Langmuir 1994,10, 2694-2698
2694
Clustering and Its Effects on Adsorption Eli Ruckenstein* and Ashok Bhakta Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received February 24,1994. In Final Form: May 31, 1994@ Two types of clustering of adsorbed molecules on homogeneous surfaces are considered one is twodimensional and the other is three-dimensional (droplet-like). The first is likely to occur when the interactions between two adatoms are sufficiently weak compared to those between an adatom and adsorbent; the second is probably associated with cases in which the above condition is inverted. The main conclusion is that at a critical pressure, for which the Langmuir isotherm predicts a low occupation of the surface, a jump from low occupation to complete coverage can occur. The complete occupation corresponds to a monolayer in the first case and to a multilayer or droplets in equilibrium with a mono- or multilayer in the second. Below the critical pressure, the contribution of clusters to the overall coverage appears to be negligible in comparison to that of singlets.
Introduction Numerous adsorption isotherms for both homogeneous and heterogeneous surfaces are available, which have been summarized extensively in literature.'-' The heterogeneity of the surface is usually attributed to energetic differences between sites. For homogeneous surfaces, isotherms have been proposed which take into account the interaction between molecules. However, the adsorbed molecules are always considered to be distributed uniformly. The goal of this paper is to show that even on homogeneous surfaces, a size distribution of clusters can occur as a result of the competition between the entropy, which tends to disperse the molecules, and the attractive interactions between molecules, which tend to cause aggregation. Two kinds of clustering are considered. In one, the interactions between two molecules of adsorbate are sufficiently weak in comparison to those between an adsorbate atom and adsorbent, in which case twodimensional clustering is assumed to occur. In the other case, the interactions between two atoms of adsorbate are strong and three-dimensional clustering in the form of "droplets" is considered to take place. Two main assumptions are involved in the calculations: (i) there are no interactions among the clusters and (ii) within a cluster, only nearest neighbor interactions are taken into account. It is shown that in certain conditions, at a relatively low pressure, the fraction occupied can jump from a small value to unity, which corresponds to a monolayer in the first case and to a multilayer or droplets in equilibrium with a mono- or multilayer in the second. Below the critical pressure at which the jump takes place, the clustering appears to have a negligible contribution to the fraction of sites occupied.
Size Distribution of Two-DimensionalClusters When the interactions between two molecules of adsorbate are sufficiently small in comparison with those between them and adsorbent, two-dimensional clusters are expected to form. It will be assumed that for a particular number of molecules in an aggregate, a single kind of cluster exists. This implies that at a given size, there is only one configuration whose free energy of formation is appreciably more negative than that of any other configuration. This assumption can be relaxed. At equilibrium, the surface of the adsorbent has N,' free sites and Ng clusters containing g molecules, where g varies from 1to,,,i i,, being the number of molecules in the largest cluster. Considering the aggregates as distinct chemical species, with negligible interactions between them, the free energy of this system is F = N,'a,p
+
h,
N,' In-
imax
+ C N i In Nt Nt
+
i=l
1=1
wherep' is the standard chemical potential, the subscript i refers to clusters formed of i molecules, a,, is the adsorbent-gas interfacial tension, a is the surface area per site of adsorbent, k is the Boltzmann constant, T the absolute temperature, and hax
N~ = N,'
+ CN, i=l
In eq 1,the entropy term was calculated as for an ideal mixture. "he quantities N,' and N, are not independent, since the total number of sites N , is constant:
* Author to whom correspondence should be addressed. Abstract published in Advance ACS Abstracts, J u l y 15,1994. (l)Brunauer, S. The Adsorption of Gases and Vapors; Oxford University Press: London, 1945. (2) De Boer, J. H. The Dynamical Character of Adsorption; Oxford University Press: London, 1953. Growel1,A. D. PhysicaZAdsorption of Gases;Butter(3) Young, D. M.; worth: London, 1962. (4) Clark, A. The Theory of Adsorption and Catalysis; Academic Press: New York, 1970. (5)Jaroniec, M.;Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: New York, 1988. (6) Kiselev, V. F.; Kryolov, 0. V. Adsorption and Catalysis on Transition Metals and Their Oxides;Springer Verlag: New York, 1989. (7) Adamson, A. W. PhysicaE Chemistry of Surfaces, 5th ed.; New York, 1990. @
N , = N,'
+ c i N i = const -
(3)
i=l
The minimization of F with respect to Ni, subject to the constraint expressed by eq 3, yields (4)
where
0743-7463/94/2410-2694$04.50/00 1994 American Chemical Society
Clustering and Its Effectson Adsorption
Langmuir, Vol. 10, No. 8, 1994 2695
Pi0
AGi" = - - p l o I
is the standard free energy change per molecule when a singlet is included in a cluster of size i . Equation 4relates the number of clusters of size i to the number of singlets and to Nt and is used in the next section to obtain the adsorption isotherm.
