Langmuir 1993,9, 2618-2623
2618
Effects of Surface Topography on Adsorption Isotherms of Mobile Molecules: Comparison of Patchwise and Random Surfaces Tomoshige Nitta* and Atsushi Yamaguchi Department of Chemical Engineering, Osaka University, Toyonaka, Osaka 560, Japan Received September 8, 1992. In Final Form: February 12,1993@
Adsorption isotherms on patchwise and random heterogeneous surfaces consisting of two energetically different sites are compared in terms of a mobile adsorption model. A hybrid isotherm equation, for mobile adsorption on a random heterogeneous surface, has been proposed by introducing an ordering function Q,which takes into consideration nonrandom distributions of adsorption pairs between sites on a surface and segments in a molecule. It is shown that the patchwise and the random surfaces are distinguishable for the mobile adsorptionseven withoutlateral interactions. A patchwisesurface consisting of two different sites shows a stepped isotherm curve for adsorption with lateral interactions, while a random surface shows a rather smooth isotherm curve. It is shown, from adsorption isothermsof digroup molecules on random surfaces, that competition between active segments in a molecule for active sites controls the shape of isotherm curves. For example, if the active-segment fraction in a molecule is less than the active-sitefraction,the isothermcurve on such a random surface resemblesthat on a homogeneous curve. On the other hand, when the active-segment fraction is greater than the active-site fraction, the active segments are forced to compete with each other, resultingin the decrease of the slope of the isotherm curve and finally forming a slightly stepped curve. 1. Introduction
The energetic surface heterogeneity is known to be an important factor influencingadsorption isotherms of gases on actual surfaces having edges, cracks, and/or various micropores. Comprehensive and critical reviews on the theoretical studies on this subject are presented in recent monographs by Jaroniec and Maday' and Rudzinski and Everett.2 A standard theoretical framework for adsorption on heterogeneous surfaces is based on the so-called integral adsorption equation consisting of the local adsorption isotherm and the energy distribution function. The local isotherm equation has been described by either the localized adsorption model, such as the Langmuir equation, or the mobile adsorption model, such as the two-dimensionalvan der Waalsequation, and both are applied mainly on patchwise heterogeneous surfaces, where sites of equal adsorption energy are grouped into patches and in each patch molecules are assumed to move as if they are on a homogeneous surface. An important role of the surface topography was first suggested by Hill? who extended the Langmuir model to describeadsorptionon a heterogeneous surfacewith lateral molecular interactions; two extreme model surfaces, the patchwise and the random heterogeneous surfaces, were clearly recognized. Steele4developed a general formula for monolayer adsorption on a heterogeneous surface, regardless of the mobile or localized adsorption; however, the theory is limited to a low surface coverage region. The effect of surface topography on adsorption isotherms was almost neglecteduntil Rudzinski and co-workers5reported that the topography of surface affects the adsorption energy distribution evaluated from an adsorption isotherm of a gas, by using the condensation approximationmethod. e Abstract published in Advance ACS Abstracts, August 15,1993. (1) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Soli&, Elsevier: Amsterdam, 1988. (2) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (3) Hill, T. L. J. Chem. Phys. 1949, 17, 762. (4)Steele. W. A. J. Phvs. Chem. 1963. 67. 2016. (5) Rudzihki, W.; L a j b , L.; Patrykiejew,'A. Surf. Sci. 1977,67,195; Surf. Sci. 1978, 77, L655.
