Langmuir 1993,9, 2583-2585
2583
Numerical Procedure for Calculating the Gas-Solid Adsorption Energy from Molecular Properties. Some Comments on Surface Irregularities and Adsorption Site Topography Joaqufn Cortbs Facultad de Ciencias Fisicas y Matemlrticas, Universidad de Chile, Casilla 2777, Santiago, Chile Received October 15,1992. I n Final Form: December 3 0 , 1 9 9 9
A numerical procedure that has been developed to evaluate the adsorptionenergy distributionfunction x(Uo)and has been illustrated for the case of relatively well-defined solid surfaces (J.Chem. Phys. 1988, 88, 8081;1989, 91, 1932. Surf. Sci. 1989,218, L461) can be extended to other surfaces simulating, for example,various kinds of irregularities. This method, which proposes calculating x(UO) from the system's molecular parameters without any a priori assumptions on the form of the distributionfunction,is applied to surfaces that have some defecte. Since the calculation of the overall adsorption isotherm for mobile adsorptionrequires both heterogeneityfactors,the energy distributionand the topographyof the adsorption sites, to be specified, the calculation of x(Uo)must be complementedwith considerations relative to the topography, which in the given examples has been referred to as periodically structured, as opposed to the random and patchwise types. This can be thought of as a particular example of what may be called contourtopography,mentioned recently by Rudzinskiand Everett (Adsorptionof Gases on Heterogeneous Surfaces; Academic Press, London, 1992;p 191).
Introduction Two basic aspects of the phenomenon of heterogeneous adsorption are the energy distribution and, if lateral interaction between the adsorbed moleculesis considered, the topography of the superficial sites. In relation to the energy distribution, two different situations should be identified. In the first one, which includes most of the papers published on this subject, the distribution function is chosen a priori and arbitrarily in formssuch as the Gaussian type,t2the Maxwell-Boltzmann type: a log-normaltype: a /35 or y6 function, etc., thereby configuring in some way global parametric solutions of the problem without a satisfactory explanation of the phenomenon at the microscopic level. In the second situation, which is less frequent in the literature, it is attempted to determine the distribution function on the basis of some type of fundamental explanationof the phenomenon. The work of Cerofolini7-9 and RudzinskiS1l must be mentioned, together with a numerical method developedby the author9J2-14that may be extended to other situations, such as those reported in this paper. Abstract published in Advance ACS Abstracts, Auguet 15,1993. (1) Hove, J.; Krumhanel, J. A. Phys. Rev. 1963, 92, 569. (2) Roee, S.; Olivier,J. P. OnPhysicaZ Adsorption; Interscience: New York, 1964. (3) Kindl, B.; Pachovsky, R. A.; Spencer, B. A.; Wojciechowaki, B. W. J. Chem. SOC., Faraday Trans. 1 1973,69,1162. (4) Hoory, S. E.; Prausnitz, J. M. Surf. Sci. 1967,6, 377. (5) My". L.;Ou,D.Y. Annu.Meet.Am.Znst. Chem.Eng. (U.S.A.) 1981. (6) Sircar, 5.J. Chem. Soc., Faraday !franu.1 1984,80, 1101. (7) Cerofolini, G. F. Surf. Sci. 1975,51, 333. (8) Cerofolini, G. F. Colloid Sci. 1983,4, 59. (9) Rudziiki, W.; Everett, D. H. Adsorption of Gases on Heterogeneou Surfaces; Academic Preas: London; 1992. (10) Morel, D.; Stoeckli, H.F.; Rudzinski, W. Surf. Sci. 1982,114,85. (11) Rudzinski,W.; Michaelek,J.; Brun, B.; Partyka,S. J. Chromutogr. 1987,406, 295. (12) CortBe, J. J. Chem. Phys. 1988,88,8081. (13) CortBe, 3. J. Chem. Phye. 1989,91,1932. (14) CortBe, J. Surf. Sci. 1989,218, L461. ~~~~
A Numerical Method for Determining the Adsorption Energy Distribution The distribution function x(UO) is defined if the fraction of the surface having adsorption energies between u" and Uo dUo is x(Uo)d u o . The procedure proposed by the author for obtaining x(Uo)consistssimply in determining it directly from the adsorption energy Uo(x,y)at each point (x,y) located on the x y plane defined on the surface. Uo(x,y)represents the minimum value in the z direction perpendicular to the surface for each site ( i ) . The procedure can be represented mathematically by defining a function F(x,y,w) of the form
+
F(x,y,w) = 1 if 0 I Uo(x,y)I w F(x,y,w) = 0 if Uo(x,y)> w
(1)
A frequency can be calculated in principle by determining the function +(w) by integrating F(x,y,w)over the surface 52 of the solid.
