) and the nearest-neighbor interaction energy, V2, in eq. (2). The ratios we choose are typical of many systems. The isosteric heat (eq. (37)) will also be useful later on to understand thermal desorption in systems with fast diffusion.
5. Closure approximations
To make the set of coupled equations of motion for the correlation functions the basis of an analytic theory of surface processes, the hierarchy must be truncated. The simplest such closure approximation is the Kirkwood approximation in which one expresses all higher correlation functions in terms of two-body correlation functions. To develop a systematic closure scheme we are guided by similar procedures in equilibrium statistical mechanics. We recall that in the quasi-chemical approximation [14] one calculates the partition function for two sites exactly and then distributes these pairs randomly over the lattice, normalizing the result to account for double counting of overlapping sites. Similarly in Hill's approximation [30] one calculates the partition function for a square of four sites (on the square lattice) exactly. Kikuchi [31] and Morita [32] generalized this approach also taking account of the environment of such clusters in a self-consistent way. Any scheme to truncate the infinite hierarchy of equations of motion for the correlation functions by factorizing higher order correlation functions in terms of lower order ones must reduce to the factorization properties in equilibrium contained, e.g., in eqs. (32)-(36).
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
249
5.1. Two-site closure
In the two-site (or Kirkwood) approximation one retains the equations of motion for the coverage, ( • ), and the two-site nearest-neighbor correlation function, ( • • ). In eqs. (20) and (21) we factorize all higher order correlation functions with single-site overlap so that only single-site functions ( . ) and ( o ) = 1 - ( • ) appear in the denominator, e.g. (...)
= (..)2/(.),
(38)
(" " o ' ) = (" " ) ( ' o ) 2 / ( " ) ( o ) .
(39)
The truncated set of equations thus reads in the two-site approximation d ( . ) / d t = W o ( o ) - WoCo(" )(1 + C1(. • ) / ( .
))2,
d ( . . ) ~ d r = 2W0(o • ) - 2WoCo(Cl + 1 ) ( ( . . ) + G ( ' " _ (..)2/(.
(40) ) 2 / ( . )) _ 2Jo[( • • )
) _ (. 0)2/(0)
+CI((. ")_ (..)2/(.
)_ ( . . ) ( O " ) 2 / ( . ) ( O ) ) ] .
(41)
repeat that eqs. (40) and (41) yield the exact equilibrium solution. On two-dimensional lattices the two-site closure yields the quasi-chemical approximation which is exact on a one-dimensional lattice. Note, however, that eqs. (40) and (41) are approximate away from equilibrium because the factorizations no longer hold exactly.
We
5. 2. Three-site closure and beyond
To go beyond the two-site closure we include all correlation functions on three sites, i.e. (.),(--),(--.),(.o.).
(42)
All higher order correlation functions appearing on the right-hand sides of eqs. (20)-(23) will be factorized in terms of three-site correlation functions in the numerator with two-site overlap. These overlapping sites appear as two-site functions in the denominator to compensate for overcounting, e.g. ( .... ) = (-.-)(...
)/(.
• ),
(" "o" ) = (" " o ) ( ' o " ) / ( ' o ) , ('oo"
• ) = ('oo)(oo")(o"
(43) (44)
• )/(oo)(o').
(45)
In the absence of diffusion this scheme produces an exact solution for desorption [22]. This arises because the equations of motion for the full n-site particle correlator, Pn, reduce to
de. dt
= - W ° C ° { [ n + 2C1(n - 1) + C2(n - 2 ) ] P , + 2(C 1 + C2)P~+I},
(46)
for n > 2. One then verifies that the time evolution of P~ for n > 4 is correctly given by that of Pn_ 1P3/P2. The ratio P3/P2 evolves independently of any correlator, i.e. d -~-f(P,,+I/P,,) = -- WoC0(1
+ C1) 2,
(47)
for n >_ 2. Hence, the equations for ( • ), ( • • ) and ( • • • ) suffice. Note that correlation funetions with holes, such as ( -o • ) cannot enter the desorption kinetics.
