Diffusion and desorption in weakly interacting adsorbate

Physica A 195 (1993) 375-386 North-Holland

SDI: 0378-4371(92)00303-9

Diffusion and desorption in weakly interacting adsorbate

M a g d a l e n a A . Z a l u s k a - K o t u r Institute of Physics, Polish Academy of Sciences, Al. Lotnik6w 32/46, 02-668 Warszawa, Poland

L u k a s z A . T u r s k i IFF der KFA Jiilich, W-5170 Jiilich, Germany and Center for Theoretical Physics, Polish Academy of Sciences, 02-668 Warszawa, Poland

Received 6 October 1992

Using a properly chosen master equation describing combined desorption and surface diffusion processes in interacting adsorbate layers, we analyze the coverage evolution and calculate the dynamical structure factor for the adsorbate in the weak intra-adsorbate interaction limit. The connection with formerly analyzed desorption-diffusion processes in noninteracting adsorbate is made. For a simplified one-dimensional model the analytically derived expression for the dynamical structure factor permits discussion of its dependence on sign of the interactions and value of the desorption rate.

1. Introduction

When a freshly cleaved solid surface is exposed to the ambient gas phase, the mutual interactions between particles of these two systems often results in format ion of a macroscopically thin layer of gas particles covering the solid surface: the adsorbate [1]. We shall restrict ourselves here to the physisorbed layers and address the question of interrelations between two dynamical

processes affecting macroscopic propert ies of the physisorbed layers, namely desorpt ion and surface diffusion [2]. There is an extensive literature in which both of these phenomena are investigated separately [3,4]. The issue of the intra-adsorbate particle interactions and their influence on either diffusion or desorpt ion is only recently attracting some attention [5,6]. The intra-adsorbate interactions are of course crucial in the description of the variety of phase t ransformations in the adsorbates [7], and in the previous publication [8] we have formula ted a variant kinetic lattice gas and /o r Potts models permitt ing us to analyze the influence of the intra-adsorbate interactions on the desorption

0378-4371/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved

376 M.A. Zaluska-Kotur, L.A. Turski / Diffusion and desorption in interacting adsorbate

time. In another publication [9] we have proposed a fluid dynamical model which takes into account the influence of the intra-adsorbate interactions on the diffusion processes.

In this paper we shall use the kinetic lattice gas model from ref. [8] together with the master equation approach to combined surface diffusion and desorption processes from ref. [10] in analysis of these processes in a weakly

interacting adsorbate. Using systematic expansion in the small parameter measuring the intra-adsorbate interactions ~ we calculate the dynamical structure factor for the adsorbate S(q, to). In the limit of 5 = 0 we will recover the dynamical structure factor calculated in ref. [10]. For a simple one-dimensional model we were able to calculate the explicit form of the dynamical structure factor and see how it depends on the values of various pertinent coefficients, rate of desorption and sign of the coupling constant ~.

2. The model

The convenient tool of description of the dynamical processes happening in the physisorbed layer of, for example noble gas on the graphite surface or carbon oxides on some metal surfaces, is the kinetic lattice gas model. The lattice gas model, close kin of the Ising model, has been extensively used in describing the equilibrium properties of the adsorbate [1,7,8]. The kinetic generalization of such a model requires a proper master equation. Construction of that master equation is an "art" in itself, but one can be guided in that endeavor by general rules, out of which the detail balance requirements are of predominant importance [11]. We shall consider the surface of the solid on which the adsorption sites form a regular Bravais lattice. The adsorption lattice points will be labeled by i and their occupation number n i equals 0 or 1. Thus multiple occupation of the same adsorption site is not permitted (this is often alluded to as blocking). The complete dynamical information of such a system is now contained in the probability density P(n 1 . . . . . nN; t) of finding lattice site i = 1 , . . . , N occupied or empty. This probability density obeys a master equation containing transition probabilities constructed such that the equilibrium solution of the master equation is the canonical distribution function for the adsorbate, corresponding to the microscopic Hamiltonian for the adsorbate

y( = _ ~ £ nink _ ~r £ ni ' (1) i,k i

where ~ is the coupling constant describing mutual interactions (attractive or repulsive, depending on the sign of ~) and ~ is an effective one-site potential related to the chemical potential (pressure) of the ambient gas phase.

