Deviation from Henry's Law: Effects of Energetic Heterogeneity and of

Effects of Adsorbate/Adsorbate Interactions and Surface Fractality on Diffusion- and Reaction-Limited Adsorption. Jordi Dachs and Steven J. Eisenreich...
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Langmuir 1997, 13, 1138-1144

Deviation from Henry’s Law: Effects of Energetic Heterogeneity and of Surface Diffusion† Massimiliano Giona‡,§ and Manuela Giustiniani*,| Centro Interuniversitario sui Sistemi Disordinati e sui Frattali nell’Ingegneria Chimica c/o Dipartimento di Ingegneria Chimica, Universita` di Roma “La Sapienza” Via Eudossiana 18, 00184 Roma, Italy Received November 17, 1995. In Final Form: January 7, 1997X This article focuses on the clustering effects of admolecules driven by surface diffusion in a heterogeneous energy landscape as a mechanism for Freundlich behavior at low pressures. The idealized preferential adsorption model is presented and the phenomenology of this model is justified in terms of surface diffusion (preferential adsorption model, PAM). Mean field analysis and Monte Carlo simulation of PAM are analyzed. The effects of temperature and of patchwise distributions of adsorption energies are discussed.

1. Introduction The complexities of geometrical structure and of surface chemical composition are the main sources of heterogeneity in adsorption phenomena. Crystallographic defects, chemisorbed impurities, dislocations, and superposition of different crystalline planes are generally responsible for surface geometric and energetic heterogeneity on crystalline, mesoporous, and macroporous materials. Experimental results on adsorption equilibria show in many cases a deviation from the linear Henry behavior that can be interpreted by means of the empirical Freundlich isotherm

θt ∼ PR

(1)

where θt is the total coverage, P the pressure, and R ∈ (0,1). The dependence of R on the size of the admolecules and on temperature (see also Rudzinski and Everett (ref 1) for a review) seems to indicate that the Freundlich exponent is the result of both geometric (size-dependent) and energetic effects. Quite recently Keller2 proposed a class of adsorption isotherms for multicomponent mixtures in which the exponent R is related to the fractal dimension of the adsorbent. The Keller model proves to be thermodynamically consistent (in the sense that it satisfies the Maxwell relations) and displays Freundlich-like behavior at low pressures. An extensive analysis of literature data interpreted according to the Keller model3,4 showed a behavior of the Freundlich exponent which is qualitatively in agreement with other experimental results5-8 in terms both of the * To whom correspondence should be addressed. † Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovakia, September 4-10, 1995. ‡ Centro Interuniversitario sui Sistemi Disordinati e sui Frattali nell’Ingegneria Chimica. § Permanent address: Dipartimento di Ingegneria Chimica, Universita´ di Cagliari, piazza d'Armi, 09123 Cagliari, Italy. | Dipartimento di Ingegneria Chimica. X Abstract published in Advance ACS Abstracts, February 15, 1997. (1) Rudzinski, W.; Everett D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (2) Keller, J. U. Physica A 1990, 166, 180. (3) Giona, M.; Giustiniani, M.; Ludlow, D. K. Fractals 1995, 3, 235. (4) Giustiniani, M.; Giona, M.; Ludlow, D. K. Ind. Eng. Chem. Res. 1995, 34, 3856. (5) Ray, G. C.; Box, E. O. Ind. Eng. Chem. 1950, 42, 1315.

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temperature dependence of the Freundlich exponent (which increases with temperature) and of its dependence on the size of the admolecules (the Freundlich exponent increases with the size of the admolecules). The latter result is a consequence in the Keller model of the fractal hypothesis, since the Freundlich exponent Ri for the ith species is related by the Avnir-Pfeifer scaling law9 of the monolayer coverage n∞,i to adparticle size