250
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
This factorization scheme is unique which is not obviously the case for any other scheme. For example, if we factorize with single-site functions in the denominator, we have the following possibilities (- - o . )---- (- - ) ( . o . ) / ( ' )
(48)
(..o.)
(49)
and = (..o)(o.)/(o).
Either choice is exact in equilibrium. However, away from equilibrium, eqs. (43)-(45), (48) or (49) yield different time evolutions. The "best" convergence as one includes more correlation functions in the basis set is achieved by the first choice above, namely two-site overlap in the three-site approximation. The generalization to larger basis sets is guided by the principle of maximum overlap. E.g., in the five-site approximation we factorize all higher correlation functions in terms of all possible five-site functions in the numerator with four-site overlap (and thus four-site functions in the denominator), e.g., < . o . - o - "> = < . o . . o > < o . . o . > < .
-o. ->/
(50)
If we were to choose less than maximum overlap, there would be a multiplicity of choices with no clear indication of which is the optimum in some sense.
6. Thermal desorption
6.1. Mobile adsorbate We begin our study of thermal desorption kinetics by looking at mobile adsorbates, i.e. at systems in which surface diffusion is much faster than adsorption and desorption. As a consequence, such adsorbates will remain in quasi-equilibrium throughout desorption, i.e. all correlation functions will attain their equilibrium values at the instantaneous coverage. The rate of desorption is then given by eqs. (1) and (25). In fig. 3 we present numerical results for temperature-programmed desorption from a one-dimensional chain of mobile atoms with repulsive nearest-neighbor interactions (solid lines). We note that for initial coverages 00 > 1/2, a second, low-temperature peak develops reflecting the drop in isosteric heat (solid lines in fig. 2). To make this connection we parametrize the desorption rate as
~t des= -S(O, T)v(O, T)
exp(--Ed(0,
r)/kBr)o
(51)
as a quasi-first-order process with a coverage-dependent rate constant, where Ed(0,
T) -kBT2-~0
In ( - l d 0dt0
des]l
(52)
is the differential desorption energy. We get, via eqs. (1), (4) and (37),
Ed( O, T)
0
= Qiso(0,
T) -kBT/2 + kBT2~-~ In S( O, T).
(53)
Because in most systems the sticking coefficient varies little with temperature (and not at all for Langmuir kinetics), we see that the desorption energy is essentially equal to the isosteric heat for mobile adsorbates. As explicitly displayed in eq. (51), the desorption rate is proportional to the sticking coefficient which for Langmuir kinetics is S(O)= S 0 ( 1 - 0). If we were to use eq. (51) as the starting point for a
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I 3.5
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Temperoture Fig. 3. T e m p e r a t u r e - p r o g r a m m e d d e s o r p t i o n rate (in units o f m o n o l a y e r s per s e c o n d ) for a m o b i l e a d s o r b a t e with repulsive i n t e r a c t i o n V 2 = 400 K. Initial c o v e r a g e s (left to right) 00 = 0.9,
0 .... !30
=r,~ 135
140
145
150
155
Temperoture Fig. 4. T e m p e r a t u r e - p r o g r a m m e d d e s o r p t i o n rate for a m o b i l e adsorbate with attractive i n t e r a c t i o n V 2 = - 4 0 0 K. O t h e r w i s e as fig. 3.
0.7, 0.5, 0.3, 0.1. Parameters: V0 = 4500 K, 1,I2= 400 K; other p a r a m e t e r s : h e a t i n g rate 1 K s -1, a s = 10 ~2, m = 28 amu. Solid lines: sticking S(O) = S0(1 - 0). D a s h e d lines: S = S O = 1.
phenomenological description, we could take any (measured) coverage dependence of the sticking coefficient, e.g., setting it equal to a constant S O up to a coverage close to a monolayer. Such a behavior is, e.g., to be expected in the presence of precursors. Temperature-programmed desorption is then given by the dashed lines of fig. 3, demonstrating the marked change of the low-temperature peak caused by the decreasing sticking coefficient. In fig. 4 we give an example of temperature-programmed desorption from a chain of atoms with attractive nearest-neighbor interactions and S(O) = S0(1 - 0), remaining in quasi-equilibrium throughout desorption (solid lines). For higher initial coverages the peaks shift to higher temperatures reflecting the fact that the isosteric heat, and thus the desorption energy, rises (dashed lines in fig. 2). The dashed lines in fig. 4 give equivalent results for a constant sticking coefficient, demonstrating again the radical effect of the sticking coefficient on desorption. In closing this section we note that in the limit of fast diffusion we necessarily get identical results to those obtained from eq. (1) in figs. 3 and 4 by using the equations of motion with large hopping rate J0 (in any approximation).
