2016
WILLIAMA. STEELE
Vol. 67
MONOLAYER ADSORPTION WITH LATERAL INTERACTION ON HETEROGENEOUS SURFACES BY WILLIAM A. STEELE Department of Chemistry, Whitmore Laboratory, Pennsylvania State University, University Park, Pennsylvania Received March 14, 1963
-4general approach to the theory of physically adsorbed monolayers on heterogeneous surfaces is given. Explicit expressions are derived which gipe adsorption properties in terms of the distribution functions for single sites, pairs of sites at a given distance, and similar higher order functions. The lateral interaction terms are primarily dependent upon the pair and higher order distribution functions. A generalized form of the Bragg-Williams equation is obtained, valid for any distribution of pairs of sites. A few specific site distribution functions ale substituted into the general equations, and equations are obtained for the low coverage adsorption isotherms and isosteric heats. These expressions are then compared with experimental data on two carbon black adsorbents (one homogeneous and one heterogeneous), and some information concerning the site distribution functions on the heterogeneous sample is deduced.
1. Introduction The theory of monolayer physical adsorption on heterogeneous surfaces has been treated by many authors. This research was reviewed in detail by Honigl in 1954; since that time, a relatively small amount of additional work has appeared. Many of these papers have been concerned with localized adsorption with no lateral interactions; the choice of this model leads to particularly simple results since the isotherms and heats can be expressed as simple integrals over the site distribution function. If either one of the simplifying assumptions of this model is removed, the problem becomes very much more difficult. For instance, in the case of localized adsorption with lateral interactions, one must specify not only the site distribution function (ie., the numbers of each kind of site), but also the distribution functions for pairs, triplets, etc., of sites. I n much of the work on this model, a patchwise distribution of sites has been assumed,1*2 although the case in which sites are randomly distributed over the surface also has been ons side red.^ I n these treatments, either the Bragg-\J7illiams or the quasi-chemical models were used to estimate the effects of lateral interactions; in these models, only the distribution function for pairs of sites is required. Alternatively, a two-dimensional van der Waals gas on a heterogeneous surface has been analyzed using patchwise4 and random5 site distributions. Thus, the treatments of this problem to date have invoked restricted models for the adsorptive behavior of single atoms, as well as highly simplified assumptions concerning the lateral interaction terms. Recently, a more general formulation of the theory of monolayer adsorption on homogeneous surfaces has been presented.6.7 The present paper consists in the extension of this formalism to include heterogeneous surfaces. Some specific expressions are derived from the general equations. As an example of the application of the theory presented here, some of the surface properties of a carbon black with a heterogeneous surface are deduced by comparing the theory with low coverage data ( 1 ) J. M. Honig, Ann. N. Y . Acad. Sca., 58, 741 (1954). (2) W. 17.1. Champion and G . D. Ilalsey. Jr., J . Phys. Chem., 87, 646 (1953); J . Am. Chem. Soc.. 76, 974 (1954). (3) T. L. Hill, J . Chem. Phys., 17, 762 (1949). (4) S. Ross, J . Phys. Chem., 65, 608 (1961). (5) F. C. Tompkins, Trans. Faraday Soc., 46, 569 (1950). (6) W. Steele and M. Ross, J . Chem. Phys., 35, 862 (1961); W. Steele, Advances in Chemistry Series, No. 33, Amerlcan Chemical Society, Washington, D. C., 1961, p. 269. (7) T. L. Hill end S.Oreenschlag, J . Chem. Phya., 84, 1538 (1961).
on a highly uniform graphitized carbon sample and on the heterogeneous adsorbent. 2. General Theory In the statistical mechanics of adsorption systems, the use of the grand partition function presents the most natural approach to the problem of computing the number of adsorbed atoms as a function of p. In general, the grand partitsionfunction %* for the atoms in a small volume V snear the surface of the adsorbent can be written6
where x is the activity of the adsorbed atoms, and is equal to p/kT if the gas is ideal far from the surface. Z$) is the configurational integral for N atoms near the surface and is given by
drl . . drN (2.2) where d s ) ( r i ) is the gas-solid interaction energy for an atom a t f i , and u(rlj) is the mutual interaction of a pair of gas atoms separated by a distance rij. Since ~ ( ~ ) ( r i ) goes to zero quite rapidly when xi, the perpendicular distance of the ith atom above the surface, becomes larger than a few atomic diameters, V , can be set equal to Ax,, where A is the area of the adsorbent and xs is about 5-10 A., depending on the sizes of the gas and solid atoms. In the homogeneous surface case considered previously, @(ri) a t fixed xi is a periodic function of ti, the two dimensional vector perpendicular to xi; however, heterogeneity implies that @)(ri) will not only have a periodic variation, but will also undergo large changes in magnitude which are associated with the gas-surface interaction near an exposed lattice imperfection, a surface impurity atom or some other similar deviation from uniformity. Formally, these variations can be taken into account by partitioning the surface into N , sites with areas which are characteristic of the area per atom in the solid surface. It is assumed that there are M I sites of type I with an area AI(^) and a gas~ ) ,that the total number of surface potential e ~ ( ~ ) ( rand different kinds of sites is W . In this case, the configurational integral can be conveniently divided up into a sum of integrals
ON HETEROGENEOUS SURFACES MONOLAYER ADSORPTION
Oct., 1963 '7
UN
(8)
pNl . . . ",
=
2017
(2.3)
N
where Zs)xl . . . N~ is the configurational integral when there are exactly N I molecules on the Ith kind of site (for 1 = 1 to W ). The sum is taken over all possible assignmenk of N molecules to W different groups of sites. The explicit equation for Z(')N~. . . N~ is Z(')Nl . . .