0743-746B/93/2409-2618$04.00/0
The adsorption characteristics on random and patchwise surfaces with a uniform distribution of adsorption energies were recently compared by Ritter et ale6by using the localized adsorption model with favorable and unfavorable lateral interactions. The patchwise surfacewas the only model that has been used to describe the mobile adsorption on heterogeneous surfaces; the monograph by Ross and Olivier' presents a detailed description of this model with the two-dimensional van der Waals equation. Tompkins8 firstly studied the isotherm equation for the mobile adsorption on random heterogeneous surfaces; however, the final expression reduced to that on a homogeneoussurface with an averaged adsorption energy. This result is, however, different from the present theory and will be touched on later in the next section. Nikitas et aleg developed lattice theories to describe the mobile adsorption on both random and patchwise heterogeneous surfaces and compared the isotherm curves on the two surfaces with a Gaussian-type energy distribution. The adsorption models cited above are based on the assumption that one molecule occupies one site when a surface is divided into lattice sites. Nitta et al."Jproposed an isotherm equation based on the localized adsorption and the multisite occupancy model, which assumed that one molecule occupied multisites; the heterogeneous multisite occupancy model inevitably reminded us of the importance of surface topography. They assumed that sites of different adsorption energies were distributed randomly; furthermore, molecules were modeled to be composed of segments interacting with sites in different energies. The quasi-chemical approximation, first introduced by Guggenheimll in studying nonrandom mixing of liquid mixtures, was used for evaluating the number of (6) Ritter,J. A.; Kapoor, A.; Yang,R. T. J. Phys. Chem. 1990,94,6785. (7) Ross, 5.;Olivier, J. P. On Physical Adsorption; Interscience Publishers: New York, 1964. (8) Tompkins,F. C. fians. Faraday SOC.1960,46,569. (9) Nikitas, P.; Anaetopouloe, A.; Jannakoudakis, D. Chem. Chron. 1985, 14, 21. (10) Nitta, T.;Kuro-oka, M.; Katayama, T. J . Chem. Eng. Jpn. 1984, 17, 45. (11) Guggenheim, E. A. Miztures; Oxford, 1952; Chapter 11.
0 1993 American Chemical Society
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Langmuir, Vol. 9, No. 10,1993 2619
nonrandom configurations distributing molecules over the surface. The quasi-chemical approximationrequires the normalizing factor and a reference equation for correcting the overcounting of the number of configurations. Since any equation useful for adsorption on a homogeneous surface is found to be applicable to the reference equation, Nitta and Yamaguchi12 recently extended the localized multisite model to the mobile multisite model and called the resultant expression as the hybrid isotherm equation. The objective of this paper is the comparison of the adsorption isotherm characteristics of mobile molecules on two model surfaces: the patchwise and the random heterogeneous surfaces. Among many factors influencing the isotherm behavior, molecular size and molecular heterogeneity are chosen to focus our attention in model calculations, which have not been investigated before. Section 2 describesthe hybrid adsorption theory for mobile adsorption on random heterogeneous surfaces, by extending the adsorption model to include molecules composed of different groups. Model systems and working equations used for the comparison of adsorption characteristics on the two surfaces are described in section 3, and the results are shown in section 4. Concluding remarks on various factors are given in the last section. 2. Mobile and Random Surface Model The basic idea of the new isotherm expression for mobile adsorption on arandom heterogeneoussurface is as follows. We first adopt the concept of sites which are regions characterizing the depth of real potential field changing with location. When a surface is heterogeneous, we imagine that molecules will spend more time on an energetically more stable site; therefore, molecules cannot be distributed randomly on each site of the surface as if they were on a homogeneous surface. This nonrandom distribution or ordering effect should result in decreasingthe total number of configurations, which will work against increasing the ordering of the system driven by the energetic advantage. Following the above idea, the canonical partition function Q for Nmolecules moving around in a monolayer on a random heterogeneous surface is written12