6.2. Immobile adsorbate If surface diffusion is slow in the temperature range of desorption we must make use of the equations of motion to describe the desorption kinetics. The largest deviations from the desorption kinetics of a mobile adsorbate obviously occur for an immobile adsorbate where surface diffusion is negligible throughout desorption. If the time scale of surface diffusion is of the order of that of desorption we expect results intermediate between those of fast and negligible diffusion. In fig. 5a we present temperature-programmed desorption spectra for an immobile adsorbate with nearest-neighbor repulsion for varying initial coverages. The dashed lines are obtained in the two-site closure, while the solid lines result from the three-site closure (which, according to section 5.2 is exact). Clearly the two-site approximation is inadequate. Evans et al. [22] have interpreted the three peaks in the
S.H. Payne et aL /
252 4.,5
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Kinetic lattice gas model in one dimension. ,
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Temperoture
Fig. 5. ( a ) T e m p e r a t u r e - p r o g r a m m e d d e s o r p t i o n r a t e f o r a n i m m o b i l e a d s o r b a t e w i t h r e p u l s i o n V 2 = 4 0 0 K. S o l i d lines: t h r e e - s i t e c l o s u r e . I n i t i a l c o v e r a g e s ( l e f t t o r i g h t ) O0 = 0.9, 0.7, 0.5, 0.3, 0.1. D a s h e d lines: t w o - s i t e c l o s u r e ( f o r initial c o v e r a g e s 00 = 0.9, 0.5; f o r initial c o v e r a g e s l e s s t h a n 0.3 t w o - a n d t h r e e - s i t e c l o s u r e s c o i n c i d e ) . (b) C o r r e s p o n d i n g e v o l u t i o n r a t e s d ( . - ) / d t (solid lines) a n d d ( • - • ) / d t ( d a s h e d l i n e s ) in t h r e e - s i t e c l o s u r e f o r 00 = 0.9, 0.7, 0.5.
desorption spectra as "staged" desorption, with the low-temperature peak reflecting desorption of particles from an environment of two occupied neighboring sites and the smaller middle peak as desorption from the ends of chains. This picture is nicely confirmed by looking at the time evolution of ( . • ) and ( . . . ), fig. 5b. We have performed an Arrhenius analysis of the desorption spectra in fig. 5a by plotting isosteric rates versus T-1. For an immobile adsorbate these curves are not expected to be straight lines. However, instead of using eq. (52) to define a coverage- and temperature-dependent desorption energy we do a linear fit for simplicity. The resulting (temperature-averaged) desorption energy and prefactor are plotted in figs. 6a and 6b for the two-site closure and the exact immobile and mobile cases. Several connections can be made to preceding graphs. First, as we know from eq. (54), for a mobile adsorbate the desorption energy equals the heat of adsorption, to within a factor kBT/2. With this correction, the long/short dashed curve of fig. 6a is just the temperature average of the (solid) curves in fig. 2. Second, the two peaks in fig. 3 correspond to the two plateaus in the desorption energy, fig. 6a, at low and high coverage. Third, although we get a two-peak structure in the desorption spectrum for an immobile adsorbate calculated in the two-site approximation, similar to the mobile case, the desorption energy looks quite different with the appearance of a peak at half coverage. The exact (three-site) calculation is even more intriguing with the appearance of two peaks in the desorption energy. In the analysis of experimental desorption data, such structures might be interpreted as arising from attractive interactions. This is clearly not the case here. Rather, because the minima in the desorption rate are much lower for an immobile adsorbate than for a mobile one, the effective desorption energy must be much higher. The corresponding prefactors also show sharp peaks, which would be interpreted as an ordering above that for equilibrium. Clearly these structures are an artifact of the Arrhenius analysis. Indeed, their form depends on the strength of the interaction, the initial conditions (initial coverage and adsorption temperature) and the range of desorption temperatures over which the (linear) analysis is performed. Such effects have already been discussed for models of desorption of two-dimensional systems [24]. As examples we display, in fig. 6c, the differential desorption energy for a different interaction value and initial coverage (prepared in equilibrium) obtained by isothermal desorption. The variation of the peak structure between the lowest and the highest temperature traces (corresponding to
S.H. Payne et aL / Kinetic lattice gas model in one dimension. I 40