Ri ∼ n∞,i ∼ ri-D

(2)

where ri is the radius of the admolecules and D the fractal dimension of the adsorbent. In the literature on adsorption, and especially in work specifically oriented toward the properties of mixture for separation purposes,10,11 it is generally accepted that a theoretical model for adsorption equilibria can be considered thermodynamically consistent if it satisfies the Maxwell relations (and ultimately the Gibbs-Duhem equation) and displays linear Henry behavior at low pressure, regardless of whether the adsorbent exhibits a homogeneous or heterogeneous energy structure. The latter condition is derived from the works of Hill12 and Myers and Prausnitz.13 The origin of empirical Freundlich behavior has been extensively analyzed.1 With the exclusion of the statistical mechanical analysis of Nitta and co-workers14,15 on random and patchwise adsorption, the usually accepted derivation of the Freundlich isotherm makes use of condensation approximation in the presence of an exponential distribution of adsorption energies.16,17 To some extent, condensation approximation can be regarded as a mathematical way to derive an energy distribution function responsible for the overall behavior of the empirical isotherms (Freundlich, Dubinin-Radushkevich, Temkin, etc.) under the simplified hypothesis of a local isotherm of step form.1 We lack a phenomenological interpretation of the mathematical simplifications underlying condensation (6) Baker, G. B.; Fox, P. G. Trans. Faraday Soc. 1965, 61, 2001. (7) Koble, R. A.; Corrigan, T. E. Ind. Eng. Chem. 1952, 44, 383. (8) Trapnell, B. M. Proc. R. Soc. London, Ser. A 1951, 206, 39. (9) Avnir, D.; Farin, D.; Pfeifer, P. Nature 1984, 308, 261. (10) Rudisill, E. N.; LeVan, M. D. AIChE J. 1988, 34, 2080. (11) Rudisill, E. N.; LeVan, M. D. Chem. Eng. Sci. 1992, 47, 1239. (12) Hill, T. L. Adv. Catal. 1952, 4, 211. (13) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (14) Nitta, T.; Shigetomi, T.; Kuro-Oka, M.; Katayama, T. J. Chem. Eng. Jpn. 1984, 17, 39. (15) Nitta, T.; Kuro-Oka, M.; Katayama, T. J. Chem. Eng. Jpn. 1984, 17, 45. (16) Cerofolini, G. F. Thin Solid Films 1974, 25, 129. (17) Cerofolini, G.; Re, N. Riv. Nuovo Cimento 1993, 16, 1.

© 1997 American Chemical Society

Effects of Energetic Heterogeneity

approximation in the form of a mechanistic description of the adsorption process and of the nature of the adphase. This has given rise to some criticism as summarized by Adamson:18 “There is no assurance that the derivation of the Freundlich equation is unique; consequently if data fit the equation it is only likely, but not proven, that the surface is heterogeneous. Basically the equation is an empirical one, limited in its usefulness to its ability to fit data.” This work attempts an approach toward a physical justification of Freundlich behavior based on the assumption of a continuous energy distribution with E ∈[0,∞) and on selective adsorption on the most energetic sites driven by surface diffusion. For this reason we call this approach the preferential adsorption model (PAM). This model gives a physical explanation of nonlinear behavior at low pressure as a consequence of the clustering of the adsorbed molecules toward the most energetic sites. The selective choice of the most energetic sites is explained in terms of the hopping of admolecules between nearest neighboring sites and depends on the structure of the energy landscape. The role of surface diffusion in adsorption and catalysis has recently been subjected to intensive investigation19-21 and many models have been proposed (e.g., the dual-site bond model22,23). The preferential adsorption model developed here is based on a simpler nearest-hopping mechanism than that assumed in the dual-site bond model in that only site energies are considered. This article is organized as follows. In the next section we discuss the idealized preferential adsorption model (IPAM), a simplified model for which closed-form results at low pressures can be obtained. We then connect the IPAM to a more realistic version (PAM) in which surface diffusion effects are introduced. Mean-field analysis of the PAM is developed on one-dimensional lattice structures. Monte Carlo simulations are also performed in order to investigate the nature of the adsorbed phase and to support the conjecture that Freundlich behavior finds one explanation in the clustering of the admolecules. The influence of the patchwise topography and of temperature is analyzed. 2. The Idealized Preferential Adsorption Model This section discusses the idealized preferential adsorption model (IPAM). In this model, adparticles can choose the most energetic sites. The selection of the most favorable sites can be physically justified in terms of particle hopping between nearest neighboring sites, i.e., by including the effect of surface diffusion driven by the structure of the adsorption energy field and depending on temperature. These aspects are discussed in the next section. The basic assumptions of the IPAM model are as follows: (1) a continuous distribution of adsorption energies, described by the probability density function (pdf) g(E), E ∈[0,∞); (2) admolecules behave like a Fermi gas at zero temperature. (18) Adamson, A. W. Physical Chemistry of Surfaces; Interscience: New York, 1982. (19) Riccardo, J. L.; Chade, M. A.; Pereyra, V. D.; Zgrablich, G. Langmuir 1992, 8, 1518. (20) Sapag, K.; Bulnes, F.; Riccardo, J. L.; Pereyra, V.; Zgrablich, G. Langmuir 1993, 9, 2670. (21) Kapoor, A.; Yang, R. T.; Wong, C. Catal. Rev.-Sci. Eng. 1989, 31, 129. (22) Mayagoitia, V.; Rojas, F.; Riccardo, J. L.; Pereyra, V. D.; Zgrablich, G. Phys. Rev. B 1990, 41, 7150. (23) Riccardo, J. L.; Pereyra, V.; Zgrablich, G.; Rojas, F.; Mayagoitia, V.; Kornhauser, I. Langmuir 1993, 9, 2730.