adsorption energy per unit pair of site m and group i. The vertical potential energy E, is assumed to be the sum of pair adsorption energies between sites and groups a
k
(3)
where N m i is the number of pairs formed by site m and groupi. By extending the notation, let Nmobe the number of vacant sites for site m. The ordering function ilis then a function of adsorption pairs {Nmi). The quasi-chemical approximation, first introduced by Guggenheimll and applied to the nonrandom distribution of adsorption pairs {Nmi) by Nitta et al.,l0J2is used here
where the asterisks denote the random distribution of adsorption pairs. The last term is introduced to normalize 52 so that it is unity when all the pairs are randomly distributed. Since eq 4 is based on the assumption that all the adsorption pairs, between sites and segments, are independent of each other,13it includes some impossible configurations for adsorption pairs when a molecular size is larger than a site size. However, at present it may serve as a useful approximationfor fl. If one molecule occupies just one site, eq 4 is exact in the framework of the present model, which combines the mobile adsorption and heterogeneous sites concepts. There are two types of constraint conditions for a set of variables ("i): the group balance and the site balance equations. 8
CNmi - niN = 0
(i = 1 , 2 , ...,k)
(5)
(m = a, b, ..., s)
(6)
m=a
- M,
$"j
0
'I
The partition function Q is then rewritten as In Q = Nln( wherejo is the molecular partition function on the surface, ~ ) de Broglie thermal Afthefree area, A ( = h / Q ~ m k T ) l /the wavelength,E, the potential energy vertical to the surface, E h the horizontal or lateral interaction potential energy, k the Boltzmann constant, and T the temperature. Q is an ordering function representing the ratio of the total number of configurations in distributing molecules over sites on the random heterogeneous surface to that on a homogeneous surface. Tompkins8 ignored this ordering function; therefore, a resultant partition function on a random heterogeneous surface was formally identical to that on a homogeneous surface when an expression for Eh was given by Eh
=-PCU/A
+
N A ~
The distribution of adsorption pairs ( N h }is determined from the principle of maximizing the partition function, eq 7, under the constraint of eqs 5 and 6. By use of the undeterminedmultiplier method,an analyticalexpression for the solution of this problem has been obtained.lO The essential feature of the solution for {Nmi] may be the Langmuir-type expressionsfor the site coverages of groups which are given
(2)
where a is a constant and A is the surface area. Let the surface be divided into M sites and Ma, Mb, ..., M8be the numbers of sites for site species a, b, ..., s, respectively. One molecule is supposed to be composed of k groups; nl, nz, ...,nk are the numbers of sites occupied by group 1,2,...,k, respectively. Let €mi be the depth of (12) Nitta, T.;Yamaguchi, A. J . Chem. Eng. Jpn. 1992,2s, 420.
-)
m = a , b,...,s;i = 1 , 2,...,k
where (13)Hill, T. L. An Introduction to Statistical Thermodynamics; Addieon-Wesley: Reading, MA, 1980; Chapter 14.
2620 Langmuir, Vol. 9, No. 10,1993
Nitta and Yamaguchi
em, = Nmi/Mm
(9)
m = a , b,...,s ; i = 1,2,...,k
and site a is a reference site chosen arbitrarily. We define new variables (ri)relating the site coverages on a reference site {Oai) to the overall surface coverages of groups (4)88 'ai
(1)-, !Z
= ri
(i = 1,2, ...,k )
'i
(1- y
j ,
where
q = ?rd2N/4A = DN/A (17) where /3(=?rd2/4)is the cross-sectionalarea of a molecule. Then p b is given as
Other thermodynamic properties of the surface phase, such as the intemalenergy, the entropy,and the spreading pressure, are derived from the partition functioneq 7 under the constraint of eqs 5 and 6; their analytical expressions are summarized as follows: Internal energy
Bi = niN/M
U=k p ( y ) N A
i = 1,2, ..., k The site coverages {OmJ are then given as rmiriei
where Ub is given as
emi = 1+ 2 j ( r m j r-j l)ej m = a , b, ...,s ; i = 1,2, ...,k
Entropy
where
S=lnQ+U/T rmi= exp[(emi- eai)/kTl
=S ,
coefficient of group i on reference site a because two terms having the form W(l- 0) in eq 10 are proportional to the activities of group i on referencesite a andona hypothetical homogenous surface. ri's are unity if the surface is homogeneous. It is noted here that the variable I'i is equivalent to the ratio Yi/Yi* defined in the previous paper;1°however, we adopt {ri} because it is easy to grasp their physical meaning. By inserting eq 12to the site balance equation, ZJmBmi = Bi where f m is the fraction of site m, we obtain k simultaneous equations in k unknowns (ri}.