M.A. Zaluska-Kotur, L .A. Turski / Diffusion and desorption in interacting adsorbate 377

Our master equation describing the adsorbate, which can simultaneously desorb from the surface and diffuse along it, has the form

O t P ( n , , . . . , n N ; t ) = ~ [W2(n ; , n ~ ) P ( n , , . . . , n ; , n ~ , . . . n N ; t ) ( i , j )

- - W 2 ( n i , n]) P ( n l , . . . , n i, n j , . . . , n N ; t)]

- Z r ( n i ) P ( n l , . . . , nN; t ) . (2) i

The first two terms on the right hand side of eq. (2) describe the adsorbate particle jumps between neighboring lattice sites i, j, and W 2 are the proper rates for those jumps. Then third term describes desorption processes, and F ( n i ) is a measure of the desorption rate. As it stands, eq. (2) is completely general and all references to the specific adsorbate and solid surface models are contained in the form of the transition probabilities W 2 and F. In ref. [10] we have used a similar master equation to describe the combined diffusion and desorption processes in noninteracting adsorbate. The transition probabilities W 2 from ref. [10] have to be modified to account for interactions, and following ref. [8] we postulate them in the form below, where the symbol ( i ~ j ~ t indicates that the lattice site i is a nearest neighbor of the site l which is no t

equal to ]:

W2(ni , n j ) = W i , j n i ( 1 - n j ) +Wj,inj(l - n i ) , (3)

with

nil ( )~ (t,~i)j

/ \ Wj, i = W 0 exp[-A/3o¢ ~ , n, + 1~/35~ ~ n k } . (4)

( l ~ i ) j ( k ~ j ) i

In the noninteracting case, i.e. when 3~ =0, those rates reduce to the simplified form of those used in ref. [10]. Similarly the desorption rate for the interacting case is postulated in the form

/~i = Fo exp(--AD/3O¢ ~ n k - - AD/3°//')

=Fexp(-AD/3O¢ E n k ) . (5) ( k ) i

As usual/3 denotes inverse temperature in energy units.


The coefficients h (/.t = 1 - X), and similarly hi~ in eqs. (3), (5), are the asymmetry coefficients for diffusion and desorption processes, respectively. Detailed discussion of these coefficients was given in ref. [8]. In a nutshell: The asymmetry coefficient tells us how much the value of the energy of the initial and final states influence the transition probabilities between these two states. There are two different A's, for we consider two independent dynamical processes: diffusion and desorption. For A = 1 the transition probabilities depend simply on the energy difference. This is what usually happens in the diffusion process, because both the initial and final states are of the same type and are relatively close in the configuration space. In case )t = 1, the energy of the initial state alone decides about the transition. Such a situation is charac- teristic to the desorption process, when a particle jumps from a bound state to the continuum.

As it stands, eq. (2) presents a formidable mathematical problem even in the simplified one-dimensional case. Attempts to solve this equation via a tech- nique called canonical expansion [5] have not been useful for spatially inhomogeneous adsorbates. The reason for this is that eq. (2) leads to the hierarchy of coupled equations for moments of P({ni}; t) and decoupling schemes for that hierarchy are neither well-developed nor understood. Indeed, using eq. (2) we obtain the hierarchy of moments equations from which we wrote down the two first equations relevant for our further discussion. These equations describe time evolution for averaged surface density, the local coverage, (n/) , and the two-site correlation function (nin s) :

d d-tt (n i ) = -- E (Wqni ) d- E (Wl.in j)

(J)i (J)i

-- E ( ( W j i - Wq)n,ni) - ( F i n i ) , (6) (J)i

d dt (ninj) = - E (Wq(n tn in j - ntni)) + ~ (Wjt(ntninj- ninj))

