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Monte Carlo studies of adsorption III: Localized monolayers on randomly heterogeneous surfaces
Thin Solid Films, 208 (1992) 189 196
189
Monte Carlo studies of adsorption III: Localized monolayers on randomly heterogeneous surfaces Andrzej Patrykiejew Department o[" Chemical Physics, Faculty of Chemistry, MCS University, 20031 Lublin (Polamt) (Received June 17, 1991; accepted September 16, 1991)
Abstract The properties of localized monolayers formed on a square lattice with randomly distributed sites of different adsorption energies are studied with the help of Monte Carlo computer simulations. The effects of random impurities on phase transitions in adsorbed layers are discussed. It is demonstrated that the commonly used approaches to adsorption on heterogeneous surfaces, based on mean field theories, are valid only at very high temperatures. In the region of low temperatures, below the critical two-dimensional temperature of an adsorbate, the predictions of mean field theories seriously underestimate the heterogeneity effects on phase transitions in adsorbed layers.
1. Introduction
Adsorption processes occurring in nature are very complex and influenced by many factors. The structure of an adsorbing surface and associated heterogeneity effects belong to the most important among them [15]. The description and characterization of heterogeneous surfaces are still unsolved problems in surface science, despite intensive studies performed over the last four decades. These studies have resulted in abundant literature (refs. 1-5 and references cited therein) and rather limited progress. From the time when Halsey and Taylor [6] proposed their famous integral equation of the adsorption isotherm, it was commonly used as a starting point in a vast majority of attempts aiming at the development of a theory of adsorption on heterogeneous surfaces [112]. The equation of Halsey and Taylor takes the following form: f'rnax /I
O(p, T) = t X(e)O(p, T,e.) de
(1)
gmhl
where O(p, T) is the overall adsorption isotherm on the heterogeneous surface, being a function of pressure p and temperature T, Z(e) is the adsorption energy distribution function (EDF), O(p, T, ~) is the so-called local adsorption isotherm describing adsorption on sites characterized by an energy of adsorption equal to e and the integration in eqn. (1) runs over the whole range of adsorption energies accessible to the system. For computational reasons it is often assumed that emir = 0 and grnax (3(3.Equation (1) is usually used either to describe =
0040-6090/92/$5.00
experimental adsorption isotherms by assuming various forms for X(e) and O(p, T, e) or to evaluate the EDF by taking various forms of O(p, T, ~) and experimental data for 3(p, T) [5, 13-17]. In both cases, the results depend on the choice of local adsorption isotherm and the model of the heterogeneous surface used. The assumption that the adsorption process is described by eqn. (1) introduces certain limitations. Strictly speaking, it is valid only when the interactions between adsorbed particles can be neglected and/or when the surface is composed of macroscopically large patches characterized by the same value of the energy of adsorption, so that boundary effects between different patches can be neglected. Both situations seem to be rather far from reality. In crystalline solids, surface heterogeneity results from lattice imperfections and defects, the presence of random impurities, steps and corners etc. Therefore the assumption of a patchwise topography of adsorption sites does not seem to be reasonable. Also, for various amorphous solids, the structure of their surfaces can be described as random lattices [ 18, 19] and the patchwise model again does not apply. The only possible real systems with a patchwise distribution of adsorption sites are large perfect crystals with different planes exposed to the gas phase and mixed adsorbents composed of materials characterized by homogeneous surfaces. In this work we shall concentrate on the properties of heterogeneous surfaces with random topography of adsorption sites and assume that all effects due to surface heterogeneity are of energetic origin only. In general, the energetic heterogeneities (e.g. resulting from the presence of random impurities) are always accompanied
',~5) 1 9 9 2 -
Elsevier Sequoia. All rights reserved
A. Patr)'kidew / Monte C a r l o studies o/ adsorpthm HI
190
by geometric heterogeneities that influence the adsorbate-adsorbate interactions in adsorbed layer. Recently, Benegas el al. [19] have presented results of Monte Carlo simulations for random lattices with gaussian distributions of both adsorbate-adsorbent and adsorbate-adsorbate interaction energies. Their calculations have shown that both effects act in the same direction and cause the adsorption isotherms to become more and more Langmuir like as the width of these distributions increases. Our main aim here was to determine the effects of small concentrations of random impurities on the behaviour of adsorption systems at low temperatures and for clarity we have neglected the geometric component of surface heterogeneity. Furthermore, we also address a more general problem of surface heterogeneity effects on phase transitions in adsorbed films and demonstrate that results of earlier studies [11, 20, 21], based on eqn. (1), seriously underestimated heterogeneity effects on the properties of adsorbed films.
2. The model and mean field theory
In this work we use a simple two-dimensional lattice gas model on a square lattice, represented by the hamiltonian 1
*:
- 5
Z
<1. f7
-
E i
(2)
where n; is the occupation variable equal to 1 (0) when the ith site is occupied (empty), UNN is the energy of interaction between a pair of nearest neighbours and ~,, is the adsorbate-adsorbent interaction energy for the ith site. By assuming different forms of the distribution of adsorbate-adsorbent interaction energies c,; over the lattice sites we can model surfaces with different heterogeneity effects. Here we shall consider surfaces with only very few types of sites, denoted by L, of different energies so that the E D F will be represented by a set of L pairs of energy and contribution values {s~, y," 72; " s;,39_} such that
(but only one) value of as and apply the condition of thermodynamic equilibrium, i.e. equality of the chemical potential for all patches. The problem of boundary effects in adsorption on patchwise heterogeneous surfaces will be a subject of our next paper. We now describe briefly a simple mean field theory of adsorption on such discretely heterogeneous surfaces [22-24]. In the framework of the Bragg-Williams approximation the adsorption on a homogeneous surface or on a single homotatic patch, consisting of sites with the energy ~;,, is represented by the well-known adsorption isotherm equation [5, 23]
fl
~;,
k.T-
(
Oi ~
k,T+ln\~-O~/
2HNN(}i
k~T
(4)
where /~ is the chemical potential, kB is the Boltzmann constant, T is the temperature and 0; is the degree of surface coverage. When the surface is composed of L patches then we have L equations like (4) representing adsorption on each patch. In the case of a heterogeneous surface with random topography we also have a set of equations describing adsorption on each type of site, but in a slightly different form: p
k];
+ln(
-
0;
\U-@
)
2UNN~1 kBr
i5)
where 0 is the total surface coverage of an entire heterogeneous surface. For a heterogeneous surface composed of L different types of sites and the EDF {';J, 72 . . . . . ,,;. } the total surface coverage is given by L
~; = ~ O~;'k k
(6)
1
independently of the surface topography. Using eqns. (4) (or (5)) and (6) we can readily calculate adsorption isotherms for any assumed distribution of adsorption sites and for different surface topographies. Within this mean field approximation the derivation of expressions for any thermodynamic quantity, e.g. internal energy, heat capacity, is also straightforward [22, 23] and will not be considered here.