(14)
Combining eqs 4 and 14, one finally obtains
A Generalized Langmuir Isotherm The chemical potential, p1, of the singlets is obtained from eqs 1 and 3 as "t
cmax
i=2
On the other hand, the chemical potential in the gas phase, at sufficiently low pressures, has the form
+
pv = pvo kT In@)
The above expressions can become useful only if explicit expressions are suggested for AGi".
(7)
where p is the pressure and p," is a standard chemical potential. The equality of the chemical potentials pl and p, leads to
A Large Cluster Approximation for AGi" For large circular clusters, the standard free energy change contains a bulk term proportional to i and a line contribution which is proportional to ill2
iAGi" = iAG"
+ yiU2
(16)
where where
y = 2 n 112a 112o,
is a temperature-dependent quantity. From eqs 2 and 8 one obtains
(10) Po
cmax
1-
CN~/N, i=l
(17)
q and a being the line tension and molecular area, respectively. The quantities AGO and y are evaluated (assuming only nearest neighbor interactions between adsorbed molecules) in terms of the interaction potential, E , between two molecules, as follows. First let us observe that, because of the strong interactions with the adsorbent, the molecules of adsorbate are almost fixed on the adsorption sites. For this reason, the free energy change iAG" can be approximated by the interaction energy between the molecules of the cluster. Denoting by z the number of nearest neighbors in a cluster
which, combined with eq 4,becomes
Nl
The fractional coverage of the adsorbent surface can be calculated using the expression
+
XiNilNt 6, =
where E represents the interaction potential between two neighboring molecules assumed fixed on the adsorption sites. Further, let us evaluate y . The line tension represents the excess free energy which appears when the attractive interactions with some molecules of the cluster are replaced by the less negative interactions which exist between them and gas molecules. In this analysis, the latter have been neglected. Denoting by /3 the fraction of the nearest neighbors in the same layer, one atom (molecule) in the line layer will have, for large clusters, (p [(l - p)/21)z = [(p 1)1212 neighbors. Consequently
+
i=l
(12)
NJNt Since eqs 2 and 3 provide the relation
Combining eqs 17, 18, and 19, yields
which for p eq 12 becomes
= 113, becomes y = -1.19AG"
-AGO
(21)
Ruckenstein and Bhakta
2696 Langmuir, Vol. 10,No. 8,1994 While eq 21 is valid for large clusters, it is extrapolated here also to small ones.
The Adsorption Isotherm Substituting iAGi' from eq 16 into eqs 11and 15 yields
D
and " I
0
05
15
1
2
25
35
3
PIPo
Figure 1. Adsorption isotherm for two-dimensional clusters with no jump.
=
Introducing the notations 0 70 60 5-
Equations 22 and 23 can be rewritten as
p -
0 40 3-
X
--
/
01-
i=2
0
0
0L.ng.ui
002
004
006
008
01
012
014
016
r
1
018
PIPo
Figure 2. Adsorption isotherm for two-dimensional clusters with a jump.
i=2
e, =
correspondingly a critical value AG," of AGO, for which = 1. The critical value & is given by the relation:
xm,y
The isotherm can be obtained by solving eq 25 for x and substituting in eq 26. Since x increases with p , the maximum value of x , denoted x,,, corresponds t o p 00 and is given by
-
(27) i=2
(29) For E > Ec a curve similar to that of Langmuir is obtained, while for 5 < E, the curve is sigmoidal.