Langmuir, Vol. 13, No. 5, 1997 1139

The latter assumption implies that all the admolecules adsorb on the most energetic sites in a deterministic way, i.e., following a completely preferential mechanism. Let us define the probability density function θ(E) in such a way that θ(E) dE is the fraction of adsorption sites with energy between E and E + dE, occupied by admolecules, and is therefore a step function

{

θ(E) )

0 E < E* 1 E g E*

(3)

where E* is defined in terms of the energy distribution function

F(E) )

∫0Eg(E) dE

by means of the relation

(4)

F(E*) ) 1 - θt

It is clear that the threshold energy E* is equivalent to the Fermi level for a Fermi gas at zero temperature. The analogy with condensation approximation is evident. By inversion of eq 4, the explicit relation between the threshold energy E* and the total coverage is obtained. In fact, if the energy pdf is exponential,

g(E))β exp-βE

(5)

with β ) β0/kBT, then

1 log θt β

E*(θt) ) -

(6)

A temperature dependence of the parameter β ) β0/kBT is not a new feature, since it is obtained in the condensation approximation in order to give a physical interpretation of a Freundlich experimental adsorption isotherm (for an extensive analysis see Chapter 4 of ref 1). To obtain the adsorption isotherm, we apply the condition of local equilibrium between adsorption and desorption in terms of a detailed balance for each energy:

kd(E)θ(E) ) ka(E)P(1 - θ(E))

(7)

By integrating with respect to the distribution function g(E) and applying eq 3, we obtain ∞ k (E)g(E) dE ∫0∞ kd(E)g(E)θ(E) dE ) ∫E*(θ ) d t

) KaP - P

∞ k (E)g(E) dE ∫E*(θ ) a t

(8)

where Ka ) ∫0∞ ka(E)g(E) dE. Equation 8 enables us to obtain the adsorption isotherm once ka(E) and kd(E) are specified. Let us consider the case of an exponential distribution of desorption rates

kd(E) ) Kde-γE

(9)

where γ ) γ0/kBT, and of a uniform adsorption rate ka ) Ka (the behavior of ka(E) is irrelevant in the low-pressure limit). For the definition of θ(E), the overall coverage θt is given by

θt )

∫0∞ θ(E)g(E) dE

By inserting eq 6 into eq 8, we obtain

(10)

1140 Langmuir, Vol. 13, No. 5, 1997

θt(γ+β)/β )

[

]

(γ + β) Ka P(1 - θt) β Kd

Giona and Giustiniani

(11)

θt ∼ P

S ) [1 + exp[-(Ei - Ei+1)/kBT] + (12)

with

R)

β γ+β

(13)