A
= 1 k
(3)
|
and ?k represents the fraction of sites with the energy s~. The model defined by the hamiltonian (2) can be used to model heterogeneous surfaces with random topography. When it is used to model surfaces with patchwise topography we meet the problem of boundary effects between patches of sites with different energies of adsorption. In order to model truly patchwise surfaces, that satisfy the patch independence condition [4], we must consider a collection of systems, each represented by the hamiltonian (2) with a different
3. Monte Carlo method
The thermodynamic properties of the model described in Section 2 have been investigated by means of Monte Carlo simulations in grand canonical ensemble using the Metropolis sampling method [25]. Thus, the grand canonical thermodynamic potential for our model is given by
i
A. Patrykiejew / Monte Carlo studies of adsorption 111
A number of thermodynamic quantities have been calculated, including fractional coverages 0~., k = 1 , . . . , L, of each type of site, the total surface coverage & mean adsorption energy U and heat capacity C. All these quantities have been normalized per lattice site and calculated as simple averages: 1 O~.=~i_~= ni, 1
k=l,2
.....
L
(8)
M
O=~
12 n,,
(9)
i--I
u = 1_ M ('Y~)
(10)
and 1 C - MkB T 2 ( ( ~ 2 ) _ ( ~ . ) 2 )
(11)
In the above equations Mk is the number of adsorption sites of type k, and M is the total number of adsorption sites. In our calculations we have used lattices of size 30 x 30 and hence M = 900. Considering that in this work we are not interested in accurate estimation of critical properties the size of our simulation box was sufficient to obtain reliable results. In order to avoid boundary effects we have used standard periodic boundary conditions in both directions [25]. In most cases it was sufficient to use about 103 Monte Carlo steps (per site) (MCSs) to obtain accurate results for surface coverage and mean adsorption energy. Calculations of heat capacity required longer runs with up to 104-105 MCSs, in particular at temperatures close to critical points. The results for heterogeneous surfaces with patchwise topography were obtained by independent calculations of thermodynamic quantities for homogeneous surfaces with given energy of adsorption and then by averaging the results using the relations L
O ~ 12 )'k0k
(12)
k=l L
u:
12 ;,~.v, k=!
{13)
and L
C=
12 )'kCk
(14)
From the grand canonical potential (7) and the hamiltonian (2) it follows that in the case of a homogeneous surface the change in adsorption energy e causes only a shift in chemical potential by the difference in adsorption energies. Hence, the results for systems with patchwise topography could be obtained using results obtained for
191
only one value of the adsorption energy (e = 0) and by further rescaling of the chemical potential. In the case of surfaces with random topography we have generated the surface structure with a given distribution of adsorption sites by using random numbers. For each type of E D F we have performed calculations for several (up to 10) independently prepared surfaces in order to average results over different configurations of adsorption sites with different energies. It was found that these effects were completely negligible for systems considered in this work. Throughout this paper we assume that the energy of interaction between nearest neighbours is given by UNN/ kB = 0.5 SO that the two-dimensional critical temperature Tc for our model, corresponding to a homogeneous surface, is about 0.28364. In order to determine phase boundaries for systems exhibiting first-order phase transitions we have calculated free energies of both coexisting phases using a standard thermodynamic integration method [26, 27].
4. Results and discussion
We begin with the presentation of results for a simple case of a surface with only two types of adsorption sites. Figure 1 shows examples of adsorption isotherms for systems characterized by 7~ = 7 2 - - 0 . 5 (L =2), e~ = 1.0 and different values of e2 at T - - 0 . 5 and obtained from Monte Carlo simulations for a patchwise and random topographies. We observe that, even at that rather high temperature, the adsorption isotherms obtained for a patchwise topography exhibit well-pronounced steps corresponding to subsequent occupation of patches of different sites. On the contrary, this effect is much weaker in the case of random topography, and the steps become visible only for a sufficiently large difference in adsorption energies of different sites and for sufficiently low temperatures. It is easy to understand these results considering the mechanism of adsorption on a surface with random topography. The more active sites are occupied first and they serve as centres of condensation for adsorption on less active sites. Therefore, for a given value of the chemical potential the net coverage of less active sites is higher in the presence of more active sites than on a homogeneous patch of only less active sites. Figure 2 shows the dependence of the fractional coverage of less active sites as a function of chemical potential on surfaces with different concentrations of random impurities of more active sites. In the range of chemical potential values used in the calculations the more active sites were completely covered by adsorbed particles. It is seen that the coverage of less active sites increases with the concentration of random impurities. When the random
A. Patrykiejew / Monte Carlo studies of ad~'orption 111
192
impurities consist of sites of lower energy of adsorption we observe an opposite effect; the net coverage of more active sites decreases with the concentration of random impurities (Fig. 3). In Fig. 4 we present a series of phase diagrams (obtained from the Monte Carlo simulations) for systems with different concentration of random impurities of higher energy. We find that an increase in the concentration of random impurities
1.0
8 qa
0
O-P
28-
0.6
O.4-
s
O~P D-R
t0.