( l~ i ) j ( l~ i ) j

- ~ (Wki(nknin/-- nkns) ) + ~, (Wik(nknin/- nin/) ) (k&j) i ( k~ j ) i

- ((Fii + Fj)nins). (7)

Note that due to the nonlinear dependence of the rates in eq. (3) on the occupation numbers, the hierarchy of moments equations, viz. eqs. (6), (7) is also highly nonlinear. For noninteracting adsorbate, corresponding to the model studied in ref. [10], ~ = 0 and eq. (6) decouples from the hierarchy and reads

M.A. Zaluska-Kotur, L.A. Turski / Diffusion and desorption in interacting adsorbate 379

O , ( n , ) = -zWo(n,) + Wo ~ < n j ) - r(ni) - - A ° ( n j ) , (J)i

(8)

where z is the adsorption-site lattice coordination number. Eq. (8) can easily be solved via the Fourier-Laplace transform in space and time. Denoting by n0(q, s) the Fourier-Laplace transform of the coverage,

n0(q, s) = f dt exp(-s t ) ~ exp( - iq , r~) ( n i ( t ) ) , i

0

and using the notation

_1 ~ exp(iq • ai) f ( q) = z

(9)

(10)

where q denotes the wave vector in the first Brillouin zone of the adsorption- site lattice, and a i are the Bravais lattice vectors at the site i, we can write the solution of eq. (9) as

~o(q, s) = n(q , t = 0) (11) s + F + zW0[1 - f ( q ) ] "

In eq. (11) n(q , t = 0) denotes the initial coverage. Following standard argu- ments we can recover from eq. (11) the expression for the Green function of the A ° operator, G(q, s). The dynamical structure factor is then given by

S(q, to) = 2 Re G(q, ico - 0 + ) . (12)

For a square lattice we obtain from the above the dynamical structure factor for the combined desorption-diffusion model discussed in ref. [10] (the one adsorption site per unit cell case),

S ( q , to) = 2 2 F + z W o ( q a i ) 2 (13) o, + [ r + z W o ( q a , ) 2 ] 2 "

Having shown that our model gives known results for the noninteracting adsorbate, in the following section we shall discuss its weak coupling limit.

3. Weak interaction limit

When the adparticles do interact among themselves the equation of motion for the coverage, eq. (6), does not decouple from the remaining hierarchy and

380 M . A . Zatuska-Kotur, L . A . Turski / Diffusion and desorption in interacting adsorbate

to proceed with the analysis we have to employ some, approximate, closure procedure. None of the proposed closures [5] works well for inhomogeneous adsorbate. Here we shall use the weak intra-adsorbate interaction approximation, which in its essence consists in expansion of the right hand side of eqs. (6), (7) in power series with respect to the coupling constant t/3 and retaining the first order terms. Our expansion might then also be viewed as high temperature approximation.

Using transition probabilities from eq. (3) we obtain the following equation of coverage evolution:

Or(n,) = -A°(ni) + ~flFA D ~ (nkn,) (kh

+ "flWo(A(z-1) ~'~ (nkni) + Ix ~ ~ (nkn,)) (k)i (J)i (k#J)i

---~flW0(A 2 "~ (njnt) + Ix 2 2 (ntni)) (J)i (l¢:i)j (J)i ( l# i ) j

+3~flWo[~ ( 1 ~ (n,ninj)--(k~-~;) (nkninj))l. '-(J)i ( )j ' "

(14)

Notice that the right hand side of eq. (14) depends on the two- and three-point correlation functions. The two-point function obeys an equation, viz. eq. (7), which can also be expanded into powers of •/3, and here we are going to make our most drastic approximation. Since the "corrections" due to (ninj) in eq. (14) are all first order in &/3, we write down equations for the pair correlation which are zero order in this expansion parameter. The resulting equation of motion for the coverage will then be correct up to terms linear in ~¢/3. Assuming the above notice that the three-site correlation function disappears from eq. (7), and we obtain a closed equation of motion for the pair function,

cOt(n in j ) = -- 2F(nin;) + Wo ~, ( n t n i ) -- Wo Z ( n i n j ) ( l~i ) j ( l~i ) j