Explanation of the Predicted Behavior
It is clear thatx,, depends on y and 5, and hence on AG". Because y = 1/5, eq 27 shows that
(28) One can note that 5 is a measure of the extent of clustering, a small value indicating that clustering is strongly favored.
Results Depending on the value of x,,, two qualitatively < different types of isotherms are predicted. Whenx,,v 1, the isotherm is similar to the Langmuir isotherm (see > 1, a sigmoidal Figure 1).On the other hand, whenx,,y curve is obtained with a sharp jump from a low surface coverage to almost complete coverage (see Figure 2). In other words, there exists a critical value, 5, of 5 or
Equation 14 indicates that the coverage Bt depends on Ci,lim-(Ni/Nt)and C,=lim-(iNJNt).The difference between these two sums depends on the extent of clustering. While the former is always less than unity, the latter increases to values much larger than unity as clustering becomes more significant. One may note that both sums are strongly dependent on yx, since (yx)' is the dominant factor in the terms of the series. Figure 3 compares them for i,, = 1000 and shows that for yx < 1the behavior of the two sums is similar, but that for yx slightly greater than unity, Llim-(iNi/Nt)exhibits a very sharp rise. Case 1,t > go. In this case, xmaxy< 1, which means that xv < 1 over the entire range of pressures from 0 to m. Under these conditions, Ci=lim-(Ni/Nt) does not differ significantly from &lim-(iNi/Nt) which means that the adsorbate exists predominantly in the form of single molecules and small clusters. As a result, the adsorption isotherm has a Langmuir-like behavior.
Clustering and Its Effects on Adsorption " imx i
.
Langmuir, Vol. 10, No. 8,1994 2697
1000 0.1
2.5-
2-
i i
1-1
Three-Dimensional Clusters When the interaction forces between two adatoms are sufficiently strong compared to those between an adatom and adsorbent, three-dimensional clustering is expected to occur. The expression for the free energy is again provided by eq 1; however, the constraint to which the quantities N,' and Ni are subject is different, because the number of sites on the surface of the adsorbent occupied by a cluster is no longer equal to the number of atoms forming the cluster. Denoting by si the number of sites occupied on the surface of the adsorbent by a cluster containing i atoms, the constraint has the form Lmax
N , = N,'
I
0
04
02
08
08
12
1
+ CszV,
(32)
i=l
The minimization of F subject to the constraint (32)leads to 0.90.8-
where
0.70 60 5-
0 4-
0.34
01 O 0
2
0
002
004
I
W
006
008
01
012
014
016
018
PIPo
Figure 4. Effect of the maximum cluster size,,i on the adsorption isotherm of two-dimensionalclusters. The ordinate values are et.
Xi=+em< 1and there
Case 2, g < &. In this case, exists a finite pressure (p,), given by Pc -_
5
(30)
Po i=l
for which x q = 1. A small increase in pressure above p e and ofxq above 1raises Xi=1i4iNJNJsharply (see Figure 3), and hence
Equation 33 contains three factors: one of them, (NlINtY, is due to the entropy decrease caused by the reduction in the number of independent adsorbate species because of clustering; the second, (Ns'INt)(sl-E), is due to the entropy increase resulting from a greater number of sites that become available because three-dimensional clusters cover fewer sites on the surface than the atoms they contain; the third factor is due to the standard free energy change involved in clustering. Treating the cluster as a macroscopic body, eq 34 can be written as
iAG," = iAG"
+ scgucg+ scs(ocs- oag)+ iaoSg
where AGO is the difference between the standard free energies of an atom within a large body and as a singlet on the surface ofthe adsorbent, u is the interfacial tension, s is the interfacial area, and the subscripts c, g, and s refer to cluster, gas, and adsorbent, respectively. Considering that each cluster has the shape of a spherical cap of radius R and wetting angle 0, we have sCg= 2?t(l - COS
O)R2
s, = ?tR2sin2 8
(31) Physically this means that a small increase in pressure above pccauses a dramatic increase in clustering. Figure 2 shows that this is accompanied by a sharp decrease in the fraction 81 occupied by singlets. The increase needed for the sharp rise in clustering decreases as i,, increases (see Figure 4)) tending to zero as ,,z becomes large. Consequently, x q = 1or p = p c , represents a threshold beyond which large clusters predominate, resulting in almost complete coverage of the surface. Equation 30 shows that as x i = 1 8 m + m 1,pdpo m. In other words, &li-.(Ni/NJ 1 as x i 1, so that the jump becomes smaller and occurs at a higher coverage. It should be noted that the interactions between clusters have been neglected in our calculations.