/
O.2-
)
P
o.6
-6}
-,~1
-£1
311
-2~1
-1~
-0:1
(c) Fig. 1. Adsorption isotherms obtained from Monte Carlo simulations for systems with ",,~=Y2 =0.5, T =0.5, c~/kB = 1.0 and the values of ~:z/kB = (a) 0.5, (b) 2.0 and (c) 3.0 for random topography (R) and patchwise topography (P). 0.2-
causes a rapid decrease in the critical temperature for the phase transition occurring in the adsorbed layer and an increasing effect of asymmetry of phase boundaries. In particular we expect that there is a certain concentration threshold Y~m and when the impurity concentration is higher than }im ,c the system does not exhibit a phase transition at non-zero temperatures at all. This is in contrast to the mean field prediction that the critical temperature changes with the concentration of any given type of sites as
0-
(a)
1,0" /t[
6 o-R
9.8 ¸
4(
T ~ ( T k ) = T~V(0)y,, 0.6.
~2
..Oj -6
(b)
-'5
-&
O -3
-2
-1
0
k = 1,2 . . . . .
L
(15)
where T~F(0) = 0.5 for our model Figure 5 presents a comparison of the critical temperature changes with the concentration of random impurities obtained from Monte Carlo simulations and predicted by the mean field theory. We find that for the systems considered here the extrapolation of our simulation results yields im ~ 0.21 From eqn. (15) it follows that at sufficiently low temperatures the mean field theory predicts the appearance of L first-order transitions on a surface with L types of sites. This is illustrated in Fig. 6 which shows examples of adsorption isotherms for the system with E D F given in Table 1 and obtained from the mean field theory (Fig. 6(a)) and Monte Carlo simulations (Fig. 6(b)). At T = 0 . 1 the adsorption isotherm obtained
A. Patrykiejew / Monte Carlo studies of adsorption III
1.o.
193
7~ (o)
8
~,0IqF
Qsiiii
0.8"
/
q6-
,'J',/;Il
o.4
x' I~
0.4.
t l]~
q2-
o.2.
o 3./,
3.2
3.0
2.8
2~6
Fig. 2. The fractional coverage of less active sites on randomly heterogeneous surfaces with different concentration of highly active sites obtained from Monte Carlo simulations for the system with e~/kB = 2.0, e2/kB = 5.0 at T = 0.3. The concentrations of highly active sites are shown in the figure.
~.o5
J
'
Fig. 4. Phase diagrams for a series of systems with different concentrations of highly active sites (shown in the figure) obtained from Monte Carlo simulations for systems with ~/kB = 2.0 and %[k B = 5.0.
"T/~ (0)
~'i,'~
. . . @--,@,
1
QI
1/45
1/g
'~
@
i'"" "" g,........,,.
Q-u.-g,-~~
/ 1/90
• .--/.,~
I//13--
P ,,"/ +,,z7 a-~l -1/9o El, ~I~,, f tl÷ x - q =1/30
--'I
,,tl
]/,! 0 0.2-
IH!
0.2
0,,4
0,6
0.8
1,0
8 Fig. 5. Dependence of the critical temperature of the phase transition in adsorbed film as a function of highly active random impurity concentration for systems with e,]k B = 2.0 and ~z/k8 = 5.0: - - results of mean field calculations; @, - - - , Monte Carlo results.
Fig. 3. The fractional coverage of highly active sites on randomly heterogeneous surfaces with different concentration of less active sites obtained from Monte Carlo simulations for the system with el/ kB = 2.0, e2/kB = 5.0 at T = 0.3. The concentrations of less active sites are shown in the figure.
f r o m t h e m e a n field t h e o r y e x h i b i t s a w e l l - s e e n v e r t i c a l step c o n n e c t e d with the first-order transition o c c u r r i n g o n sites w i t h 7 = 0.5 a n d c r i t i c a l p o i n t s f o r a d s o r p t i o n o n t w o t y p e s o f sites w i t h ? = 0.2. B y f u r t h e r l o w e r i n g o f t e m p e r a t u r e b e l o w T = 0.05 w e f i n d five v e r t i c a l
A. PatrykRjew / Monte Carlo studies ql" adsorption 111
t94
TABLE 1. The energy density function used in calculations of adsorption isotherms shown in Figs. 6 and 7
1.0-
i ~:i ;'i
J
6
0,6-
0.2-
J
-55
-&6
-,3.7
-2-.,8
-~9
-4,9
-39
-2.9
-tO
(a)
0.8
C~6
0.4
- 5.9
(b) Fig. 6. A comparison of adsorption isotherms for a system with EDF given in Table 1 at T = 0 . 1 and obtained from (a) mean field calculations and (b) Monte Carlo calcula~tions ( -- , contributions to the total surface coverage corresponding to the occupation of different types of adsorption sites).
I 1.0 0.05
2 2.0 0.20
3 3.0 0.50
4 4.0 0.20
5 5.0 0.05
steps corresponding to five first-order transitions. On the contrary, the adsorption isotherm at T = 0.1 and obtained from Monte Carlo simulations (Fig. 6(b)) does not show any trace of a phase transition. When the temperature increases, however, the differences between the mean field and Monte Carlo results become smaller and smaller. Figure 7 shows two adsorption isotherms for the system with the E D F from Table 1 at T = 0.5; they are only slightly different. Now we shall show that the results of our Monte Carlo simulations find a confirmation in experiment. Recently, Heidberg et al. [28] have published results of adsorption measurements for carbon dioxide on NaCl(100) surface at T = 85 K. The experimental adsorption isotherm exhibits a vertical step connected with the first-order phase transition and a well-seen asymmetry of phase boundaries characteristic of a system with a low concentration of random impurities. In Fig. 8 we show the experimental adsorption isotherm obtained by Heidberg et al. and examples of adsorption isotherms obtained from the present Monte Carlo simulations. Although the experimentalists say that the sample of NaCI crystal had highly homogeneous surface, it is possible that small effects of surface heterogeneity due to lattice imperfections of the cleaved crystal introduce some r a m d o m heterogeneity effects that escaped detection. It will be very interesting to compare the results of Monte Carlo simulations with experimental data for other thermodynamic properties of adsorbed films when experimental data are available [29]. Concluding this paper, we would like to comment briefly on the effects of surface heterogeneity on multilayer adsorption, a situation not allowed by the present model. From the mean field calculations [30-32] it follows that heterogeneity influences the properties of higher adsorbed layers, provided that the adsorption energy distribution is sufficiently broad. In particular, it has been observed that randomly distributed heterogeneity has a considerably smaller effect than the heterogeneity distributed in a patchwise manner. Of course, any mean field results must be treated cautiously. As has been demonstrated in this work, any mean field approach is bound to underestimate effects due to surface heterogeneity in the submonolayer region, and this is surely the case in the multilayer region as well. We can recall here the recent ellipsometric measurements of adsorption of noble gases and nitrogen on what was believed to be a highly homogeneous
A. Patrykiejew / Monte Carlo studies of adsorption HI
195
8
"]0"
(
8
0.8
0.6
0.S0,4
J
0.2-
Log [P/( nbor)] O-
(a)
-&5
-~S
-4.5
-&5
-&5
-1~
-q5
(a) 1.0-
rX2~-IB--123.d2-
(b)
0B
,~ .,./~...----,~
,,,~z~-
-4
-3.s
-3
-is
(b) 6.5
5.5
4.5
3.5
2.5
1'5
d,5
(b) Fig. 7. As for Fig. 6 but at T =0.5.