9 W = JJ&x,y,w)
dx dy
(2)
The d9/dw vs w plot would give the required distribution function. The procedure employed to get numerical solutions of eq 2 consists in subdividing the solid's surface into a grillwork of adsorption sites, and calculating for each one the value of Uo(x,y). This calculation can be made semiempiricallyby means of the additivity in pairs approximation, e.g., in the form used by the author to calculate x ( Uo)for the systems of argon adsorbed on K C P and NaC1.I4 In this way a matrix of values of Uo(x,y)that corresponds to the whole surface is obtained, and from them the distribution curve can be evaluated numerically by tabulating the frequency of sites that correspond to each interval of values of the adsorption energy. Here, in order to show some interesting situations, simplified systems are considered that include inert particles whose interactions Uij between the adsorbate
Q743-7463/93/2409-2583$04.QQ~Q 0 1993 American Chemical Society
2584 Langmuir, Vol. 9, No.10,1993
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Figure 2. Adsorption energy distribution function of helium with the (100) face of a xenon lattice with the first layer having various fractions, 8, of argon atoms located randomlyon the xenon atoms, repreaenting surface impurities: (-1 pure solid xenon with no impurities (8 = 0), (- -) xenon completely covered with argn (8 = l),(- -) xenon covered with an argon fraction (8 = s//~e), xenon covered with an argon fraction (8 = '*/a).
5
4
.
3
-
(-e)
i
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Some Applications Figure 1 shows the distribution functions determined by our numerical method using eq 3 to calculate U i j , for the system of helium adsorbed on compact solid xenon,
and for different Xe atom densities in the first superficial layer, as shown schematically in the figure. Because of the symmetry of all these cases, it is easy to choose a sort of superficial unit cell that is representative of the various examples, and make the computational calculation with high precision. In all cases the cell was taken to be a square whose side was equal to the Xe atom diameter, and each cell was divided, in turn, into 1681 superficial points ( i ) to determine x(Uo). For each point and z value, Ujj was calculated considering 512 atoms of the solid, and then incrementing z by 0.25 A outward from the xy plane to find the adsorption energy minimum U". Figure 1 is interestingbecause it showsthe surface energy distribution calculated directly, without a priori assumptions, and with a microscopic view of the various surface structuresconsidered in each example. Figure 2, however, is certainly a better example of what must happen in some real situations of surfaces that have varying proportions of impurities. These cases simulate the adsorption of He over solid Xe having on its surface various fractions of impurities assumed to be Ar atoms randomly distributed on the Xe atoms. The precision achieved in the calculations is certainly somewhat lower than that for the cases of Figure 1,because the representative cell considered in the latter examples had 49 Xe atoms forming a square shape on which the required fraction of Ar atoms was distributed randomly. This cell was then divided into 11 881 superficial points to carry out the calculations, incrementing z by only 0.05 A to determine x ( U")in the same way as for the previous cases. Figure 3 shows, for two interesting examples, the plots of adsorption potential energy against the x coordinate, keeping y constant for the equilibrium values of distance z to the surface. Figure 3b shows what happens at the microscopic level in the fraction of the surface having high adsorption energy values that correspond to holes on the surface, somewhat similarly to the concept of superficial geometric heterogeneity introduced by Benegas et al.9J6 These curves are also basic for studying transitions from
(15) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon: Oxford, 1974.
L647.
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X*
1 Y.
5 4
3
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111
Figure 1. (a) Adsorption energy distributionfunctionof helium with the (100)face of a xenon lattice, and the same but considering a deficiency of xenon atoms in the first layer of the solid (-) (pure xenon) drawing I from (b), (- - -) drawing I1 from (b), (.-) drawing I11 from (b), (- -) drawing IV from (b). (b) Schematic representation of the first layer (z= 0) of the solidscorresponding to the distributions of (a): xenon atom, representative surface cell.
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molecule at site i and each atom j of the solid can be expressed by means of Lennard-Jones' well-known expression15 of the form
where uBpis the lattice parameter, U, is the depth of the energy well, and u@, the distance between the gas atom and a single atom of the solid, is calculated from the diameter u,of the adsorbate,assumingthat the combining rule for atomic distances of closest approach is that of the arithmetic mean. The reduced distance r i j is defined as F i j / u m where Fij is the distance between i and j .
(16)Benegae, E. I.; Pereyra, V. D.; Zgrablich, G. Surf. Sci. 1987,187,
Langmuir, Vol. 9,No. 10, 1993 2585
Gas-Solid Adsorption Energy
called this a periodicallystructured topography1214instead of a homotattic surface, a term sometimes found in the literature.27128 The distribution functions x(Uo)of Figure 1 can be used to calculate the overall adsorption isotherm u(P,T) for mobile adsorption according to the equation
u(P,T) = Je(U"P,T) x(U")dU"
c
J
(
5
6
X*
(b)
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x*
Figure 3. Equilibrium distance z (z* = z/u,) from the surface, and potential energy of adsorption Uo(kcal/mol)when the atom of the gas moves parallel to the surface in the x direction (x* = x/uu) and y is kept constant (r* = y/uu): (a) case I of Figure l a (pure xenon, segment AB). (b) case I11 of Figure l a (segment
AB). localizedto mobile adsorption, e.g., in the classical case in Hill's hindered translation model" which was also used by House and Jaycock,'BJg in which a simple analytical functionis assumed for the bidimensionalpotentialenergy, and also in the determination of the quantum mechanical state of adsorbed particles, from Steele et al.'s first papers,15*20*21 followed by the work of Ricca et and Milford and Novaco22*26*28 on the same type of system.