Equation 13 is the basic result of the IPAM model in the low-pressure limit. As already mentioned, the IPAM model is strictly analogous to condensation approximation. On the other hand, the well-known theoretical foundation of the Freundlich isotherm proposed by Cerofolini24,25 is based precisely on preferential adsorption arising from a particular structure of the solid surface, the so-called “equilibrium surface”. We shall argue in the next section that this preferential and highly selective behavior can be justified in terms of the surface motion of adparticles. A final comment is related to the influence of the two assumptions on which the IPAM is based. Let us remove condition 2 while maintaining a heterogeneous (nonuniform) energy distribution. In this case, by applying eqs 7 and 10 and making use of the constitutive relations eqs 3 and 9, the total coverage is expressed as

θt ) KP

∫0∞

exp[(γ - β)E] dE 1 + KP exp(γE)

(14)

where K ) Ka/Kd. From eq 14 it follows for KP f 0 (see Appendix) that

{

βγ

(16)

where S is the normalization factor

At low coverage, (1 - θt) = 1, and therefore R

hifi(1 ) S exp[-(Ei - Ei(1)/kBT]

(15)

Therefore, even by removing assumption 2 it is possible to obtain a Freundlich behavior depending on the relative value of β and γ, i.e., for β < γ. However, the result expressed by eq 15 should properly be regarded as a mathematical singularity related to the relative asymptotic behavior for large E of g(E) and kd(E) rather than a physical explanation of Freundlich behavior. On the contrary, in the IPAM, Freundlich scaling at low pressures is observed for every β and γ, under the condition that γ > 0. 3. Mean-Field Analysis and Clustering Effects This section proposes a physical mechanism underlying the selection mechanism of the most favorable sites based on surface diffusion, i.e., on surface migration of the admolecules. A general approach to describe surface diffusion can be given by means of a random-walk model grounded on the hopping of adparticles from one site to its neighboring sites.21,26 Let us consider for simplicity a one-dimensional lattice model. An adparticle adsorbed on a site i with energy Ei can move randomly toward the nearest neighboring sites {i ( 1} with the hopping probabilities (24) Cerofolini, G. F. Surf. Sci. 1975, 51, 333. (25) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (26) Havlin, S.; Ben-Avraham, D. Adv. Phys. 1987, 36, 695.

exp[-(Ei - Ei-1)/kBT]]-1 (17) Adparticle mobility depends on its characteristic rate compared to the characteristic adsorption rate. The ratio of these rates controls diffusion and adsorption. Therefore, it is convenient to introduce an adsorption Thiele modulus φa, the square of which is the ratio between the characteristic diffusion time τd and the characteristic time for adsorption τa, φa2 ) τd/τa. We call φa an adsorption Thiele modulus by analogy with the reaction Thiele modulus introduced in chemical kinetics to describe the mutual and competitive effect of reaction and diffusion.27 Low values of the adsorption Thiele modulus imply that diffusion is faster than adsorption, and therefore adsorption is the rate limiting step. The opposite holds for high values of φa, in which diffusion is rate controlling. The surface-diffusion model considered here could be regarded as a simplified description of the interplay between surface diffusion and a rough energy landscape. In particular, if compared to the dual site-bond model,22,23 in which a distribution of site energies (associated with adsorption) and bond energies (saddle-point energies between nearest neighboring sites) are introduced, the hopping model, eqs 16 and 17, entails exclusive consideration of site energies and neglect of saddle-point effects. In the present analysis, we regard the adsorption Thiele modulus as an independent parameter of the model. According to the hopping model, eqs 16 and 17, adparticles tend to move toward the most favorable energy sites. Local particle motion depends on temperature and on the value of the adsorption Thiele modulus. The presence of surface diffusion as a random siteselection mechanism governed by the probabilistic rules, eqs 16 and 17, constitutes the basic assumption of the PAM. A nonuniform distribution of the adsorption energies, eq 5, is the second requisite. The mean-field equations describing the PAM are therefore given by

dθi ) P(1 - θi) - K-1 exp(-γEi)θi + ri+1,i + ri-1,i dτ ri,i+1 - ri,i-1 (18) where τ is the dimensionless time τ ) tKa, and the dimensionless hopping rates ri,j are given by

ri,j ) φa2hifjθi(1 - θj)

j)i(1

(19)

where hifj are the hopping probabilities defined in eq 16. The equilibrium condition can be obtained from dθi/dτ ) 0 (i ) 1, ..., N), where N is the lattice size. The parameters describing the model are φa2, β, γ, and the dimensionless temperature parameter m ) Emax/kBT, where Emax is the maximum adsorption energy in the simulation of the exponential distribution eq 5. This value of adsorption energy was chosen as a characteristic energy in the simulations to make the temperature dimensionless. In all the simulations, without loss of generality, we assume K ) 1. Another important feature describing the model surface is correlation in the spectral distribution of adsorption energies. Patchwise and random surface (27) Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design; John Wiley & Sons: New York, 1979.