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0 < 0 0 and 0 --* 0 in the T P D of fig. 5a, respectively) is significant on the scale of the interaction. The temperature average of both sets of traces in fig. 6c will yield peaks of Ed(O) at 0 = 0.4 noticeably different from that in fig. 6a. The Arrhenius analysis should be reserved for systems that remain in quasi-equilibrium where the interpretation of the desorption energy as the heat of adsorption and the prefactor as a measure for the entropy is rigorous. In fig. 7a we show desorption from an immobile adsorbate with attractive interactions, with the curves of the mobile case included for comparison. We note that the peak desorption from an immobile adsorbate occurs at a higher temperature than from a mobile adsorbate. This can be traced to the fact that during desorption there are more nearest-neighbor pairs and fewer holes (e.g. ( . o . >) in the immobile system, compare figs. 7b and 7c, respectively. Details of this evolution also depend on the initial conditions, as for repulsive interactions, but in a more marked manner. For instance, at low initial coverages a low-temperature s h o u l d e r / p e a k develops in the rate at a position corresponding to the
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
254
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Solid lines: t e m p e r a t u r e - p r o g r a m m e d d e s o r p t i o n rate for an i m m o b i l e adsorbate with attraction V 2 = - 4 0 0 K , in three-site closure for initial c o v e r a g e s (top to b o t t o m ) O0 = 0 . 9 , 0 . 7 , 0 . 5 , 0 . 3 , 0 . 1 . D a s h e d lines: m o b i l e adsorbate f r o m solid lines o f f i g . 4 . ( b ) C o r r e s p o n d i n g e v o l u t i o n o f < - • > for i m m o b i l e (solid) and m o b i l e (dash) systems. (c) E v o l u t i o n o f < . o . > for both cases. Fig.
7. (a)
desorption peaks for a non-interacting system. This is the result of the initial desorption of isolated particles in the adsorbate (there are relatively fewer in number at higher coverage) which are not replenished by diffusion as in the mobile case, see fig. 8a [33]. Such a feature clearly alters the interpretation of the spectra, and the Arrhenius analysis in particular. As another demonstration we show in fig. 8b isothermal rates for three initial coverages, prepared in equilibrium. If we now (against better judgement) perform an Arrhenius analysis of isothermal rates we obtain vastly different results for the desorption energy depending on the initial conditions, fig. 8c. For the case 00 --- 1, Ea(O) is essentially independent of 0 for the value of V 2 assumed. ( U p o n increasing 11.I2 I we find Ea(O) - Ed(0) ~ -- 1.5V2, the limit deduced by Evans et al. [22].) This behaviour can only persist for lower initial coverages, if the initial distributions < • • >, < . . . > have those (non-equilibrium) values which would have evolved from 0 o = 1. Again, one can associate the peaks of Ed(0) with the temporary reduction of the desorption rate, in this case for the low 00 traces of fig. 8a. Yet another, and more complicated picture would emerge if we were to do an isosteric analysis of T P D data, obtained either by the method of variation of initial
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
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TemperQture 44 ,
%
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It r e m a i n s to l o o k at the role o f diffusion. Firstly, in the a b s e n c e o f desorption, fig. 9 shows the time d e p e n d e n c e o f a n u m b e r o f correlation functions calculated at the three-, four- and five-site truncation levels in the approach to equilibrium, starting initially from an ideal gas. W e n o t e that the e v o l u t i o n is slower the larger the basis. For the two- and three-particle correlators o f fig. 9a c o n v e r g e n c e to the
256
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I .4
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x
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Time (sec)
.03
.04
Time
.05
.06
.07
(sec)
Fig. 9. I s o t h e r m a l e v o l u t i o n o f adsorbate c o r r e l a t i o n functions towards e q u i l i b r i u m f r o m a r a n d o m distribution in the a b s e n c e o f desorption, c a l c u l a t e d in three-site ( . . . . ), four-site (- - -), and five-site ( ) closure approximations. O r d i n a t e is n o r m a l i z e d deviation f r o m e q u i l i b r i u m value. 0 = 0.9, V 2 = 400 K, and h o p p i n g rate J0 = 1 s -1. (a) n-particle correlations (top to b o t t o m at t = 0): ( . . . . . ), ( . . . . ), < . . . >, < . . >, (b) c o r r e l a t i o n s with holes: ( . o o - > x 10 - 2 , < . . o - . >, < - o . >, < -o . o . >.