+ W o ~_~ (nknj)-- W o ~ (ninj). (15) (k~j) i (k~j) i

The same can in principle be done with the three-point correlation function, viz. the last term in eq. (14). This leads to a quite complicated equation, thus to simplify the final results we will express the three-point correlation function by the two-point ones using the Kirkwood approximation, cf. ref. [5],

( n k n i n / ) = 1_~ (nkni)(ninj) (ngn~), (16) no


where n o is the equilibrium mean density of the surface layer. Using eq. (16) we have effectively closed the hierarchy.

4. The solution

The two-point correlation function of the noninteracting system is obtained by the solution of eq. (15). To do so we use the Fourier-Laplace transform.

Let

o o

~b(q, p, s) = f dt exp ( - s t ) ~ ~ e x p [ - i q , r t - ip . ( r j - r/)] ( n t n j ) . l j - t

0

(17)

After some manipulations eq. (15) yields

exp( - ia i • p) - 1 ~ b ( q ' p ' s ) = O ( q ' p ' s ) + 2 W ° ~ Y ( q , p , s ) {~b}a'(q's) '

ai

(18)

where

r ( q , p, s) = s + 2 r + zW0[2 - f ( p ) - f ( q -p)], (19)

and

O(q , p, s) : ~b( q, p, t = 0)

Y(q , p , s )

r~0( q, s) ( F + zW0[1 + f ( q ) - f ( p ) - f ( q - p ) ] } + r ( q, p, s)

(20)

In the above we have used the notation

Ir / a

- ~ r / a

dap ' O(q, p', s) exp(iai • p ' ) . (21)

Notice that the function O ( p , q, s) depends on the value of the single-site distribution function r~0(q, s) viz. eq. (9). The reason for this is that the summation on the fight hand side of eq. (17) is over all the lattice sites j, l. For equal sites, due to the assumption n i = 0, 1 we have ( n i n i ) = ( n i ) = no(i, t). We have to carefully take that into account in the algebra leading to eq. (18).

Eq. (18) is an integral equation with separable kernel and is solved by inverting the matrix ( I - / ( / ) , whose i, j matrix element reads


( 1 - / 9 ) / j = 6 / j - 2Wflr e x p ( - i .-j -,r z "~ "'~ - 1[,;~, (q, s) t Y(q, p , s) (22)

For s ~ 0 or q ~ 0, matrix (22) is nonsingular, hence we can write

{0}o,(q, s) = ~ (1 - A/)~l(/2}°,(q, s ) . (23) ]

On substituting solution (23) into eq. (18) we obtain the explicit expression for the two-point correlation function for the noninteracting adsorbate, which in turn can be inserted into the Fourier-Laplace transform of eq. (14) resulting in the closed expression for the coverage,

~(q, s) = n(q, t = 0) +/3JAD/-~ 1 + ~JWo(fg z + ~q3) (24) s + F + zW0[1 - f(q)]

where

"~/a

(a) f ~1 : z ~ ddp ~b( q, p, s) f( p) , -~r/a

¢n'/a

~2 = ( ~ ) d f ddp~b(q,p,s) -~r/a

× {Az2f(p) [1 - f(q)] + txz[zf(q - p ) f(p) - zfZ(p) - f ( q ) + 1])

(25)

(26)

and

"~/a '~/a Z2( a'} 5a

- - ' ~ / a - - ' ~ / a

x [~b(q _p5 _p3, ql _ q3, s) - ~b(q _p5 _p3, ql _ q2, s)]

X V/(p 3 __p4, p2, S) ~ (p4 ..b ps, ps, S) f ( p 3 _ p 2 ) f ( p l ) . (27)