Effects of Energetic Heterogeneity

Langmuir, Vol. 13, No. 5, 1997 1141

Figure 1. θt vs P ˜ for β ) 1, γ ) 0.5, φa2 ) 10-4, and m ) 1150. The number of lattice sites is N ) 105. The dots are the meanfield results. The lines are as follows: a, θt ∼ P; b, θt ∼ PR (with R ) 0.75 ); c, θt ) 1/N, indicating that the crossover between behavior a and behavior b occurs at θt,c ) 1/N.

topographies are usually adopted for the spatial description of energetic heterogeneity in adsorption models. In the usual picture, homogeneous patches with the same energy are regarded as independent adsorption subsystems (homotattic surfaces). A patchwise structure with patches of different average size 〈s〉, spatially distributed at random, such that the spatial distribution of adsorption energies is correlated within each patch and different energy patches are uncorrelated is analyzed in order to study the effect of patch size on the Freundlich exponent. The average patch size 〈s〉 can be defined as the number of sites in each patch. A regular (spatially deterministic) distribution of adsorption energies corresponds to a single patch with 〈s〉 ) N, in which the energy landscape is a smooth regular curve (or surface) as a function of the position. This corresponds to a single non-homotattic patch. Starting from such a picture, we can ideally divide the structure into (N/〈s〉) 〈s〉-sized patches and mix them randomly. The spatial disorder of the structure increases as the patch size 〈s〉 decreases, conserving a spatially ordered but energetically heterogeneous structure within each patch. The case 〈s〉 ) 1 yields a completely random spatial distribution of adsorption energies. Let us first consider the regular case 〈s〉 ) N. Figure 1 shows the behavior of the total coverage with the dimensionless pressure P ˜ (the pressure is made dimensionless as P/Pref, the reference pressure being Pref ) K) for β ) 1, γ ) 0.5 for a very small value of φa2, and a large value of m f ∞. This corresponds to an almost deterministic structure of the random walk, since particles tend to move toward the most energetic site in the neighborhood. As can be observed from this figure, a crossover occurs at θt,c ) 10-5 ) 1/N between a linear Henry behavior for θt < θt,c and a Freundlich scaling for θt > θt,c. The initial linear behavior corresponds to the filling of the first and most energetic site. Correspondingly, this linear scaling can be attributed to finite-size effects due to the finite lattice size N. This observation is supported by the results shown in Figure 2, in which the normalized coverage y ) θtN is plotted against the normalized pressure x)P ˜ /P*, where P* is the pressure corresponding to θt ) 1/N for different lattice sizes N. As can be observed, independently of N, the crossover point is yc ) 1, which implies that

θt,c )

1 N

(20)

Figure 2. Normalized representation of finite-size scaling ˜ /P*, where P* is the pressure effect. y ) θtN vs x ) P corresponding to θt ) 1/N. The simulation conditions are the same as Figure 1. The data are as follows: b, N ) 105; ], N ) 2 × 104; O, N ) 104; 4 N ) 103. The crossover point between linear Henry behavior due to the filling of the most energetic site line (a) and the Freundlich behavior line (b) occurs at (xc,yc) ) (1,1).

Figure 3. Freundlich exponent R vs m ) Emax/kBT (Emax ) 6.9 a.u.) for β ) 1, γ ) 1. Line R ) 0.5 is the prediction of the IPAM.