"exact" evolution has already been achieved at the four- and five-site levels, respectively. This is confirmed for the hole functions of fig. 9b: those containing 2 particles, e.g. < • o . > and ( • o o . >, are accurately calculated at the four-site closure level. Obviously multi-site correlators require a truncation within a larger basis, but out results for diffusion alone suggest a rule of thumb for repulsive interactions: correlations on m sites with n < rn particles require closure at the (n + 2)-site level for "exact" evolution. Qualitatively similar results are obtained for attractive interactions, but on a longer time scale. Next we look at desorption from adsorbates with finite hopping rates, J0, larger and smaller than the desorption rate constant, WoC o, in the midrange of desorption temperatures. We display, in fig. 10, TPD spectra for a system with repulsive interactions. The variation of J0 between the immobile and mobile 4
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81
90
100
110
120
130
140
150
TemperQture
Fig. 10. T e m p e r a t u r e - p r o g r a m m e d d e s o r p t i o n rate for adsorbate with repulsion, V 2 = 400 K, for 00 = 0.9 and varying hopping rates H o p p i n g barrier Q = 0, a n d v o = 10 . 4 s - 1 (solid, t h r e e - p e a k e d ) (effectively immobile), 10 . 3 s - 1 ( - - - ) , 10 - z s - t (__ _ _ __), 10 - 1 s - 1 (__ _ _ __), 1 s - 1 ( . . . . ), and 10 s - ] ( ) (effectively mobile).
Jo = vo e n d ( - Q / k B T ) .
S.H. Payne et a L / Kinetic lattice gas model in one dimension. I
257
limits is approximately five orders of magnitude. These results for the desorption rate are obtained in the three-site approximation, there is negligible improvement in the four-site approximation. This does not contradict our earlier statement that isothermal and isosteric diffusion towards equilibrium can only be described accurately within the four- or even five-site approximation: the exponential decay of the coverage in desorption eliminates the need for a detailed knowledge of the long-time evolution of the multi-site correlators due to diffusion, thus making the factorization scheme almost exact.
7. Sticking coefficient
Everything done so far in this paper is based on the "Langmuir kinetics" contained in the transition probabilities eqs. (9)-(12). They imply in particular that the sticking coefficient is given trivially by S(O) = S0(1- 0). For most systems this simple form, based on site exclusion only, does not apply. In particular, if precursors mediate adsorption one expects a sticking coefficient that is initially coverage independent. One might also argue that the adsorbing particle experiences the local environment in a way analogous to the desorbing particle. In such cases lateral interactions will affect the adsorption probability, e.g., repulsive nearest-neighbor interactions will aid desorption and inhibit adsorption. For such a scenario, which we term interaction kinetics, we can choose the transition probabilities for adsorption and desorption to read [11] Wa(n'; n) = W o E ( 1 - n / ) [ 1 - Cl(ni+l-I-ni_x) -C2ni+lni_l]~(n~, i
Wd(n';
n) = WoCo~ni[1
+ Cl(ni+l
--1-n i _ l )
d- C 2 n i + l n i _ l ] t ~ ( n ~ ,
i
1-ni)l-Is(n~, l~i
1 - ni) I"Is(n~, l~i
nl).
nt),
(54) (55)
Detailed balance gives C1 = tanh(V2/2kBT ) ,
(56)
C2 = tanh( V2/kBT
(57)
) - 2C 1. 1.1
....