graphite surface [33, 34] at temperatures below the roughening temperature. It has been observed that onsets of the layering transitions in the first, second and also third layer were rounded in the same way as observed in the submonolayer region (Fig. 8). This can
Fig. 8. (a) Experimental adsorption isotherm for carbon dioxide on NaCI(100) at 85 K [28], (b) adsorption isotherm obtained from Monte Carlo simulations for a system with y~ = 0.9, "/2 = 0.1, e~ = 2.0 and s2 = 3 . 0 at T = 0 . 1 2 .
be interpreted as an influence of weak surface heterogeneity, present in the adsorption system owing to the finite size of graphite crystallites, steps at its cleaved surface etc. This experimental finding clearly demonstrates that surface heterogeneity effects can easily ex-
196
A. Patrykiejew / Monte Carlo studies of adsorption 111
tend over the first adsorbed layer. The predominantly occupied highly active sites may act as nucleation centres for the formation of molecular clusters extending to second and higher adsorbed layers. This effect of the promotion of multilayer adsorption by surface heterogeneity has been recently demonstrated by molecular dynamics simulations performed by Bojan and Steele [35]. These workers have considered a model surface with steps and found that adsorbed particles tend to develop a one-dimensional order along these steps and that the local density of an adsorbate at distances corresponding to the second layer exhibits clear maxima in the vicinity of these steps.
Acknowledgments The author wants to express his thanks to Professor K. Knorr and Dr. U. G. Volkmann (Institute of Physics, University of Mainz, F.R.G.) for helpful discussions.
References 1 S. Ross and J. P. Olivier, On Physical Adsorption, Wiley, New York, 1964. 2 A. Clark, The Theory' of Adsorption and Catalysis, Academic Press, New York, 1970. 3 J. G. Dash, Films on Solid Surfaces, Academic Press, New York, 1975. 4 M. Jaroniec, A. Patrykiejew and M. Bordwko, Prog. Surf Memb. Sci., 14(1981) 1. 5 M, Jaroniec and R. Madey, Physical Adsorption on Heterogeneous Solids. Elsevier, Amsterdam, 1988. 6 G. D. Halsey, Jr., and H. S. Taylor, J. Chem. Phys., 15 (1947) 624. 7 L. B. Harris, Sur£ Sci., 13 (1969) 377. 8 R. H. van Dongen and J. C. P. Broekhoff, Surf. Sci., 18 (1969) 462. 9 G. F. Cerofolini, Surf Sci., 51 (1975) 333.
10 W. Rudzifiski and L. Lajtar, J. Chem. Soc., Faraday Trans. 2, 77 (1981) 153. 11 W. Rudzifiski and J. Jagietto, J. Low Temp. Phys., 45 (1981) 1. 12 V, Mayagoitia, F. Rojas, V. D. Pereyra and G. Zgrablich, Surf. Sci., 221 (1989) 294. 13 M. Jaroniec, Surf Sci., 50(1975) 553. 14 W. A. House and J. M. Jaycock, Colloid Polym. Sci., 256(1978) 52. 15 S. Ross and I. D. Morrison, J. Colloid Interface Sci., 52 (1975) 103, 16 C. H. W. Vos and L. K. Koopal, J. Colloid Interface Sci., 105 (1985) 183. 17 P. Brauer, M. Fassler and M. Jaroniec, Chem. Phys. Lett., 125 (1986) 241. 18 R. Zallen, The Physics of Amorphous Solids, lnterscience, New York, 1983. 19 E. 1. Benegas, V. D. Pereyra and G. Zgrablich, Surf Sci., 187 (1987) L647. 20 W. Rudzifiski and J. Jagiet|o, J. Low Temp. Phys., 48 (1982) 307. 21 J. Jagietlo, Ph.D. Thesis, UMCS, Lublin, 1984. 22 P. Brauer, Wiss. Z. Friedrich-Schiller-Univ., Jena, Math.-Naturwiss. Reihe, 27(1978) 701. 23 A. Patrykiejew, Thesis, 14th Post-graduate Course in Chemistry and Chemical Engineering, Tokyo Institute of Technology, Tokyo, 1979. 24 P. Brauer, H. Dunken and R. Hafer, Wiss. Z. Friedrich-SchillerUniv., Jena, Math.-Naturwiss. Reihe, 30 (1981) 505. 25 K. Binder (ed.), Monte Carlo Methods in Statistical Physics, Topics in Current Physics, Vol. 7, Springer, Berlin, 1978. 26 K. Binder, Z. Phys. B, 45(1981) 61. 27 A. Milchev and K. Binder, Surf Sci., 164 (1985) 1. 28 J. Heidberg, E. Kampshoff, R. Kiihnemuth, O. Sch6nekfis, H. Stein and H. Weiss, Surf Sci. Lett., 226(1990) L43. 29 J. Heidberg, E. Kampshoff, O. Sch6nek/is, H. Stein and H. Weiss, submitted to Ber. Bunsenges. Phys. Chem. 30 W. M. Champion and O. D. Halsey, Jr., J. Phys. Chem., 57 (1953) 646. 31 W. M. Champion and G. D. Halsey, Jr., J. Am. Chem. Soc., 76 (1954) 974. 32 D. Nicholson and R. G. Silvester, J. Colloid lnterJhce Sci., 62 (1977) 447. 33 J. W. O. Faul, U. G, Volkmann and K. Knorr, Surf Sci., 227 (1990) 390. 34 K. Knorr and U. G. Volkmann, personal communication, 1991. 35 M. J. Bojan and W. A. Steele, Surf S('i., 199(1988) L395.