Some Observations on Surface Topography It is well known that the most widely studied surface topographies correspond to the extreme random and patchwise cases, mostly due to their mathematical simplicity. In the excellent recent book by Rudzinski and Everett? other highly interesting intermediate situations are studied. For example, in what may be called contour topography, the sites of equal energy can be joined to form contours. The examples shown in Figure 1of this paper, which refer to relatively well-defined solid surfaces, can be considered as extreme cases of contour topography in which each contour map corresponds to adsorption sites that are not distributed randomly, but are related to the solid's periodicity. For that reason we have previously ~~
~
(17)Hill, T.L. J. Chem. Phys. 1946,14,441. Faraday Trans 1 (18)House, W.A.; Jaycock, W. J. J. Chem. SOC., 1974,70,1710. (19)House, W.A.; Jaycock, M. J. J. Chem. SOC.,Faraday Trans. 1 1975,71, 1597. (20)Steele, W.A.; Roes, M. J . Chem. Phys. 1961,36,850. (21)Rose, M.;Steele, W. A. J . Chem. Phys. 1961,35,862. (22)Ricca, F.;P h i , C.; Garrone, E. Proceedings of the 2nd International Symposium on Adsorption-DesorptionPhenomena,New York, 1972. _. .-
(23)Ricca, F.;Pisani, C.; Garrone, E. J. Chem. Phys. 1969,51,4079. (24)Ricca, F.;Garrone, E. Tram. Furaday SOC.1970,66959. (25)Novaco, A. D.; Milford, F. J. J. Low Temp. Phys. 1970,3, 307. (26)Milford, F.G.; Novaco, A. D. Phys. Rev. 1971,A4, 1136.
(4)
where 6( UoQ,T)is a "local" isotherm for mobile adsorption. The choice of the 6(UoP,T)must take into account, however, the surface topography. In Figure 2 the situation has been extended to more global cases in which the calculation of the distribution is not restricted to a given adsorption site, so that the use of x( U")has a wider applicability. It is thus possible to understand that our numerical procedure for calculating the distribution may finally be extended to the general situation of any heterogeneous surface and any adsorption mechanism. Approaches to Surface Heterogeneity An approach for studying adsorption on heterogeneous solids, based on a recent idea by Bakaev,has been continued in a series of interesting papers by Bakaev and Steele.3m7 It is assumed that surface heterogeneity is a consequence of the amorphous atomic structure of the first layers of the solid's surface, even when the solid is crystalline. On the basis of that hypothesis, Bakaev and Steele build amorphoussurfaces,e.g., with a dense random packing of hard spheres, and subsequently perform on them mathematical adsorption experiments of the Monte Carlo type, reproducing a number of characteristics observed in experimental systems similar to those whose simulation is attempted. In some of our previous p a p e r ~ , lon ~ -the ~ ~other hand, the heterogeneity obtained for mobile adsorption systems has been explored by calculating the surface energies at each point of the surface. In this paper, and followingthe same approach, we have extended our method to solids that additionallyhave a heterogeneityof the type described by Bakaev, as shown in the example of Figure 2, in which an amorphous layer of impurities formed by Ar atoms has been placed on the Xe surface. Both kinds of heterogeneityare certainly present in gassolid adsorption systems. A combination of both approaches, and their relationships and consequences,which are suggested in a preliminary manner in Figure 2, shall be analyzed in future reports from this laboratory.
Acknowledgment. The author thanks FONDECYT (Grant 92-948) and DTI (Universidad de Chile) for financial support of this work. (27)Sandord, C.; Roes, S. J. Phys. Chem. 1954,58,288. (28)House, W.A. Colloid Sci. 1983,4,1. (29)Bakaev, V. A. Surf. Sci. 1988,198,571. (30)Bakaev, V. A. Dokl. Akad. Nauk SSSR 1984,279,115. (31)Bakaev, V. A. Izv. Akad. Nauk SSSR, Ser. Khim. 1988,1478. (32)Bakaev, V.A.;Dubinin,M. M. Dokl. Akad. Nauk SSSR 1987,296, 369. (33)Bakaev, V.A.; Chelnokova, 0. V. Surf. Sci. 1989,215,521. (34)Bakaev, V. A.; Voit, A. V. Zzv. Akad. Nauk SSSR, Ser. Khim. 1990,2007. (35)Bakaev, V. A.;Steele, W. A. Langmuir 1992,8,148. (36)Bakaev, V. A,; Steele, W. A. Langmuir 1992,8,1372. (37)Bakaev, V. A.; Steele, W. A. Langmuir 1992,8,1379.
