Langmuir 1991, 7, 1310-1313
1310
Quasi-Chemical Model for Adsorbate Interactions on a Heterogeneous Surface: Analysis through Pressure Derivatives A. S. Datar and S. D. Prasad’ Physical Chemistry Division, National Chemical Laboratory,’ f i n e 41 1008, India Received November 21,1990. In Final Form: February 20, 1991 A judicious study of pressure derivatives is shown to help identify the type of adsorbate interactions and supplement the information about site energy distribution obtained from adsorption isotherms and differentialheat measurements. Within the framework of the quasi-chemical approximation, the attractive forces predict a maximum in the pressure derivative plot. Since this can occur at very low values of surface coverages, especially for negative exponential distribution, a method is suggested based on the product of the pressure derivative and the surface coverage (DP). However both attractive and repulsive forces predict a maximum in the DP plot when plotted against the mean surface coverage. The location of the maximum, which occurs at appreciable surface coverages, can reveal the nature of adsorbate interactions.
Introduction The primary role of adsorbate interactions1+ in causing adsorbate islanding, surface phase transformations,H and other novel dynamic phenomena has been vigorously pursued. However, in the literature only a few studies deal with the intruding role of surface heterogeneity,*14 which is a real complexity. The superimposed roles of surface heterogeneity and adsorbate interactions and how one can identify them by a judicious study of the pressure derivatives of isotherms is the main objective of the present work. Different degrees of approximationshave been employed to account for the adsorbate interactions. The most popular of these is the so-called mean field approximation in which the assumption of random distribution of adsorbed molecules on a checkerboard of lattice sites is en~isaged.~J~ A higher order approximation would be that one which postulates the random distribution of molecular pairs (viz. correlations higher than that of a pair are neglected) which often is named the quasi-chemical approximation (QCA).1~4Js-uThe pressure derivatives of isotherms are
* To whom all correspondence should be addressed. f
NCL Communication No. 5050.
(1) Silverberg, M.; Benshaul, A.; Robentrost, F. J. Chem. Phys. 1985, 83, 6501. (2) Barteau, M. A.; KO,E. I.; Madix, R. J. Surf. Sci. 1981,104, 161. (3) Gland, J. L., Fisher, G. B.; Kollin, E. B. J. Catal. 1982,87, 263. (4) Silverberg, M.; Benshaul, A. J. Chem. Phys. 1987,87,3179.
( 5 )Lagally, M.; Wang, G. C.; Lu, T. M. Crit. Rev. Solid State Sci. Mater. Sci. 1988, 7, 233. (6) Roelofs, L. D. In Chemistry and Physics of Solid Surfaces; Vanselow, R., Howe, R., Eds.; Springer Press: Berlin, 1982; Vol. 4, Chapter
computed by using the QCA; the chief objective is to know whether surface heterogeneity smudges out the finer distinctions between the attractive and repulsive type interactions, especially when we study the pressure derivatives.
Formulation For quantifying surface heterogeneity, the most tractable model is the random patch model in which adsorption sites of a given energy are grouped in a patch; the patches themselves are randomly distributedemZ2 The other model, in which total random distribution of sites is assumed, is not analyzed here.22929 All statistical averages are computed by taking appropriate integrals over the site energy distribution. The site energy distribution thus in a sense reflects the energy spectrum of the surface. Although the site energy distribution could in principle have any analytical form of dependence on the adsorption energy, for the sake of illustration only three commonly employed site energy distributions are adopted. Thus three popular site energy distributions, viz. the positive exponential distribution, the negative exponential distribution, and the constant distribution, are analyzed. These three distributions are listed in Table I.12 The analysis and the numerical techniques can be extended in a straightforward manner to any arbitary distribution. In light of what has been said above, the basic equations for the mean surface coverage, 8, and the pressure derivative are easily written
10. (7) Woodruff, D. P.; Wang, G. C.; Lu, T. M. .In The Chemical Physics of Solid Surfaces and Heterogeneoue Catalysur; King, D. A., Woodruff, D. P., Eds.; Elsevier: Amsterdam, 1983; Vol. 2. (8)Sahani, P. S.;Gunton, J. D. Phys. Rev. Lett. 1981,47, 1754. (9) Ertl, G. In Catalysis Science and Technology; Andereon, J. R., Boudart, M., Eds.; Springer: Berlin, 1983.