189
Monte Carlo studies of adsorption III: Localized monolayers on randomly heterogeneous surfaces Andrzej Patrykiejew Department o[" Chemical Physics, Faculty of Chemistry, MCS University, 20031 Lublin (Polamt) (Received June 17, 1991; accepted September 16, 1991)
Abstract The properties of localized monolayers formed on a square lattice with randomly distributed sites of different adsorption energies are studied with the help of Monte Carlo computer simulations. The effects of random impurities on phase transitions in adsorbed layers are discussed. It is demonstrated that the commonly used approaches to adsorption on heterogeneous surfaces, based on mean field theories, are valid only at very high temperatures. In the region of low temperatures, below the critical two-dimensional temperature of an adsorbate, the predictions of mean field theories seriously underestimate the heterogeneity effects on phase transitions in adsorbed layers.
1. Introduction
Adsorption processes occurring in nature are very complex and influenced by many factors. The structure of an adsorbing surface and associated heterogeneity effects belong to the most important among them [15]. The description and characterization of heterogeneous surfaces are still unsolved problems in surface science, despite intensive studies performed over the last four decades. These studies have resulted in abundant literature (refs. 1-5 and references cited therein) and rather limited progress. From the time when Halsey and Taylor [6] proposed their famous integral equation of the adsorption isotherm, it was commonly used as a starting point in a vast majority of attempts aiming at the development of a theory of adsorption on heterogeneous surfaces [112]. The equation of Halsey and Taylor takes the following form: f'rnax /I
O(p, T) = t X(e)O(p, T,e.) de
(1)
gmhl
where O(p, T) is the overall adsorption isotherm on the heterogeneous surface, being a function of pressure p and temperature T, Z(e) is the adsorption energy distribution function (EDF), O(p, T, ~) is the so-called local adsorption isotherm describing adsorption on sites characterized by an energy of adsorption equal to e and the integration in eqn. (1) runs over the whole range of adsorption energies accessible to the system. For computational reasons it is often assumed that emir = 0 and grnax (3(3.Equation (1) is usually used either to describe =
0040-6090/92/$5.00
experimental adsorption isotherms by assuming various forms for X(e) and O(p, T, e) or to evaluate the EDF by taking various forms of O(p, T, ~) and experimental data for 3(p, T) [5, 13-17]. In both cases, the results depend on the choice of local adsorption isotherm and the model of the heterogeneous surface used. The assumption that the adsorption process is described by eqn. (1) introduces certain limitations. Strictly speaking, it is valid only when the interactions between adsorbed particles can be neglected and/or when the surface is composed of macroscopically large patches characterized by the same value of the energy of adsorption, so that boundary effects between different patches can be neglected. Both situations seem to be rather far from reality. In crystalline solids, surface heterogeneity results from lattice imperfections and defects, the presence of random impurities, steps and corners etc. Therefore the assumption of a patchwise topography of adsorption sites does not seem to be reasonable. Also, for various amorphous solids, the structure of their surfaces can be described as random lattices [ 18, 19] and the patchwise model again does not apply. The only possible real systems with a patchwise distribution of adsorption sites are large perfect crystals with different planes exposed to the gas phase and mixed adsorbents composed of materials characterized by homogeneous surfaces. In this work we shall concentrate on the properties of heterogeneous surfaces with random topography of adsorption sites and assume that all effects due to surface heterogeneity are of energetic origin only. In general, the energetic heterogeneities (e.g. resulting from the presence of random impurities) are always accompanied
',~5) 1 9 9 2 -
Elsevier Sequoia. All rights reserved
A. Patr)'kidew / Monte C a r l o studies o/ adsorpthm HI
190
by geometric heterogeneities that influence the adsorbate-adsorbate interactions in adsorbed layer. Recently, Benegas el al. [19] have presented results of Monte Carlo simulations for random lattices with gaussian distributions of both adsorbate-adsorbent and adsorbate-adsorbate interaction energies. Their calculations have shown that both effects act in the same direction and cause the adsorption isotherms to become more and more Langmuir like as the width of these distributions increases. Our main aim here was to determine the effects of small concentrations of random impurities on the behaviour of adsorption systems at low temperatures and for clarity we have neglected the geometric component of surface heterogeneity. Furthermore, we also address a more general problem of surface heterogeneity effects on phase transitions in adsorbed films and demonstrate that results of earlier studies [11, 20, 21], based on eqn. (1), seriously underestimated heterogeneity effects on the properties of adsorbed films.