1
- 1= 0
The local activitycoefficients{ri)are determined by solving the above simultaneous equations numerically. The chemical potential p is then derived as
k
+ x n i ( k T In ri- eai)
Nmi
(21)
Nmi*
+
S d N = k ln(-) j2eAf kT d lnOa/A2) dT N A ~ Spreading pressure
where
(22)
is given as
(14)
i = 1,2, ...,k
= p, - 2 d / A
-)
where Sb is given as
A variable I'i defined by eq 10may be called a local activity
1+ Zj(rmjrj- l)ej
k
m-a a-1
m = a, b, ...,s ; i = 1 , 2 , ...,k
fmrmiri
s
- k z c N m iln(
(15)
t=l
where p b is the chemical potential of hard-core molecules, having the repulsive potential which characterizes the molecular shape; it is issued from the free area Af in eq 7. In the present work, we use the scaled particle theory for Af, that is
where q is the dimensionlesssurface density for hard disks of diameter d
3. Model Systems and Isotherm Equations The effects of surface topography on adsorption isotherms may be clearly understandable by comparing the theoretical isotherms of simple systems which are welldefiied to have the same parameters between solidmolecule and molecule-molecule interactions but to differ only in the surface topography. The energy distribution functions used hitherto for comparing isotherm characteristics between two extreme model surfaces, the patchwise and the random heterogeneous surfaces, are continuous functions of energy: a uniform function by Ritter et al.s and a Gaussian function by Nikitas et al.9 Instead of a continuous function, however, we use a two-site surface model because a simple and discrete distribution function seems suitable for understanding the effects of surface topography and other factors influencing adsorption isotherms. Figure 1 shows schematic diagrams of three model systems selected for case studies to show the effects of several factors on adsorption isotherms. The figures in the left-hand column depict a random heterogeneous surface and those in the right-hand column depict a patchwise heterogeneoussurface. Case A is characterized
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Langmuir, Vol. 9, No. 10,1993 2621
m=a,b B
0
0
- E Fb a T
Comparisons of isotherms for mobile adsorptions between two model surfacesare made for the same interaction parameters (e) and molecular size parameter (n). All the parameters for the three cases shown in Figure 1are listed in Table I. The energetic heterogeneity is represented by the parameter r b instead of the difference of pair adsorption energies € b - ea. The adsorption equilibrium constants are given as follows
a
I,
K[RI RANDOM SURFACE
PATCHWISE SURFACE
Figure 1. Schematic diagrams of mobile adsorption models on heterogeneous surfaces of random and patchwise topography having two sites a and b (Cb > c3: (A) one molecule occupies one site, (B) one molecule occupies multisites;(C) digroup molecule is composed Of groups 1 and 2
(Cb2
> Chi).
by n = 1, which represents one molecule occupying one site, and case B is characterized by n > 1 due to the larger molecular size than the site size. In case C one molecule is composed of two different groups, 1and 2, which may differ in pair adsorption energy between two sites, a and b. In all cases the site fractions are fixed at f a = 0.7 and f b = 0.3, and site b is assumed to be more active than site a, that is, t b > ta, In case C, only the pair adsorption energy between site b and group 2 is assumed to be larger than other pair energies (eb2 > ea2 = eel = ebd. An adsorption isotherm equation is obtained by equating the chemical potentials in the surface phase and in the gas phase. We assume an ideal gas and disklike molecules, though the shape may be somewhat in conflict with the models in cases B and C. The isotherm equation for mobile adsorption on a random heterogeneous surfaceis then given as
where the adsorption equilibrium constant KLRland the lateral interaction parameter u are given as
and
u = 2ar//3 (27) The isotherm equation for mobile adsorption on a patchwise heterogeneous surface is given as 9 = xmfmqm
(28)
where qm's are determined from isotherm equations on homogeneous surfaces, which are given as
m=a,b where
In KbrP1=
= K [PI = K a
In K + n In rb (cases A and B) In K + n2 In rb2 (case C)
The overall surface coverage 8 is calculated as 8 = t/qmax, where vmar = 0.907. 4. Comparison of Patchwise and Random Surfaces The important role of surface topography has been recognized when the lateral interactions between adsorbed molecules are taken into consideration. Figure 2 shows the effect of the lateral interaction, represented by the parameter u/kT, on the adsorption isotherm on a homogeneous surface, not on a heterogeneous surface. The ordinate is the surface coverage 8 and the abscissa is the logarithmic dimensionless pressure Kp. The numbers in Figure 2 stand for u/kT varying from 0 to 10. The curves for 0 represent no lateral interactions; the curves for 10 are close to the isotherms at the two-dimensional critical point, u/kT,fzD1= 11.68. In Figure 3 are shown isotherm curves for adsorption of monogroup molecules of n = 1,which corresponds to the illustration of Figure 1A. The solid lines represent the random heterogeneous surface and the dotted lines the patchwise heterogeneous surface. The surface heterogeneity parameter q,varies as 1, 10, and 100, where the curves for q, = 1refer to the homogeneous surface. Figure 3A shows the isotherms with no lateral interaction (u = 0), where the solid lines are usually higher than the dotted lines when q, > 1. That is, adsorptions on the strongly interacting sites b in the patchwise heterogeneous surface are suppressed more than those in the random surface when the local surface coverage becomes high, the major reason of which is the repulsive interactions between adsorbed molecules. This result differs from that of the localized adsorption model, which predicts the identical isotherms for the two surfaces in the case of n = 1 with no lateral interactions.6 This is because each site in the localized adsorption (i.e. lattice model) is considered to be an independent patch regardless of the topography of energy distribution on a surface. Figure 3C shows the isotherm curves for a large lateral interaction,ulkT = 10. The dottedlines showa remarkable stepped-shape, which indicates that, on the patchwise heterogeneous surface, the strongly interacting patch consisting of site b is filled at the early stage due to the favorable lateral interactions. However, the filling rate in patch b becomes slow due to repulsive forces between adsorbed molecules which work against further filling at high packing density. On the other hand, the solid lines do not change shape as much as the dottedlines do, though they gradually change to resemble a step-shaped c w e .