Eq. (24) represents a formal solution for the coverage evolution in the weak interaction limit. In the two-dimensional case inversion of the matrix (I - /~/) in eq. (22) cannot be done analytically and requires considerable numerical work. We can capture essential features of the modification of the combined diffusion-desorption processes brought by the intra-adsorbate interactions by analyzing the simplified one-dimensional version of eq. (24). Furthermore we


consider only the lowest order in coverage fluctuations contributions to the right hand side of eq. (24). After considerable algebra we obtain a closed expression for the Fourier-Laplace transform of the d = 1 coverage,

n( q, t = O) + f l •ADFM K( q, s) = (28)

s + F + Wo[1 - cos2(qa)] '

where M reads

4 cos2(qa/2)

(x÷ - 2)(s + 2F + 4W0)

x (~b(q , q / 2 , t = O) + n( q, t O)

s + F + W0[1 - cos2( qa)]

x {2Wo[1 + cos(qa)] + F - x _ / 2 - 2W o c o s ( q a / 2 ) } ) , /

(29)

and

x~_ = (s + 2F + 4W0) -7- ~/(s + 2F + 4W0) 2 - [ 4 W o cos(qa/2)] 2 . (30)

Eq. (28) represents the final result of our analysis. The following section will contain its discussion.

5. Discussion

Eq. (28) represents a complicated analytical expression for the coverage evolution in the 1D adsorbed layer. Following standard procedures we obtain from this expression that for the dynamical structure factor of the adsorbate with concomitant desorption and diffusion processes. This expression is valid in the weak-interaction limit, that is for small values of the coupling constant 5~ and /o r high temperatures. In figs. 1, 2 we have shown the dynamical structure factor S(q, to) obtained from eq. (28) for two values of the desorption rate F. In fig. 1 F = 1 while in fig. 2 it also assumes a higher value equal to 10. Similarly as in the noninteracting case these curves are Lorentzians, but now the width of them depends on the value of the coupling constant ~. Notice that the attractive character of intra-adsorbate interactions (~ > 0) results in decrease in the width of the Lorentzian. To see these changes more clearly in fig. 3 we have plotted the hwhh of the structure factor as a function of the wave vector within the first Brillouin zone and for various values of the coupling constant. One sees that the changes of the width are different for attractive

384 M.A. Zaluska-Kotur, L .A . Turski / Diffusion and desorption in interacting adsorbate

1.0

0.8

0.6

0.4

0.2

0.0

-10

i i • i , i

i i q/i

,,T/ • s , i

i i i i

-8 -6 -4 -2

i

I

0

CO

i - i , i • i •

L

L ~

i',\

i n i i

2 4 6 8 10

Fig. 1. Dynamica l s t ructure factor for d = 1 model of weakly interacting adsorbate plot ted as a funct ion of f requency to for qa = ½ and F = 1. ( . . . . )-curve: /33.`9 = - 0 . 1 ; solid curve: /33.`9 = 0; b r o k e n curve: /33.,9 = 0.1; and ( . . . . )-curve: /33.,9 = 0.3.

i i i i 1

1.0 .,.,,,,. ,~. ~,,~

""~ "I1 " %

./> I ~ "."-. ,.;" t i " , i . 0.8 ./.,.,. ; , .,.\.

, ' t I ~ "-x. x

0.6 / " , t ' -,. x.

." , " i ~ " - -' I '~

/ i /

1 \ , \ 0.2 / \

0.0 . . . . . . . . . . . . . . . . . . . . " . . . . . . . . . . . . . . . . . . . . . I I I I I

-30 -20 -10 0 10 20 30

CO

Fig. 2. Dynamica l s t ructure factor for a d = 1 model of weakly interacting adsorbate plot ted as a funct ion of f requency to for qa = ½. ( . . . . )-curve: F = 10, /3A`9 = 0. ( . . . . )-curve: F = 10, /33.`9 = 0.1. B roken curve: F = 1, /3t`9 = 0. Solid curve: F = 1,/33.`9 = 0.1.