(10)Jaroneic, M.; Madey,R. InPhysical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988, Chapter 2. (11) Prasad, S. D.; Doraiswamy, L. K. Phys. Lett. A 1983,94,219. (12) Prasad, S. D.;Doraiswamy, L. K. Chem. Phys. Lett. 1983,99,129. (13) Prasad, S. D.; Doraiswamy, L. K. Chem. Phys. Lett. 1984,104,
315. (14) Ritter, J. A.; Kappor, A.; Yang, R. T. J. Phys. Chem. 1990,94, 6785. (15) Bhat, Y. S.; Prasad, S., D.; Doraiswamy, L. K. J. Catal. 1984,87, 10. (16) Zhdanov, V. P. Surf. Sci. 1981,111, 63. (17) Zhdanov, V. P. Surf.Sci. 1982, 123, 106.
(18) Zhdanov, V. P. Surf. Sci. 1983, 133, 469. (19) Zhdanov, V. P. Surf. Sci. 1984, 137, 615. (20) Rose, S.; Oliver, J. P. In On Physical Adsorption; John Wdey and Sone: New York, 1964. (21) Steele, W . A. In The Solid-Gaa Interface; Flood, E. h n , Ed.; Marcel Dekker, Inc.: New York, 1967; Vol 1, Chapter 10, p 344 In The Interaction of Gases with Solid Surfaces; Pergamon: New York, 1974. (22) Hill, T. L. J. Chem. Phys. 1949,17, 762; Statistical Mechanics; McGraw-Hill: New York, 1966. (23) Ricardo, J. L.; Pereyra, V.; Rezzano, J. L.; Rodriguezeaa, D.A,; Zgrablich, G. Surf. Sci. 1988,204, 289.
0743-7463/91/2407-1310$02.50/0 0 1991 American Chemical Society
Letters Table I. Table of Site Energy Distributions positive exponential ex~(Q/Qm)Cp constant V(Qz - 61) negative exponential exP(-Q/Qm)Cn Cp
= 1/Q&x~(QdQi) - exp(Qi/Qm)l
1/Qmlexp(-Qi/Qm)- eXp(-Qz/Qm)l
Cn
9
0.3 2
I
I
I
I
I
0.7 1
1
I
1
28
1 0.02
e
0.5
0.001
e
0.15
o.oooos
e
0.0071
Figure 2. Plot of d0/dp vs 0 with attractive interactions for three distributions: Qm = 2 kcal mol-', Q1 = 17 kcal mol-', QZ = 31 kcal mol-'.
interested in the derivative product 8 d8/dp (which henceforth will be referred to as DP) which is nonmonotonic when plotted against 8.
I
0.1
I
I
I
I
I
I
e
I
I10 0.6
Figure 1. Plot of de/dp vs 0 with attractive interactions for two interaction energy parameters for a homogeneous surface: bo = 1E + 8 Torr, QO= 18 kcal mol-', 2 = 4. where 8, is given implicitly by the following equation:
P = bo exp(-Q/RW,/(1- o,))g(e,)
(3) (4)
p = (1- 48,(i - e,)[i - e x p ( - w / R ~ ) ] ) ~ . ~ (5) Q1 and QZ denote the lower and upper limits of the differential heats of adsorption in the absence of lateral interactions and 2 denotes the coordination number. &(T,p) represents the local adsorption isotherm for the ith patch, S( Q) represents the energy distribution function, and w is the interaction energy parameter. Experimentally one conveniently measures the mean surface coverage 8. The pressure derivative can be computed by numerical differentiation after fitting a cubic spline to it." Alternatively if we know the energy distribution function S(Q) experimentally (from differential heat measurements)then by assuming the functional form of the local isotherm 8, d8/dp can be predicted a priori for comparison with the experiment. The physical significance of the pressure derivative is that it is related to the isothermal compressibility,which is a very good diagnostic tool for detecting phase transitions. Besides, since many rival distributions may integrate to give the same mean surface coverage,1° discrimination between them becomes difficult without additional information in the form of derivatives (specific heat, compressibility, etc.). But some caution is advocated in using it, because it is very sensitive to experimental errors and sharply depends on the limits of heats of adsorption, especially for a heterogeneous surface. As has been shown for the homogeneoussurface (wherein S(Q) has a Dirac-Delta function form)," we are more
(24) Datar, A. S.;Praead, S.D.; Doraiswamy, L.K.Chem. Phys. Lett. 1980,139,337.