-
--
-
(35)
(36) (37)
In addition
where v is the volume per atom in the cluster. Consequently
iAGio= iAG""+ aiw3
(39)
where
(40) and
Ruckenstein and Bhakta
2698 Langmuir, Vol. 10,No. 8, 1994 AGO" = A,G"
+ aa,,
(41)
The Adsorption Isotherm for Three-Dimensional Clusters Equation 33 can be rearranged as
( )(
N i - N , i N,' r [ e x p [ s ) ] [ e x p ( $ ) l @
(42)
N,-"
I
I
1.4-1
I
'1
I
I
Bf
3
i
1I
0 84
I
I
I
Introducing the notations
I
where 6 and K represent quantities analogous to and 6 used for the two-dimensional clusters, and combining with eq 8, eq 42 becomes
(44)
PIPo
Figure 5. Adsorption isotherm for three-dimensional clusters for 6 = 5 and K = 0.1.
The value of x is provided by the equation
1
-p_-
X
.0.1
I
144
(45)
I
/ , -
r
/.
/
I
12-
This equation is analogous to eq 25 used earlier. The adsorption can be monitored by computing the fractional coverage of the surface Os 041
(46)
and the total amount adsorbed per site
i=l
eta
N+
Note that 6, 5 1, while Ot, can exeed unity. By use of eq 43, eq 47 becomes
Since si < i, the dominant factor is (plp0)d. Considerations similar to those used in the two-dimensional case demonstrate that a jump occurs when (plp0)dis slightly larger than unity (and equal to unity as i,, m). The equation
-
Pc -= 8 1 = exp(-)AGO" Po kT
(49)
provides the critical value of p c above which clustering becomes dominant. As 6 increases, the jump occurs at
"0
I
I
0 05
0 15
01
02
0 25
PiPo
Figure 6. Adsorption isotherm for three-dimensional clusters for 6 = 10 and K = 0.1. smaller pressures. For a given 6, the fractional coverage a t which the jump occurs depends on K. As K increases, (C,=lLm=(N,lNt))p,e decreases which means that the jump occurs at lower surface coverages. Figures 5 and 6 present isotherms computed for two cases. Note that in these figures, Ot, is somewhat larger than unity and that this may imply droplets in equilibrium with a monolayer. Larger values of et, may correspond to multilayers or droplets in equilibrium with mono- or multilayers. Free energy considerations can provide a choice between the two-dimensional and the three-dimensional clustering, since that clustering will occur for which the total free energy is the smallest. While possible in principle, the calculations require values for numerous parameters which are not easily available. Similar considerations can be extended to adsorption on liquid-gas or liquidliquid interfaces.
Conclusion Langmuir isotherm predicts saturation at relatively large pressures. If clustering of the adsorbate molecules is taken into account, a jump in the occupied fraction can occur at a relatively low pressure from a small value to complete coverage. While clustering plays a minor role for pressures smaller than a threshold value p c , it plays a major role for pressures exceeding p c . Two kinds of clustering are considered: two- and three-dimensional clustering. In the first case, the fraction of sites occupied jumps from a low coverage to almost complete monolayer coverage; in the second, the jump leads to a multilayer or droplets in equilibrium with a mono- or multilayer.