2. The model and mean field theory
In this work we use a simple two-dimensional lattice gas model on a square lattice, represented by the hamiltonian 1
*:
- 5
Z
<1. f7
-
E i
(2)
where n; is the occupation variable equal to 1 (0) when the ith site is occupied (empty), UNN is the energy of interaction between a pair of nearest neighbours and ~,, is the adsorbate-adsorbent interaction energy for the ith site. By assuming different forms of the distribution of adsorbate-adsorbent interaction energies c,; over the lattice sites we can model surfaces with different heterogeneity effects. Here we shall consider surfaces with only very few types of sites, denoted by L, of different energies so that the E D F will be represented by a set of L pairs of energy and contribution values {s~, y," 72; " s;,39_} such that
(but only one) value of as and apply the condition of thermodynamic equilibrium, i.e. equality of the chemical potential for all patches. The problem of boundary effects in adsorption on patchwise heterogeneous surfaces will be a subject of our next paper. We now describe briefly a simple mean field theory of adsorption on such discretely heterogeneous surfaces [22-24]. In the framework of the Bragg-Williams approximation the adsorption on a homogeneous surface or on a single homotatic patch, consisting of sites with the energy ~;,, is represented by the well-known adsorption isotherm equation [5, 23]
fl
~;,
k.T-
(
Oi ~
k,T+ln\~-O~/
2HNN(}i
k~T
(4)
where /~ is the chemical potential, kB is the Boltzmann constant, T is the temperature and 0; is the degree of surface coverage. When the surface is composed of L patches then we have L equations like (4) representing adsorption on each patch. In the case of a heterogeneous surface with random topography we also have a set of equations describing adsorption on each type of site, but in a slightly different form: p
k];
+ln(
-
0;
\U-@
)
2UNN~1 kBr
i5)
where 0 is the total surface coverage of an entire heterogeneous surface. For a heterogeneous surface composed of L different types of sites and the EDF {';J, 72 . . . . . ,,;. } the total surface coverage is given by L
~; = ~ O~;'k k
(6)
1
independently of the surface topography. Using eqns. (4) (or (5)) and (6) we can readily calculate adsorption isotherms for any assumed distribution of adsorption sites and for different surface topographies. Within this mean field approximation the derivation of expressions for any thermodynamic quantity, e.g. internal energy, heat capacity, is also straightforward [22, 23] and will not be considered here.
A
= 1 k
(3)
|
and ?k represents the fraction of sites with the energy s~. The model defined by the hamiltonian (2) can be used to model heterogeneous surfaces with random topography. When it is used to model surfaces with patchwise topography we meet the problem of boundary effects between patches of sites with different energies of adsorption. In order to model truly patchwise surfaces, that satisfy the patch independence condition [4], we must consider a collection of systems, each represented by the hamiltonian (2) with a different
3. Monte Carlo method
The thermodynamic properties of the model described in Section 2 have been investigated by means of Monte Carlo simulations in grand canonical ensemble using the Metropolis sampling method [25]. Thus, the grand canonical thermodynamic potential for our model is given by
i
A. Patrykiejew / Monte Carlo studies of adsorption 111
A number of thermodynamic quantities have been calculated, including fractional coverages 0~., k = 1 , . . . , L, of each type of site, the total surface coverage & mean adsorption energy U and heat capacity C. All these quantities have been normalized per lattice site and calculated as simple averages: 1 O~.=~i_~= ni, 1
k=l,2
.....
L
(8)
M
O=~
12 n,,
(9)
i--I
u = 1_ M ('Y~)
(10)
and 1 C - MkB T 2 ( ( ~ 2 ) _ ( ~ . ) 2 )
(11)
In the above equations Mk is the number of adsorption sites of type k, and M is the total number of adsorption sites. In our calculations we have used lattices of size 30 x 30 and hence M = 900. Considering that in this work we are not interested in accurate estimation of critical properties the size of our simulation box was sufficient to obtain reliable results. In order to avoid boundary effects we have used standard periodic boundary conditions in both directions [25]. In most cases it was sufficient to use about 103 Monte Carlo steps (per site) (MCSs) to obtain accurate results for surface coverage and mean adsorption energy. Calculations of heat capacity required longer runs with up to 104-105 MCSs, in particular at temperatures close to critical points. The results for heterogeneous surfaces with patchwise topography were obtained by independent calculations of thermodynamic quantities for homogeneous surfaces with given energy of adsorption and then by averaging the results using the relations L
O ~ 12 )'k0k
(12)
k=l L
u:
12 ;,~.v, k=!
{13)
and L
C=
12 )'kCk
(14)
From the grand canonical potential (7) and the hamiltonian (2) it follows that in the case of a homogeneous surface the change in adsorption energy e causes only a shift in chemical potential by the difference in adsorption energies. Hence, the results for systems with patchwise topography could be obtained using results obtained for
191
only one value of the adsorption energy (e = 0) and by further rescaling of the chemical potential. In the case of surfaces with random topography we have generated the surface structure with a given distribution of adsorption sites by using random numbers. For each type of E D F we have performed calculations for several (up to 10) independently prepared surfaces in order to average results over different configurations of adsorption sites with different energies. It was found that these effects were completely negligible for systems considered in this work. Throughout this paper we assume that the energy of interaction between nearest neighbours is given by UNN/ kB = 0.5 SO that the two-dimensional critical temperature Tc for our model, corresponding to a homogeneous surface, is about 0.28364. In order to determine phase boundaries for systems exhibiting first-order phase transitions we have calculated free energies of both coexisting phases using a standard thermodynamic integration method [26, 27].
4. Results and discussion
We begin with the presentation of results for a simple case of a surface with only two types of adsorption sites. Figure 1 shows examples of adsorption isotherms for systems characterized by 7~ = 7 2 - - 0 . 5 (L =2), e~ = 1.0 and different values of e2 at T - - 0 . 5 and obtained from Monte Carlo simulations for a patchwise and random topographies. We observe that, even at that rather high temperature, the adsorption isotherms obtained for a patchwise topography exhibit well-pronounced steps corresponding to subsequent occupation of patches of different sites. On the contrary, this effect is much weaker in the case of random topography, and the steps become visible only for a sufficiently large difference in adsorption energies of different sites and for sufficiently low temperatures. It is easy to understand these results considering the mechanism of adsorption on a surface with random topography. The more active sites are occupied first and they serve as centres of condensation for adsorption on less active sites. Therefore, for a given value of the chemical potential the net coverage of less active sites is higher in the presence of more active sites than on a homogeneous patch of only less active sites. Figure 2 shows the dependence of the fractional coverage of less active sites as a function of chemical potential on surfaces with different concentrations of random impurities of more active sites. In the range of chemical potential values used in the calculations the more active sites were completely covered by adsorbed particles. It is seen that the coverage of less active sites increases with the concentration of random impurities. When the random
A. Patrykiejew / Monte Carlo studies of ad~'orption 111
192
impurities consist of sites of lower energy of adsorption we observe an opposite effect; the net coverage of more active sites decreases with the concentration of random impurities (Fig. 3). In Fig. 4 we present a series of phase diagrams (obtained from the Monte Carlo simulations) for systems with different concentration of random impurities of higher energy. We find that an increase in the concentration of random impurities
1.0
8 qa
0
O-P
28-
0.6
O.4-
s
O~P D-R
t0.