Results Figure 1illustrates the typical behavior of the attractive interaction forces when d8/dp is plotted against 8, the surface coverage, which shows a distinctive maximum, which is absent for repulsive forces. The two curves shown are for differentvalues of w / wc, where wc denotes the critical interaction energy for the onset of the phase transitions. As is expected on theoretical grounds since attractive interactions cause cooperative adsorption, the pressure derivative (as well as its maximum value) increases is monotonically with w/wc. The locus of the 8determined by the set of equations
+
+ [(I+ Zg4@&?6+ Zgl(g4 + @,)l(l-
6) = 0 (6)
(6.1)
Two facts are immediately obvious. The maximum in the derivative is observed only above a threshold value of w/wc even for attractive interactions. Repulsive interactions can never give a positive real root for the approximate models like quasi-chemicaland Bragg-William lattice gas and hence no maximum, whereas the exact Ising model can predict (dpld8) = 0. The analysis is carried out for w/wc C 1, i.e. at temperatures higher than the phase transition temperature. The intrusion of surface heterogeneity is illustrated in Figure 2 where derivatives corresponding to three distributions are plotted. As is obvious the positive exponential distribution predicts the highest and the negative expo-
1312 Langmuir, Vol. 7, No. 7,1991
Letters
2Et5r F 9'6c*rl
NEGATIVE
POSITIVE
.
"'
CONSTANT
~/u,.r.o
t
c
- - - ------
IEt4
0.I
e
0.5
002
e
Figure 3. Plot of 0 deldp w 0 with attractive interactions for three distributions. T w o values of interaction parameters are chosen.
nential distribution the lowest, with the constant distribution displaying intermediate values. For numerical values of the limits of heats of adsorption, typical values chosen are Q1= 17 kcal/mol and Q2 = 31 kcal/mol for the lower and upper limits, respectively. Mention may be made here that for computations of the pressure derivative in Figure 1 for the uniform surface, a value of Q = 18 kcal/mol has been used; this corresponds very close to the lower limit Q1 of the heterogeneous surface. The mean pressure derivative values are correspondingly orders of magnitude higher for the heterogeneous surface as the higher energy sites are weighted strongly, especially for the positive exponential distribution. However 19- is much lower compared to the homogeneous surface. Thus to illustrate for the positive exponential distribution, if we compare the homogeneous and heterogeneoussurface, the 19- are 0.49 and 0.116, respectively. With attractive interactions present in the adsorbed phase, surface heterogeneity may be thought to act in opposition in decreasing the pressure derivatives. Nevertheless the attractive interactions are still strong enough to cause the same qualitative behavior as the homogeneous surface, i.e. nonmonotonic behavior in the derivative vs surface coverage plot. In Figure 3 the derivative product is plotted vs 0 for the three distributions. Like in the derivative plots they also display a maximum, but the maxima are considerably broadened in comparison to the derivative. The spread in the plots parallels the numerical values of derivatives, thus the positive exponential distribution predicts the highest derivative and, therefore, the narrowest plot. The opposite is true for the negative exponential with the constant distribution displaying intermediate values. The P are however higher for the DP in comparison to the derivative plots as is understandable. The behavior of the repulsive interactions is very much at variance with that of the attractive forces. Thus the derivative for the homogeneous surface shows monotonic decrease, which is obvious from set of equations (6),as this has no real positive roots for w > 0. If we superimpose surface heterogeneity, the qualitative behavior of the curves is not altered (Figure 4) and still predicts monotonic fall with 8. If we compare the slopes of the falling portion of the derivative plot for attractive case (see Figure 2) with that of Figure 4, it is obvious that the slope of the curves in Figure 4 will be steeper, as surface heterogeneity and repulsive forces act in the same direction to decrease the pressure derivative.
Figure 4. Plot of deldp vs 0 with repulsive interactions for three distributions. 0.15
POSITIVE
NEGATIVE
- - - w/wc 1.0
I
l
e
0'1
_ _OG01
0.07
001
e
005
9
0-35
Figure 5. Plot of 0 dOldp w 0 with repulsive interactions for three distributions. Two values of interaction parameters are
chosen.