M.A. Zaluska-Kotur, L .A . Turski / Diffusion and desorption in interacting adsorbate 385

i i t i i i

5

4

3 t ' q

<1

2

l

0 I I I I I I

0.0 0.5 1.0 1.5 2.0 2.5 3.0

qa Fig. 3. Half width on half height (hwhh) of the dynamical structure factor for weakly interacting adsorbate, hwhh is plotted as the function of a dimensionless wave vector k = qa. In all cases the desorption rate F = 1. Solid curve: noninteracting adsorbate. Broken curve: /3,~ = 0.3. Densely dotted curve: /3A~ = 0.1. Sparsely dotted curve: fl,~5~ = -0 .1 .

(positive values of ~) and repulsive (negative ~) interactions between the adparticles. The attractive interactions (densely dotted and broken curve) show decrease of the value of the width as compared with non-interacting case, while repulsive interactions (sparsely dotted curve) indicate widening of the S(q, oJ). In analysing these results one should keep in mind that these are approximate results for one dimension. We believe they capture the essential physics of the processes in two dimensions in the parameter region where our weak-interactions approximation is valid. This is clearly not the case close to the phase transition, where one should use a more elaborate method of analyzing the hierarchy of moments equations.

6. Conclusions

In this paper we have analyzed the influence of the intra-adsorbate interactions on the combined diffusion and desorption processes in the adsorbate layer. For this purpose we have used the kinetic lattice gas model [5] with the transition probabilities containing the asymmetry coefficient discussed in ref. [8]. To carry out our analysis we have used the weak coupling approximation, i.e. we have assumed that fl~ ~ 1. In the noninteracting limit we have recovered the results of previous analysis of combined diffusion and desorption

386 M.A. Zaluska-Kotur, L .A . Turski / Diffusion and desorption in interacting adsorbate

processes from ref. [10]. The weak-interaction limit is analyzed using the Kirkwood ansatz for the triple-correlation function. Explicit expressions for the dynamical structure factor of the weakly interacting adsorbate are obtained in the one-dimensional case and shown in the figures. The two-dimensional case requires considerable numerical analysis and will be presented elsewhere.

Acknowledgements

We would like to acknowledge fruitful discussions on diffusion and adsorbate properties with professors Richard Bausch, Zbigniew W. Gortel, Jiirgen Kreuzer, Heiner Miiller-Krumbhar and Andrzej Wierzbicki.

This work was also partially supported by the Polish Council for Scientific Research (KBN) grant Np. 704/45/92.

References

[1] A. Zangwill, Physics at Surfaces (Cambridge Univ. Press, Cambridge, 1988). [2] W. Schommers and E Von Blanckenhagen, eds., Structure and Dynamics of Surfaces I, II,

Topics in Current Physics, vols. 41, 43 (Springer, Berlin, 1986, 1987). [3] H.J. Kreuzer and Z.W. Gortel, Physisorption Kinetics, Springer Series in Surface Sciences,

vol. 1 (Springer, Berlin, 1986). [4] M. Grunze, H.J. Kreuzer and J.J. Weimer, eds., Diffusion at Interfaces: Microscopic

Concepts, Springer Series in Surface Sciences, vol. 12 (Springer, Berlin, 1988). [5] H.J. Kreuzer, Surf. Sci. 231 (1990) 213; Phys. Rev. B 44 (1991) 1232. [6] M.C. Tringides and R. Gomer, Surf. Sci. 265 (1992) 283.

R. Gomer, Rep. Prog. Phys. 53 (1990) 917. [7] A.N. Berker, S. Ostlund and F.A. Putnam, Phys. Rev. B 17 (1978) 3650. [8] M. Zaluska-Kotur, Surf. Sci. 265 (1992) 196. [9] Z.W. Gortel and L.A. Turski, Phys. Rev. B 45 (1992) 9389.

[10] Z.W. Gortel and L.A. Turski, Phys. Rev. B 43 (1991) 4598. [11] N.G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam,

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Diffusion and desorption in weakly interacting adsorbate