Nitta and Yamaguchi
2622 Langmuir, Vol. 9, No. 10, 1993 Table I. Valuer of the Parameters for Model Systems on Random and Patchwise Surfaces Composed of T w o Sites a and b (f- = 0.7 and I$ = 0.3; t < fa) case molecule n ulkT ~b figure A1 monomouD 1 0,5,10 la 2 0 1,10,100 3A A; monogroup 1 5 1,10.100 3B 1 1 10 1; 10;100 3c 5 100 4 B monogroup 1,2,3,4 nl+nz=4; 5 % = 1, 5 C digroup %=loo n2 = 0, 1, 2 , 3 , 4 a
Homogeneous surface.
.4
.2
O-4
-3
-2
-1
0
1
1% (KP1 Figure 2. Effect of lateral interaction on adsorption isotherm on a homogeneous surface. The numbers stand for u / k T varying from 0 to 10. At the critical point, u/kTJml = 11.68.
Molecular adsorptions on active sites b on the random heterogeneous surface proceed without harsh repelling and they attract molecules around them through the favorable lateral interactions. Figure 3B represents the curves for ulkT = 5. They look like the upper figures, but they show a step-shaped characteristic as can be seen in the dotted line at high value of r,, = 100. It is noted here that the critical temperature of (three-dimensional)Lennard-Jonesfluids isestimated14askTJeu = 1.35. Since thelateral parameter u is related to cuas u = 4 . 8 e ~the , temperature specified by vlkT = 5 is converted to the reduced temperature TIT, = 0.71, which is close to the normal boiling point of a simple fluid. Figure 4 shows a comparison of isotherm curves of monogroup molecules,which means each molecule consists of a single and energetically same group, having a larger molecular size than site size of a random heterogeneous surface. The value of n,the number of sites occupied by a molecule, varies from 1 to 4. The lateral interaction parameter vlkT is fixed at 5, and the heterogeneity parameter r,, at 100. The dotted lines show a remarkable stepped-shape,while the solid lines have only a slight stepshaped curve even for a large value of n. Since the adsorption energy is the sum of energies between segments and sites, an increase in n results in a large increase in adsorption energy per molecule. Therefore, the dotted lines in Figure 4 magnify the shape of isotherms with large difference in adsorption energy per molecule on the two patches. Figure 5 shows the isotherm curves of digroup molecules composed of four segments, under the condition that nl n2 = 4. The number of n2 segments varies from 0 to 4. Other parameters are fixed at rbl= 1,rb2 = 100, and ulkT = 5; the interaction between active site b and active group 2 is presumed to be a hundred times larger than others.
+
(14) Nicolm, J. J.; Gubbim, K. E.; Streett,W. B.;Tildesley,D. J. Mol. Phye. 1979,37,1429.
log ( 0 1
Figure 3. Isotherm curves of molecules (n = 1)on heterogeneous surfaces composed of two sites a and b cf, = 0.7 and f b = 0.3): random (-) and patchwise (- - -) surfaces; (A) ulkT = 0; (B) ulkT = 5; (C) ulkT = 10. The heterogeneity parameter fi is related to the difference of adsorption energies, h = exp[(tb c3IkTl.
log (KP1
Figure 4. Comparison of isotherm curves of monogroup molecules of a larger size than a site size (n = l, 2, 3, and 4) on random (-) and patchwise (- - -1 surfaces consisting of two sites and b cf. = 0.7 and f b = 0.3). Other parametera are f i e d at ulkT = 5 and rt, = 100.