Therefore, in the limit of large N, the crossover θt,c tends to zero and the linear behavior observed for θt < θt,c is purely a consequence of a finite value of N. Having proved that Freundlich behavior is an intrinsic property of the PAM for arbitrarily small coverages, we shall focus on the other properties of the PAM. The Freundlich exponent obtained from the data of Figures 1 and 2 is R ) 0.75 ( 0.04 and should be compared with the theoretical result of the IPAM eq 13, R ) 2/3 ) 0.666. The percentage error of about 10% between the theoretical IPAM prediction and the PAM value can probably be attributed to finite-size effects in the representation of the exponential distribution of adsorption energies. In the simulations, the maximum energy is Emax = 10.0 a.u., and this truncation in the tail of the exponential distribution may be the cause of this difference. Other simulations performed for different values of β and γ indicate that the deviation of the observed Freundlich exponent obtained from the mean-field equation and the IPAM result (eq 13) is 3-5% on average. It is important to observe that for β ) 1 and γ ) 0.5 the corresponding model without surface diffusion, eq 15, gives rise to linear Henry behavior, eq 15. The effect of temperature on the Freundlich exponent R is shown in Figure 3. Actually m, for fixed Emax, is inversely proportional to temperature. As expected, the Freundlich behavior tends to disappear at high temperature m f 0. This increase in the Freundlich exponent finds confirmation in experimental observations. See for example the data reported in ref 1 by Baker and Fox,6 showing a linear temperature dependence of the exponent

1142 Langmuir, Vol. 13, No. 5, 1997

Figure 4. Effect of the characteristic time for diffusion (i.e., of φa2) on the Freundlich exponent R. β ) 1, γ ) 1, m = 700.

Figure 5. Snapshot of a rough energy landscape (β ) 1) with patch size 〈s〉 ) 100 lattice units.

Figure 6. R vs 〈s〉. Influence of spatial energetic disorder in the energy landscape, β ) 1, γ ) 1.0, m ∼ 700. Line R ) 0.5 is the prediction of the IPAM.

R for xenon and krypton adsorbed on Pyrex and nickel films in ultrahigh vacuum at low temperature 77-90 K and the data of Trapnell,8 of carbon monoxide adsorbed on charcoal. An increasing temperature dependence of R has also been found by Giustiniani et al.4 in the analysis of experimental adsorption data by means of the Keller model. The functional dependence of the Freundlich exponent R on the adsorption Thiele modulus φa is shown in Figure 4. The faster the diffusion, the closer the value of R to the IPAM solution (R ) 1/2 for the data of Figure 4). To complete the analysis, let us consider the effect of patch size. Figure 5 shows a snapshot of an energy landscape with 〈s〉 ) 100. Figure 6 shows the behavior of R with patch size. The reduced correlation induced by smaller patch size increases the value of the Freundlich exponent. This effect is caused by the trapping of admolecules on the local maxima of a rough energy landscape, which induces adparticle localization. Indeed,

Giona and Giustiniani

Figure 7. 〈R2(t)〉 vs t (in lattice time units). Effects of temperature on molecular mobility, 〈s〉 ) 10, β ) 1, N ) 105. The unit time is associated with the unit displacement of a particle toward its nearest neighboring sites.

