Energetic Topography Effects on Mobile Adsorption on

Langmuir 2003, 19, 6737-6743

6737

Energetic Topography Effects on Mobile Adsorption on Heterogeneous Surfaces at Low Coverage M. Nazzarro† and G. Zgrablich*,‡ Laboratorio de Ciencias de Superficies y Medios Porosos, Universidad Nacional de San Luis, San Luis, Argentina, and Departamento de Quı´mica, Universidad Auto´ noma Metropolitana, Iztapalapa, P.O. Box 55-534, 09340 Me´ xico D.F., Me´ xico Received February 10, 2003. In Final Form: May 28, 2003 Mobile adsorption on heterogeneous surfaces at low coverage is modeled via a gas-solid virial expansion, following the ideas of the generalized Gaussian model6,12 to incorporate energetic topography effects. Adsorbate molecules interact among them via Lennard-Jones interactions. Model predictions are compared to Monte Carlo simulations of adsorption on heterogeneous solids obtained by doping a pure crystalline solid with different concentrations of impurities. Energetic topography effects are shown to be important and they are correctly predicted by the model at low coverage.

1. Introduction The role of the adsorptive surface characteristics in many processes of practical importance is a topic of increasing interest in surface science. Adsorption, surface diffusion, and reactions on catalysts are some of the phenomena which are strongly dependent upon surface structure. Most materials have heterogeneous surfaces which, when interacting with gas molecules, present a complex spatial dependence of the adsorptive energy. It is of substantial interest to attempt a complete characterization of such heterogeneity. Through the last 50 years physical adsorption has been used for determining energetic properties of heterogeneous substrates, but this still remains an open problem in many aspects.1-4 For a very long time in the history of the studies of heterogeneous adsorbents, the adsorptive energy distribution (AED) was considered as the only important characteristic to be known in order to describe the behavior of adsorbed particles, and much effort was dedicated to its determination by inverting the integral equation5

θ h (T,µ) )

∫θ(T,µ,)f() d

(1)

where θ h is the mean total coverage at temperature T and chemical potential µ, θ is the local coverage (usually called the local isotherm) corresponding to an adsorptive energy , and f() is the AED. It should be noticed that eq 1 is strictly and generally valid only for noninteracting particles, which is a quite unrealistic case. If adsorbed particles interact with each other, then the local coverage at a point with a given * Corresponding author, on Sabbatical leave from Universidad Nacional de San Luis, Argentina. E-mail: giorgiozgrablich942@ hotmail.com. † Universidad Nacional de San Luis. ‡ Universidad Auto ´ noma Metropolitana. (1) Steele, W. A. The interaction of Gases with Solid Surfaces; Pergamon: New York, 1974. (2) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Surfaces; Elsevier: Amsterdam, 1988. (3) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (4) Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997. (5) Jaroniec, M.; Bra¨uer, P. Surf. Sci. Rep. 1986, 6, 65.

adsorptive energy depends on the local coverage on neighbor points with different adsorptive energies and, in general, eq 1 should be replaced by a much more complex one, like6

θ h (T,µ) )

∫ ... ∫θ(T,µ,1,...,M)fM(1,...,M) d1 ... dM (2)

where now θ depends not only on the adsorptive energy at a single point on the surface but also on the adsorptive energy at (in general) M neighbor points, and fM(1,...,M) is a multivariate probability distribution which specifies how adsorptive energies are spatially distributed, or in other words, the energetic topography of the surface. We remark that, even for interacting particles, eq 2 reduces to eq 1 for two extreme topographies: (a) random sites topography (RST), where adsorptive energies are distributed totally at random among adsorbing sites, and (b) large patches topography (LPT), where the surface is assumed to be a collection of homogeneous patches large enough to neglect border effects between neighbor patches with different adsorption energies. Of course, the local adsorption isotherm will be different for these two extreme topographies. It is by now clear that RST and LPT are particular limiting cases (occurring only rarely in real systems) of heterogeneous surfaces with more general topographies and that the topography affects strongly many molecular processes occurring on such surfaces, like adsorption, surface diffusion, and reactions,6-12 thus making the simple determination of the AED not enough to characterize the heterogeneity. It is then necessary to obtain the (6) Riccardo, J. L.; Chade, M. A.; Pereyra, V. D.; Zgrablich, G. Langmuir 1992, 8, 1518. (7) Zgrablich, G.; Mayagoitia, V.; Rojas, F.; Bulnes, F.; Gonza´lez, A. P.; Nazarro, M.; Pereyra, V.; Ramı´rez-Pastor, A. J.; Riccardo, J. L.; Sapag, K. Langmuir 1996, 12, 129. (8) Zgrablich, G.; Zuppa, C.; Ciacera, M.; Riccardo, J. L.; Steele, W. A. Surf. Sci. 1996, 356, 257. (9) Bulnes, F.; Nieto, F.; Pereyra, V.; Zgrablich, G.; Uebing, C. Langmuir 1999, 15, 5990. (10) Bulnes, F.; Pereyra, V.; Riccardo, J. L.; Zgrablich, G. J. Chem. Phys. 1999, 111, 1. (11) Bulnes, F.; Ramı´rez-Pastor, A. J.; Zgrablich, G. Phys. Rev. E 2002, 65, 31603. (12) Ripa, P.; Zgrablich, G. J. Phys. Chem. 1975, 79, 2118.