/
O.2-
)
P
o.6
-6}
-,~1
-£1
311
-2~1
-1~
-0:1
(c) Fig. 1. Adsorption isotherms obtained from Monte Carlo simulations for systems with ",,~=Y2 =0.5, T =0.5, c~/kB = 1.0 and the values of ~:z/kB = (a) 0.5, (b) 2.0 and (c) 3.0 for random topography (R) and patchwise topography (P). 0.2-
causes a rapid decrease in the critical temperature for the phase transition occurring in the adsorbed layer and an increasing effect of asymmetry of phase boundaries. In particular we expect that there is a certain concentration threshold Y~m and when the impurity concentration is higher than }im ,c the system does not exhibit a phase transition at non-zero temperatures at all. This is in contrast to the mean field prediction that the critical temperature changes with the concentration of any given type of sites as
0-
(a)
1,0" /t[
6 o-R
9.8 ¸
4(
T ~ ( T k ) = T~V(0)y,, 0.6.
~2
..Oj -6
(b)
-'5
-&
O -3
-2
-1
0
k = 1,2 . . . . .
L
(15)
where T~F(0) = 0.5 for our model Figure 5 presents a comparison of the critical temperature changes with the concentration of random impurities obtained from Monte Carlo simulations and predicted by the mean field theory. We find that for the systems considered here the extrapolation of our simulation results yields im ~ 0.21 From eqn. (15) it follows that at sufficiently low temperatures the mean field theory predicts the appearance of L first-order transitions on a surface with L types of sites. This is illustrated in Fig. 6 which shows examples of adsorption isotherms for the system with E D F given in Table 1 and obtained from the mean field theory (Fig. 6(a)) and Monte Carlo simulations (Fig. 6(b)). At T = 0 . 1 the adsorption isotherm obtained
A. Patrykiejew / Monte Carlo studies of adsorption III
1.o.
193
7~ (o)
8
~,0IqF
Qsiiii
0.8"
/
q6-
,'J',/;Il
o.4
x' I~
0.4.
t l]~
q2-
o.2.
o 3./,
3.2
3.0
2.8
2~6
Fig. 2. The fractional coverage of less active sites on randomly heterogeneous surfaces with different concentration of highly active sites obtained from Monte Carlo simulations for the system with e~/kB = 2.0, e2/kB = 5.0 at T = 0.3. The concentrations of highly active sites are shown in the figure.
~.o5
J
'
Fig. 4. Phase diagrams for a series of systems with different concentrations of highly active sites (shown in the figure) obtained from Monte Carlo simulations for systems with ~/kB = 2.0 and %[k B = 5.0.
"T/~ (0)
~'i,'~
. . . @--,@,
1
QI
1/45
1/g
'~
@
i'"" "" g,........,,.
Q-u.-g,-~~
/ 1/90
• .--/.,~
I//13--
P ,,"/ +,,z7 a-~l -1/9o El, ~I~,, f tl÷ x - q =1/30
--'I
,,tl
]/,! 0 0.2-
IH!
0.2
0,,4
0,6
0.8
1,0
8 Fig. 5. Dependence of the critical temperature of the phase transition in adsorbed film as a function of highly active random impurity concentration for systems with e,]k B = 2.0 and ~z/k8 = 5.0: - - results of mean field calculations; @, - - - , Monte Carlo results.
Fig. 3. The fractional coverage of highly active sites on randomly heterogeneous surfaces with different concentration of less active sites obtained from Monte Carlo simulations for the system with el/ kB = 2.0, e2/kB = 5.0 at T = 0.3. The concentrations of less active sites are shown in the figure.
f r o m t h e m e a n field t h e o r y e x h i b i t s a w e l l - s e e n v e r t i c a l step c o n n e c t e d with the first-order transition o c c u r r i n g o n sites w i t h 7 = 0.5 a n d c r i t i c a l p o i n t s f o r a d s o r p t i o n o n t w o t y p e s o f sites w i t h ? = 0.2. B y f u r t h e r l o w e r i n g o f t e m p e r a t u r e b e l o w T = 0.05 w e f i n d five v e r t i c a l
A. PatrykRjew / Monte Carlo studies ql" adsorption 111
t94
TABLE 1. The energy density function used in calculations of adsorption isotherms shown in Figs. 6 and 7
1.0-
i ~:i ;'i
J
6
0,6-
0.2-
J
-55
-&6
-,3.7
-2-.,8
-~9
-4,9
-39
-2.9
-tO
(a)
0.8
C~6
0.4
- 5.9
(b) Fig. 6. A comparison of adsorption isotherms for a system with EDF given in Table 1 at T = 0 . 1 and obtained from (a) mean field calculations and (b) Monte Carlo calcula~tions ( -- , contributions to the total surface coverage corresponding to the occupation of different types of adsorption sites).