the effect of energetic disorder on surface mobility is analogous to the intrinsic properties of localization in disordered lattices. All the normal modes and eigenfunctions of a disordered linear chain are localized.28 To conclude, mean-field analysis shows that a Freundlich behavior is observed in the PAM as a consequence of adparticle mobility. Moreover the PAM driven by surface diffusion tends to the IPAM discussed in the previous section in the limit of T f 0, φa f 0 and 〈s〉 f ∞. This is in agreement with assumption 2 of the IPAM. The analysis developed in this section gives a physical explanation of the IPAM, although the conditions for which the PAM converges toward the IPAM are rather extreme. Monte Carlo simulation of the PAM was performed in order to highlight the properties of the adphase and to describe the clustering effect induced by preferential adsorption. First, an exponential distribution of adsorption energies, eq 5, is assigned on the surface. The temperature dependence is included in the β parameter. Correspondingly, the kinetic constant distribution, eq 9, is assumed. The β and γ parameters are fixed. This basically means that adsorption energies are nonuniformly distributed on the surface, at each temperature value. Assuming an exponential distribution for E/kBT and fixing β0 and γ0 is perfectly equivalent. Therefore, the exponent R (see eq 13) in the idealized PAM is obviously independent of temperature, because of hypothesis 2 of the IPAM. As the preferential adparticle motion of the idealized PAM is replaced by the hopping probability, eqs 16 and 17, the resulting R exponent depends on temperature. The Monte Carlo simulation is the standard one:29 (1) a surface site is chosen at random; (2) the adsorption step occurs with probability ka (we assume ka ) 1) if the site is empty; (3) the desorption step occurs with probability kd(Ei); (4) the hopping mechanism occurs with the hopping probabilities hifj between nearest neigboring sites; (5) the number of diffusion steps performed per unit-adsorption step is equal to φa2, according to the definition of the adsorption Thiele modulus; (6) the unit time in Monte Carlo simulation is such that each site is visited once on average. For the same values as the model parameters, meanfield equations and Monte Carlo simulation furnish comparable results. (28) Ziman, J. M. Models of Disorder; Cambridge University Press: Cambridge, 1979. (29) Seri-Levy, A.; Avnir, D. J. Phys. Chem. 1993, 97, 10380.

Effects of Energetic Heterogeneity

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the coverage distribution function θ(E,T) as a function of temperature. Figure 7 shows the mean square displacement 〈R2(t)〉 as a function of the time t of random walkers moving on the surface according to the hopping rates, eq 16, for different values of m ) Emax/kBT in a patchwise energy landscape, with patches of 10 lattice units (l.u.). At low temperatures, the motion is constrained, i.e., 〈R2(t)〉tf∞ ) constant. Conversely, at high temperatures the motion is Brownian, 〈R2(t)〉 ∼ t. This result provides simple confirmation that even in the presence of only excludedvolume interactions, an adphase exhibiting “ordered properties” induced by surface diffusion on an energetically disordered structure can appear. The clustering effects driven by surface diffusion in the proposed model can be interpreted in terms of delocalization/localization properties of surface diffusion as a function of temperature, as well as can be obtained for eigenvalue/eigenfuntion analysis of surface diffusion in the presence of a random distribution of bond strength (hopping diffusivities). The localization effects of adparticles justifies the fact that the heat of adsorption is infinite for P f ∞. This result can be regarded as a phase-transition in the adphase induced by the localization behavior at very low temperatures. The clustering of admolecules toward the most energetic sites macroscopically inducing the deviation from Henry’s law can be better analyzed by means of two-dimensional lattices. Figure 8 shows snapshots of a 80 × 80 two-dimensional lattice subdivided into eight periodic 20 × 20 cells. In each cell the adsorption energies decrease radially, approximately following an exponential distribution. The transition from an ordered structure (Figure 8a), in which adparticles are clustered on the most energetic centers, to a random distribution (Figure 8c) as the temperature increases is clearly evident. 4. Concluding Remarks

Figure 8. Snapshot of the two-dimensional distribution of adparticles in the PAM, β ) 1, γ ) 1: (a) m ) 338.0, θt ) 0.29; (b) m ) 6.7, θt ) 0.25; (c) m ) 0.07, θt ) 0.23.

The ordering field is represented by the difference in site-adsorption energies driving adparticle surface migration. A transition between delocalized and localized adsorption occurs as temperature decreases. In fact, at very low temperatures admolecules are adsorbed on the most energetic sites: the adphase behaves like an ordered solidlike phase. As temperature increases, there appears a mobile, disordered gas-like phase since the hopping rates (eq 16) become practically insensitive to the energy differences between nearest-neighboring sites Ei(1 - Ei. The adphase properties can be further highlighted by studying the mobility properties of the adparticles and