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, ....
, ....
, ....
, ....
, ....
, ....
, ....
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, ....
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~,?..,,,
.1
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.1
.2
.3
.4
.5
.6
.7
.8
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Coverage (ML) Fig. 11. Normalized sticking coefficient, S(O)/So, in the interaction kinetics for attractive interactions (dashed lines) and repulsive interactions (solid lines). Values of V2 / k n T (top to bottom at 0 = 0.4): - 2.0, - 0.5, - 5.0, - 10.0, 0.0, 0.5, 2.0, 5.0, 10.0.
258
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
From eq. (54) we get the effective coverage-dependent sticking coefficient as
S(O, T)=S0((o)-2C1<'o)-C2<'o')).
(58)
For adsorption, the correlation functions on the right of eq. (58) are those in equilibrium if adsorption occurs at temperatures where the adsorbate is mobile. On the other hand, at very low temperature adsorption is a r a n d o m process so that the correlation functions are ( ' o ) = ( - ) ( o ) and ( ' o ' ) =
<. >2(o>.
Numerical examples of sticking coefficients for a mobile adsorbate are given in fig. 11. For strong repulsive nearest-neighbor interactions the sticking coefficient falls initially as S(O)= S0(1 - 30) as both neighboring sites must be empty as well. Above half a monolayer sticking is essentially zero. For attractive interactions the sticking coefficient is always larger than 1 - 0. It rises above its value at zero coverage, as was also observed for a square lattice in the quasi-chemical approximation [11]. We also note that in the limit of very strong attraction the sticking coefficient approaches the limit 1 - 0 again due to the fact that all particle correlators are then equal to 0, cf. eqs. (26)-(29). Due to the fact that the transition probabilities must satisfy detailed balance in equilibrium, the rather complicated coverage dependence of the sticking coefficient will reflect itself in the desorption kinetics as well. Thus an attempt to fit desorption data only without considering the adsorption characteristics and also the equilibrium properties of the system might lead to a wrong model.
8. Conclusions
In this p a p e r we have studied the one-dimensional kinetic lattice gas model to examine the convergence of closure approximations for the hierarchy of equations of motion governing the n-site correlation functions. We have shown that a unique and in a sense " b e s t " factorization scheme involves maximum overlap in which an n-site correlator is factorized with ( n - 1)-site correlators in the numerator and (n - 2)-site correlators in the denominator. In a subsequent p a p e r we will see that the analogous closure scheme is also " b e s t " for the time-dependent generalization of the grand-canonical approach. For strongly interacting one-dimensional systems in which the time scale of diffusion is of the order of that of desorption, three-site closure is sufficient for an accurate description of the desorption kinetics, whereas isothermal and isosteric diffusion is only accurate in the four- to five-site closure. In systems with slow diffusion, the kinetics depend on initial conditions and non-equilibrium effects show up strongly in desorption. This implies, as we have shown in numerical examples, that an isosteric Arrhenius analysis of desorption data is inappropriate and more often than not, misleading. However, a differential desorption energy can be defined and, indeed, measured by the threshold method [34,35], as demonstrated for rare gases on metals where non-equilibrium effects due to slow diffusion are known to be present [36]. The lesson to be learnt is that one should model equilibrium and kinetic data directly. The implications for the two-dimensional kinetic lattice gas model remain to be explored. In a previous p a p e r [13] we have developed a systematic closure approximation for a square lattice including all correlation functions on a square of four sites, i.e.
<->, (- • >, <..>, <:.>, <::>,
(59)
yielding good results. However, it has remained a mystery so far why this scheme did not work well for the diffusional part of the equations of motion. The present study of the one-dimensional lattice gas suggests a solution, namely that correlation functions of the form ( • o • ) must be kept. We intend to do this in a systematic way by including all correlation functions on a four- and on a nine-site cluster.