Nw
---
sr'
,
A
... J .... J"". . .. J Aw
I = l
i = l
In this express,ion there are NI integrations which are carried out over the total area associated with the Ith . it is convenient kind of site AI (= M p 4 1 ( ~ ) ) Finally, to define (2.5) (2.6)
When E* is written in terms of the quantities given in eq. 2.4-2.6, one has
(2.7)
(2.12)
Note that the integrations in each term in the sum of the numerator of eq. 2.12 are to be carried out over the area of an I site and a J site which are separated by a distance ai. The factor of two arises from the fact that there are two ways of placing atoms 1 and 2 on a pair of different sites; however, when ai = 0, atoms 1 and 2 are on the same site and two must be replaced by unity. Formal equations for PIJK and higher coefficients can be written down; however, these expressions involve factors such as the number of triplets of sites containing one site each of types I, J, K and having separations Gj, ajk, and %k, and other more complex configurational parameters, and will not be considered further in this paper. It is possible t o simplify eq. 2.5 and 2.12 if one assumes that : the surface heterogeneity is characterized by different values of €I*, the minimum value of E I ( ~ ) , but that the variation in potential energy over the site (EI(S)(r)- eI*) is the same for all I, for values near the minimum; and that the temperature is low enough and - q*large enough so that the only appreciable contributions to the integrands in eq. 2.5 and 2.12 come when xi is very near to the value of x corresponding to the maximum attractive energy. When these approximations are invoked, one can show that
ZO(') = V,") exp [- eo*/kT] At relatively small values of p*, one can expand E* in powers of p* and derive an expression for In E* by the well known techniques used in the theory of imperfect gases. In this way, it can be shown that
(2.13)
(2.14) PI*
=
PO*
exp[-(€I*
- E ~ * ) / ~ T(2.15) ]
where PI
=MI
PIJ
= ZIJ* =
LBLB
(2.9)
- 2.M1Ms I # J 211'' - il/lr2 I =J
JA1 JAJ
}
+
(2.10) (2.17)
+
exp [- (EI(*)(rl) EJ(')(rZ) u(rlz))/kTldrldr2
fiz
=
exp[--u(m)/kT]
-1
(2.18)
Ineq. 2.16, it has beenassumed that exp[-u(n2)/kT] = 0 for all configurations of a pair of atoms adsorbed on exp [ - (dS)(rl)+ the same site. Finally, the grand partition function e~(')(r~))/kT]dr~dr2can be written (2.11) W 1nZ* = PO* MI exp[--AeI*/kT) The total number of ways of arranging a pair of atoms 1 = 1 with one on a type I site and one on a type J site is ~ M I M for J I 4C J and is MI^ for I = J . With this in mind PIJ can be written in terms of a sum of integrals over pairs of sites. M I J ( ~is~ defined ) to be the number where of pairs of sites separated by a distance Q which contain A ~ I *= €I* - EO* (2.20) one site of type I and one of type J. PIJ becomes
ZIJ*
JeB sa. JAI(a!j-AJ(e)
= -___
+
WILLIAMA, STEELE
2018
Vol. 67
The isotherm equation is readily obtained from eq. 2.19 since N,, the number of gas molecules in the volume V,, is given by6
(-)b l n p
=
N a = b In X *
T,V,
(--) b In Z* blnpO*
T,V.
A ~ I *> - a
(3.8)
and the gaussian4
where Pz is given by an equation analogous to eq. 2.17, but with the range of integration of q in the numerator equal to the total area of each patch (assumed to be large enough to eliminate edge effects). If the random site distribution function is substituted into eq. 2.32, one obtains 1-
f~ exp[-2AeI*/kT]
The second term on the right accounts for the contribution of atoms on different sites, and it can be seen that the factor multiplying pz 1 is just equal to the square of the leading term, in agreement with Hill's statement. The last term in eq. 3.7 arises from the requirement that only one atom be placed on a site; it will be shown in the following section that the inclusion of this term can have an important effect on the calculated magnitude of the coefficientof ~ 0 for " a~real system. I n order to obtain explicit expressions for adsorption properties from the equations of sec. 2, one requires not only the distribution of pairs of sites, but also the distribution function for M I (or fr). Two simple analytic distribution functions which have been used by other authors are the exponential8
f~ d ( A e ~ * )=
-
+
W
Po*'
The sums in eq. 3.2 can be evaluated to give =
f~exp[-Ae~*/kT]
I = l J = l
-
MIJ(ai)
2019
MIMJ exp[-(Aa*
MI exp[-2Ae1*/kT] 1 = 1
......
(v'%h)-lexp[-(Ae~*/u)~]d(Ae~*)
+
0
(3.9)
Both of these functions are physically unreal in some respects : in particular, the gaussian distribution predicts that a finite number of sites of positive energy will be present; and the exponential predicts that the most probable sites will be those with the smallest attractive energy (ie., there is no possibility that a small number of weakly attractive sites will be present). However, as approximations to the general form off^, either function appears to give a, reasonably adequate representation. Therefore, the sums in eq. 2.25, 3.4, and 3.6 were replaced by integrals and evaluated with the aid of these distributions. In the case of the exponential function, the results are
+ +
(3.6) where it has been assumed that the mutual interaction of a pair of atoms is 4- m for all configurations in which both atoms are on the same site. In his discussion of the random site distribution, Hill3
> AEI*> -
1 - (E/kT)'
[e,,
pltohwise]
=
Po *
1 - (E/kT)
Po*2Pz
1 - 2(E/kT)
+
* ' + ,
E/W