I 1.0 0.05
2 2.0 0.20
3 3.0 0.50
4 4.0 0.20
5 5.0 0.05
steps corresponding to five first-order transitions. On the contrary, the adsorption isotherm at T = 0.1 and obtained from Monte Carlo simulations (Fig. 6(b)) does not show any trace of a phase transition. When the temperature increases, however, the differences between the mean field and Monte Carlo results become smaller and smaller. Figure 7 shows two adsorption isotherms for the system with the E D F from Table 1 at T = 0.5; they are only slightly different. Now we shall show that the results of our Monte Carlo simulations find a confirmation in experiment. Recently, Heidberg et al. [28] have published results of adsorption measurements for carbon dioxide on NaCl(100) surface at T = 85 K. The experimental adsorption isotherm exhibits a vertical step connected with the first-order phase transition and a well-seen asymmetry of phase boundaries characteristic of a system with a low concentration of random impurities. In Fig. 8 we show the experimental adsorption isotherm obtained by Heidberg et al. and examples of adsorption isotherms obtained from the present Monte Carlo simulations. Although the experimentalists say that the sample of NaCI crystal had highly homogeneous surface, it is possible that small effects of surface heterogeneity due to lattice imperfections of the cleaved crystal introduce some r a m d o m heterogeneity effects that escaped detection. It will be very interesting to compare the results of Monte Carlo simulations with experimental data for other thermodynamic properties of adsorbed films when experimental data are available [29]. Concluding this paper, we would like to comment briefly on the effects of surface heterogeneity on multilayer adsorption, a situation not allowed by the present model. From the mean field calculations [30-32] it follows that heterogeneity influences the properties of higher adsorbed layers, provided that the adsorption energy distribution is sufficiently broad. In particular, it has been observed that randomly distributed heterogeneity has a considerably smaller effect than the heterogeneity distributed in a patchwise manner. Of course, any mean field results must be treated cautiously. As has been demonstrated in this work, any mean field approach is bound to underestimate effects due to surface heterogeneity in the submonolayer region, and this is surely the case in the multilayer region as well. We can recall here the recent ellipsometric measurements of adsorption of noble gases and nitrogen on what was believed to be a highly homogeneous
A. Patrykiejew / Monte Carlo studies of adsorption HI
195
8
"]0"
(
8
0.8
0.6
0.S0,4
J
0.2-
Log [P/( nbor)] O-
(a)
-&5
-~S
-4.5
-&5
-&5
-1~
-q5
(a) 1.0-
rX2~-IB--123.d2-
(b)
0B
,~ .,./~...----,~
,,,~z~-
-4
-3.s
-3
-is
(b) 6.5
5.5
4.5
3.5
2.5
1'5
d,5
(b) Fig. 7. As for Fig. 6 but at T =0.5.
graphite surface [33, 34] at temperatures below the roughening temperature. It has been observed that onsets of the layering transitions in the first, second and also third layer were rounded in the same way as observed in the submonolayer region (Fig. 8). This can
Fig. 8. (a) Experimental adsorption isotherm for carbon dioxide on NaCI(100) at 85 K [28], (b) adsorption isotherm obtained from Monte Carlo simulations for a system with y~ = 0.9, "/2 = 0.1, e~ = 2.0 and s2 = 3 . 0 at T = 0 . 1 2 .
be interpreted as an influence of weak surface heterogeneity, present in the adsorption system owing to the finite size of graphite crystallites, steps at its cleaved surface etc. This experimental finding clearly demonstrates that surface heterogeneity effects can easily ex-
196
A. Patrykiejew / Monte Carlo studies of adsorption 111
tend over the first adsorbed layer. The predominantly occupied highly active sites may act as nucleation centres for the formation of molecular clusters extending to second and higher adsorbed layers. This effect of the promotion of multilayer adsorption by surface heterogeneity has been recently demonstrated by molecular dynamics simulations performed by Bojan and Steele [35]. These workers have considered a model surface with steps and found that adsorbed particles tend to develop a one-dimensional order along these steps and that the local density of an adsorbate at distances corresponding to the second layer exhibits clear maxima in the vicinity of these steps.
Acknowledgments The author wants to express his thanks to Professor K. Knorr and Dr. U. G. Volkmann (Institute of Physics, University of Mainz, F.R.G.) for helpful discussions.
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10 W. Rudzifiski and L. Lajtar, J. Chem. Soc., Faraday Trans. 2, 77 (1981) 153. 11 W. Rudzifiski and J. Jagietto, J. Low Temp. Phys., 45 (1981) 1. 12 V, Mayagoitia, F. Rojas, V. D. Pereyra and G. Zgrablich, Surf. Sci., 221 (1989) 294. 13 M. Jaroniec, Surf Sci., 50(1975) 553. 14 W. A. House and J. M. Jaycock, Colloid Polym. Sci., 256(1978) 52. 15 S. Ross and I. D. Morrison, J. Colloid Interface Sci., 52 (1975) 103, 16 C. H. W. Vos and L. K. Koopal, J. Colloid Interface Sci., 105 (1985) 183. 17 P. Brauer, M. Fassler and M. Jaroniec, Chem. Phys. Lett., 125 (1986) 241. 18 R. Zallen, The Physics of Amorphous Solids, lnterscience, New York, 1983. 19 E. 1. Benegas, V. D. Pereyra and G. Zgrablich, Surf Sci., 187 (1987) L647. 20 W. Rudzifiski and J. Jagiet|o, J. Low Temp. Phys., 48 (1982) 307. 21 J. Jagietlo, Ph.D. Thesis, UMCS, Lublin, 1984. 22 P. Brauer, Wiss. Z. Friedrich-Schiller-Univ., Jena, Math.-Naturwiss. Reihe, 27(1978) 701. 23 A. Patrykiejew, Thesis, 14th Post-graduate Course in Chemistry and Chemical Engineering, Tokyo Institute of Technology, Tokyo, 1979. 24 P. Brauer, H. Dunken and R. Hafer, Wiss. Z. Friedrich-SchillerUniv., Jena, Math.-Naturwiss. Reihe, 30 (1981) 505. 25 K. Binder (ed.), Monte Carlo Methods in Statistical Physics, Topics in Current Physics, Vol. 7, Springer, Berlin, 1978. 26 K. Binder, Z. Phys. B, 45(1981) 61. 27 A. Milchev and K. Binder, Surf Sci., 164 (1985) 1. 28 J. Heidberg, E. Kampshoff, R. Kiihnemuth, O. Sch6nekfis, H. Stein and H. Weiss, Surf Sci. Lett., 226(1990) L43. 29 J. Heidberg, E. Kampshoff, O. Sch6nek/is, H. Stein and H. Weiss, submitted to Ber. Bunsenges. Phys. Chem. 30 W. M. Champion and O. D. Halsey, Jr., J. Phys. Chem., 57 (1953) 646. 31 W. M. Champion and G. D. Halsey, Jr., J. Am. Chem. Soc., 76 (1954) 974. 32 D. Nicholson and R. G. Silvester, J. Colloid lnterJhce Sci., 62 (1977) 447. 33 J. W. O. Faul, U. G, Volkmann and K. Knorr, Surf Sci., 227 (1990) 390. 34 K. Knorr and U. G. Volkmann, personal communication, 1991. 35 M. J. Bojan and W. A. Steele, Surf S('i., 199(1988) L395.