S.H. Payne et aL / Kinetic lattice gas model in one dimension. I
259
Acknowledgement F u n d i n g for this r e s e a r c h h a s b e e n p r o v i d e d by t h e N e t w o r k o f C e n t r e s o f E x c e l l e n c e in M o l e c u l a r a n d I n t e r f a c i a l D y n a m i c s , o n e o f t h e f i f t e e n N e t w o r k s o f C e n t r e s o f E x c e l l e n c e s u p p o r t e d by t h e Government of Canada.
Appendix In addition to the equations of motion in eqs. (20)-(23) we have the following for the remaining c o r r e l a t i o n f u n c t i o n s o n f o u r sites.
d( ....
) / d t = 2W0(( • • • o) + (- -o" )) - 2WoCo(1 + C,) [(2 + Ct)( +C,(
.....
)] + 2J0[(.o
....
)
. - - > + C l < - "O " ' " ) - - (1 + C 1 ) < o . . . .
d(" "o- ) / d t = Wo((O'O" ) + ( "oo" ) + (" -oo) - (" "o" )) - WoCo[(3 + 2C1)(" -(1
+ C1)2(
+J0[("
-oo->
+ ( 1 -I..-C 1 ) ( < o
.
.
.
.
) +C1(''0""
) +C,(1
+ < "o " o " > + C l ( ( " '''
+C,)(
>],
(A.1)
"o" )
. o . . . )]
"oo" • ) + ( " " o ' o " >)
> .--I- < "o ' ' " >) - ( 3 + C 1 ) ( " o " • ) - ( C 1 - 1 ) ( -
"o'"
)], (A.2)
d( "oo" )/dt
= 2Wo((OOO" ) - ("oo"))
- 2WoC0[(
"oo" ) - ("o"
" ) + CI(( "oo" • )
-('o''>)] +2]o[<'ooo'>-2('oo'>-('o''>+('o')+('oo'') +C1( < "ooo""
> - < "oo""
>)].
(A.3)
References [1] For a review of the kinetic Ising model, see K. Kawasaki, in: Phase Transitions and Critical Phenomena, Vol. 2, Eds. C. Domb and M.S. Green (Academic Press, New York, 1972). [2] D.J.W. Geldart, H.J. Kreuzer and F.S. Rys, Surf. Sci. 176 (1986) 284. [3] J. Luscombe, Phys. Rev. B 29 (1984) 5128. [4] E. Oguz, O.T. Vails, G.F. Mazenko and J.H. Luscombe, Surf. Sci. 118 (1982) 572. [5] See, e.g., J.D. Gunton, in: Kinetics of Interface Reactions, Eds. M. Grunze and H.J. Kreuzer (Springer, Berlin, 1987). [6] K. Binder, Ed., Monte Carlo Methods (Springer, Berlin, 1986). [7] K. Wada, M. Kaburagi, T. Uchida and R. Kikuchi, J. Stat. Phys. 53 (1988) 1081. [8] A. Surda and I. Karasova, Surf. Sci. 109 (1981) 605. [9] V.P. Zhdanov and K.I. Zamaraev, Sov. Phys. Usp. 29 (1985) 755. [10] J.W. Evans, J. Chem. Phys. 87 (1986) 3038, and references therein. [11] H.J. Kreuzer and Zhang Jun, Appl. Phys. A 51 (1990) 183. [12] H.J. Kreuzer, Surf. Sci. 231 (1990) 213. [13] A. Wierzbicki and H.J. Kreuzer, Surf. Sci. 257 (1991) 417. [14] T.L. Hill, Statistical Mechanics (McGraw-Hill, New York, 1956). [15] H.J. Kreuzer, Phys. Rev. B 44 (1991) 1232. [16] R.H. Lacombe and R. Simha, J. Chem. Phys. 61 (1974) 1899. [17] A. Silberberg and R. Simha, Biopolymers 6 (1968) 479. [18] G. Schwarz, Ber. Bunsenges. Phys. Chem. 75 (1971) 40. [19] R. Simha and R.H. Lacombe, J. Chem. Phys. 55 (1971) 2936.
260
[20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
[34] [35] [36]
S.H. Payne et al. / Kinetic lattice gas model in one dimension. I
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