The dotted lines again show a step-shaped curve, and the solid lines gradually change to resemble a slight stepshaped curve. It is interesting to see that the solid curve 1 almost coincides with solid curve 0, which is the isotherm curve for a homogeneous surface, by shifting the abscissa by the magnitude of 1.5. This result formally coincides with the final expression of Tompkins? who suggested that the expression for mobile adsorption on a random heteroge-
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Langmuir, Vol. 9, No. 10, 1993 2623
* * e
log (KP)
Figure 5. Comparison of isotherm curves of digroup molecules and composed of four segments, nl+ na = 4, on random (-1 patchwise (- -1 surfaces consisting of two sites a and b cf. = 0.7 and f b = 0.3). The number of segments na varies from 0 to 4. Other parameters are fixed at u/kT = 5, = 1,and = 100.
-
neous surface is the sameas that on a homogeneoussurface with some averaged adsorption energy, though his theory ignored the ordering effect introduced in the present work. The parallel shift of an isotherm will be common only when the fraction of active group 2, nd(n1 + nz), in a molecule is less than the active-site fraction f b . In other words, if the number of active segments are less than the number of active sites, then the active segments do not need to compete, and the isotherm curve resembles that of the homogeneous surface. On the other hand, when the fraction of active segments in a molecule is greater than the fraction of active sites, they are forced to compete with each other. Then the slope of the isotherm curve decreases and a slight step-shaped curve is formed. The importance of the competetive character in adsorption has also been pointed out for mixed-gase adsorptions on zeolite in terms of the positive and negative site correlations.12 5. Conclusions The hybrid isotherm equation has been proposed for describing the mobile adsorption on a random heterogeneous surface by introducing an ordering function s2 which reduces the number of configurations distributing molecules over the surface due to preferential adsorption on energetically more stable sites. When the ordering effect is ignored, as was done in deriving an isotherm equation by Tompkins? the resultant equation becomes the equa-
tion of mobile adsorption on a homogeneous surface with some averaged adsorption energy. On the contrary, the hybrid isotherm equation yields various shapes of isotherm curves, characterized by a small slope and even a slightly stepped-curve, for the mobile adsorption on random heterogeneous surfaces. Among comparisons of isotherm curves for mobile adsorptions on the random and the patchwise heterogeneous surfaces, it is noted first that in the case of no lateral interactions between admolecules the two surfaces are distinguishable for mobile adsorptions while Langmuirtype localized adsorptions predict that the two surfaces are indistinguishable. There are two features for mobile adsorptions on a patchwise heterogeneous surface: the molecular repulsion at local high density in a strongly interacting patch, which prevents full occupation of more active sites, and the molecular attraction due to favorable lateral interactions, which enhances the local filling at a low pressure region. Therefore, a stepped curve is a remarkable feature for an adsorption isotherm on a patchwise heterogeneoussurface. For mobile adsorption on a random heterogeneous surface, molecular occupation of more active sites proceeds without repelliig each other and isotherm curves do not change shape as much as those on a patchwise surface do, although the slope of the isotherm curves decreases according to the surface heterogeneity. The surface topography is important for adsorption of molecules composed of different groups interacting with several active sites on a surface. The patchwise heterogeneous surface does not recognize the different groups in a molecule; therefore, the isotherm curve is determined by the fraction of patches and adsorption energy per molecule on each patch. For a random heterogeneous surface, however, competitive adsorption of segmentsfor active sites on the surface plays a key role for the isotherm curve. From a case study of digroup molecules on a random surface consisting of two different sites, it is suggested that the isotherm curve resembles that of a homogeneous surface when the fraction of active segmentsin a molecule is less than the fraction of active sites. On the other hand, when the fraction of active segmenta in a molecule is greater than the fraction of active sites, they are forced to compete with each other, resulting in the decrease of the slope of the isotherm curve and finally forming a slightly stepped curve.