At first sight it may appear that since the maximum in d8/dp occurs (Figure 2) at very low surface coverages, especially for the negative and constant distributions, this may not provide a convenient way of distinguishing between attractive and repulsive forces (Figure 4)) as experimentally most of the time we are in the "falling" region of the derivative. We bear in mind, however, that what is plotted on the abscisa is the mean surface coverage. Even though this may be small, the local surface coverage of the high-energy patches is often large. This multiplied by the corresponding value of the energy distribution function will give the contribution of the high-energy sites to the mean surface coverage. While absolute surface coverages are indeed hard to measure, differential increments in surface coverage can probably be measured, employing a pulsed microcalorimeter. A sequence of pulses can give differential heat values and the corresponding small increments can be summed up, to give the mean surface coverage. This can be subjected to experimental errors, therefore pressure derivative study alone may not be a good way to discriminate between attractive and repulsive forces. The qualitative behavior of the DP will be similar for the repulsive forces and attractive interactions as both predict maxima in the DP plots (Figure 5). However few quantitative differences are self-evident. The DP values are always higher for the attractive interactions in comparison to the repulsive ones irrespective of the type of site energy distribution. Also Bmu at which DP shows
Langmuir, Vol. 7, No. 7, 1991 1313
Letters 1.7 E t 5
\
t ' ' ' ' ' ' ' '
7.5Et2
e
0.0 1 0.15
0
0.5
0.0ot
0
04
Figure 6. Effect of the variation of surface heterogeneity parameter Qm on 0 d0/dp vs 0 plot with attractive interactions. maxima gets shifted to appreciably lower values in comparison to the attractive forces. This behavior parallels that of the homogeneous surface and once again we conclude that the subtle difference between attractive and repulsive forces is still not masked by surface heterogeneity. Once again the broadest curves are for the positive exponential distribution and the narrowest for the negative exponential distribution, with the constant distribution predicting values in between. We could now like to make some statements about the surface heterogeneity parameter Q m which makes its appearance in both the positive and negative exponential distributions (see Table I). As is obvious from the functions defining both these distributions the effect of variation of Q m has opposite effects on the values of the site energy distributions. The apparent choice of the Q m parameter and its fine tuning to match experimental value can be done by a careful analysis of the differential heat measurements. Figure 6 shows the trends for the positive and negative exponential distributions with attractive interactions. As the definition implies, constant distribution is devoid of any Qm parameter influence. For a larger value of Q m the DP value is higher for the positive exponential. An exactly opposite trend is manifested by the negative exponential distribution. The difference between the attractive and repulsive forces is once again quantitative and the same trends are shown by repulsive forces, when variation of the surface heterogeneity parameter is considered as is seen in Figure 7. The only distinguishing feature is that for the negative exponential distribution the DP has rising edge and a trailing tail, whereas for the positive exponential distribution the curve has a much more smoother rise and fall. Some statements are in order as to the influence of experimental error on the prediction of derivative and DP for comparison between the various models. The applications of these methods to known data are somewhat restricted due to the absence of fine-grained experimental data. To test for the sensitivity to experimental errors, Gaussian distributed random numbers with known variance and zero mean (which would resemble experimental errors) were added to the "theoretical isotherm" points to generate experimental data. Then the derivative as well as DP were deduced from the generated experimental data by fitting a cubic spline to the experimental data. By
\
"
0.021
0.0001
0.15
0.05
Figure 7. Effect of the variation of surface heterogeneity parameter Qm on 0 d0/dp vs 0 plot with repulsive interactions. 253 E t
3
-
-
$1," 0
-
THEORETICAL
o
0.01
EXPERIMENTAL POINTS
e
0.15
Figure 8. Illustration of influence of experimentalerror on the location of maximum in the 0 de/dp vs 0 plot. numerical differentiation, the derivative and DP plots are generated. The DP can also be generated theoretically. Figure 8 shows the comparison between the theoretical and experimental values for the repulsive interaction model, the site energy distribution being positive exponential distribution. Once again the qualitative trends are fairly obviously seen and are not masked by experimental error. Summary
A careful analysis of the pressure derivatives is shown to lead to the identification of adsorbate interactions, viz. attractive or repulsive, even for a heterogeneous surface. This is basically a method based on experiment, and the presence of a maximum in the pressure derivative plot is a distinguishing feature of attractive interactions. However both types of interactions display a maximum when the product of surface coverage and pressure derivative (derivative product) is plotted against the surface coverage. The location of the maximum on the abscisa gives clear pointers to the nature of adsorbate interactions for the case of the quasi-chemical approximation model of adsorption isotherms.