We have presented a model of preferential adsorption driven by surface diffusion giving rise to a Freundlich behavior at low pressure. Closed-form results for the idealized preferential adsorption model have been obtained and the connection with the PAM discussed. We want to stress that the PAM is a reasonable model (although a more realistic description of surface diffusion can be achieved by applying more refined approaches such as the dual-site bond model), i.e., not inconsistent with any fundamental principle, which leads to a Freundlich behavior, to results which are qualitatively in line with experimental observations (the temperature dependence of R), and to extremely interesting statistical mechanical phenomena. The PAM should properly be regarded as a thought experiment like many other models in statistical mechanics (the Ising model for thermal transition is probably the most evident example), which by their simplicity highlight very complex phenomenological (thermodynamic) manifestations. It is interesting to compare the results of the PAM with the hypothesis of Hill12 and Myers and Prausnitz13 (or rather of all the subsequent works originated by these articles) on the universal Henry behavior in adsorption at low pressures. In the analysis of Hill, Henry behavior arises, in the presence of a heterogeneous energy landscape, as a consequence of uncorrelated adsorption on different patches, so that a principle of linear superposition applies

1144 Langmuir, Vol. 13, No. 5, 1997

Giona and Giustiniani

(as in eq 14). This is perfectly consistent with the results obtained for the PAM, since in this case the presence of surface migration induces a strong correlation in adsorption on patches with different energies, and adsorption on different patches cannot therefore be considered as uncorrelated. In the work of Myers and Prausnitz, and in the subsequent analysis stemming from this work, Henry’s law at low pressures is a consequence of gaslike behavior in the adsorbed phase (which justifies the functional form of the chemical potential chosen by these authors). Monte Carlo simulations and mean-field analysis reveal that the Freundlich behavior in the PAM is a manifestation of the presence of clustering effects. The rich phenomenology of the PAM justifies its further investigation. The description of the surface diffusion has been deliberately considered in a simplified way in order to avoid overloading the model with too many parameters and fine details and to keep the original formulation of the model as simple as possible. Of course, the inclusion in the model of more refined description of surface diffusion and of the spatial structure of the energy landscape (e.g., by introducing a detailed characterization of the correlation properties of the energy distribution by means of its correlation function CE(i) ) 〈(Ej+i - 〈E〉)(Ej - 〈E〉)〉/(〈E2〉 〈E〉2) and by considering models including both site and bond energies as in the dual-site bond model) is a subsequent and natural step in the analysis of PAM.

where a(β/γ) is a finite, nonzero quantity and o(P) is a term arriving at zero for P f 0. By substituting this result into eq 24, it follows at very low pressures θ ∼ (KP)β/γ. In the case β ) γ, the integral can be readily calculated and θ ∼ -(KP) log(KP) for KP f 0. The case β/γ > 1 is easily obtained from eq 14, which is the usual way to demonstrate the validity of Henry’s law for a local Langmuir isotherm, i.e.,

θt ) KP

which, for KP f 0, gives a linear behavior directly. By starting from eq 24, the same results can be demonstrated in the following way. For β/γ > 1, the integrand is not summable in the neighborhood of y ) 0. Let  be an arbitrary fixed quantity greater than zero and smaller than 1, and

A() )

A() +

1 2

∫

∞

dy KP (1 + y)yβ/γ

∫KP ydyβ/γ e ∫KP∞ (1 +dyy)yβ/γ e A() + ∫KP ydyβ/γ

(23)

and therefore

1 β - 1 (KP)1-β/γ e 2 γ

(

By substituting y ) KP exp(γE), eq 14 becomes

β θt ) (KP)β/γ γ

∫∞ dy/((1 + y)yβ/γ)

For KP <  it follows that

A1() +

Appendix

exp[-βE] dE

∫0∞ exp[-γE] + KP

)

∫KP∞ (1 +dyy)yβ/γ e A2() +

(γβ - 1) (KP) (21)

a(β/γ) - o(P) (22)

(24)

where A1() ) A() - (β/γ - 1) 1-β/γ/2, A2() ) A() - (β/γ - 1) 1-β/γ. From eqs 26 and 23 it follows that

In the case of β/γ < 1, the integrand is summable in the range [0,∞). Therefore,

∫KP∞ (1 +dyy)yβ/γ ) ∫0∞ (1 +dyy)yβ/γ - ∫0KP (1 +dyy)yβ/γ )

1-β/γ

θt ) a1KP + O(Pβ/γ)

(25)

where a1 is a constant and O(x) is a quantity of order of magnitude of x for x f 0. Equation 15 is therefore proved. LA951045U