10.1021/la034227u CCC: $25.00 © 2003 American Chemical Society Published on Web 07/09/2003

6738

Langmuir, Vol. 19, No. 17, 2003

Nazzarro and Zgrablich

multivariate probability distribution, or at least the AED plus the spatial correlation function. At this point we can precisely see the difficulties involved in the characterization of a general heterogeneous surface. As is well-known, eq 1 for the simple cases of the two extreme topographies is an ill-posed problem for the determination of the AED f() due to the form of the kernel of the integral equation determined by the local isotherm. The determination of the AED from experimental adsorption isotherms requires elaborate computational methods, which have been developed with much effort in many years.5 When treating with more general topographies, eq 2 must be used, where the local isotherm is a much more complex equation (if available at all), and we must deal with a multiple integral on the energy, and the unknown quantity to be calculated is the multivariate adsorptive energy distribution. Even in the simplest case in which the topography could be described by a two-point correlation function, the problem cannot be solved by inverting the multidimensional integral equation. It is then of great importance to develop simple models capable of describing the energetic topography on the basis of a few parameters and to study the effects of these parameters on several surface processes, with the hope that, in such a process, methods to obtain the relevant parameters from experimental data will be envisaged. These models can be of two kinds: lattice-gas models or continuum models. The former are more suited to localized adsorption (for example, chemisorption), while the latter are more suited to mobile adsorption, generally physisorption, and then more closely related to the surface energetic characterization problem. In the present work we review the generalized Gaussian model, a continuum model based on a bivariate energy distribution with spatial correlations, extending it to deal with particles interacting through Lennard-Jones potential, and compare its predictions to Monte Carlo simulations of mobile adsorption on solids with well-controlled heterogeneity. In section 2 the basic concept of the adsorptive energy surface is introduced on the basis of a simple example and the characteristics determining the topography are discussed. In section 3 the generalized Gaussian model is reviewed and extended to deal with Lennard-Jones interacting particles. A Monte Carlo simulation method to obtain adsorption isotherms for solids with well-characterized heterogeneity is then developed in section 4. Results from simulations and from the model are presented and compared in section 5, and finally, conclusions are given in section 6. 2. The Adsorptive Energy Surface (AES) To base our analysis on a well-defined simple system, let us consider an heterogeneous solid consisting of a regular crystal of atoms A (for example a hexagonal close pack crystal) where a small fraction is substituted by impurity atoms B. We move a probe atom P on the (X,Y) surface of the crystal; the probe interacts with atoms A and B with a Lennard-Jones potential

[( ) ( ) ]

UPS(r) ) -4PS

σPS r

6

σPS r

function of Z by summing up all pairwise interactions with the substrate’s atoms within a cutoff distance rc ) 4σPS

E(X,Y,Z) )

∑ UPS(rij)

(4)

rijerc

Then, by finding the minimum in the coordinate Z, we obtain the equilibrium height Z0 and the adsorptive energy at position (X,Y) on the surface. What we get in this way is the AES seen by the probe atom, defined as E(X,Y,Z0) ) minZ{E(X,Y,Z)}. Figure 1 shows this energy surface for a crystal with 20% of impurity atoms with PA/kB ) 160 K, PB/kB ) 320 K, and σPS ) 0.35 nm; darker regions represent stronger adsorptive energy, while brighter ones correspond to weaker adsorptive energy. Significant correlation is seen to be present, in the sense that strong adsorptive regions appear to be quite larger than one lattice size despite the low density of impurity atoms, reflecting the fact that the probe atom interacts with many atoms of the substrate at once. As a first rough approximation the energy surface could be considered as a collection of irregular patches of different strengths. However the energetic topography shows a quite greater complexity and such a picture could lead to an oversimplified model not reflecting important behaviors in molecular processes occurring on the surface. The cuts on the borders of the sample give the adsorptive energy profiles along X and Y directions, reinforcing the idea of a high complexity. The problem is: how to model in a simple, and still realistic way, such a complex behavior? In other words, which are the characteristic (and relevant) quantities necessary to construct simple models capable of reproducing in a statistical sense the main topographic features? In a very general way, we can say that the AES is mathematically described by a stochastic process,13,14 i.e., a random function depending on some parameter. In our case, the adsorptive energy is a random function of the position on the surface, E ˆ (R B ), where the circumflex indicates a random quantity and R B is the position vector on the surface whose components are (X,Y). A particular realization of the stochastic process E ˆ (R B ) is the function E(X,Y) represented in Figure 1 (we can drop the dependence with Z0). The statistical description of such a stochastic process could be very complex. However, some simplifying assumptions, based on physical grounds, may reduce greatly this complexity. In fact, it is reasonable to assume that the surface is statistically homogeneous, i.e., any macroscopic portion of the surface has all the meaningful information, and that the adsorptive energy distribution can be approximately described by a multivariate Gaussian distribution depending on the distance between pairs of points on the surface. This approach leads to the generalized Gaussian model (GGM),6,12 which is capable of describing the energetic topography on the basis of the mean and the dispersion of the adsorptive energy and a correlation function depending on the distance on the surface. 3. Generalized Gaussian Model (GGM)

12

(3)

where S stands for the substrate atom, A or B, and and σ are the usual energy depth and particle-diameter parameters, respectively. At each point i ) (X,Y) the total interaction energy of the probe atom is calculated as a

The GGM was introduced and developed in refs 6 and 12 for particles interacting through a square well potential. Here we extend it for Lennard-Jones interacting particles. (13) Feller, W. An Introduction to Probability Theory and its Applications, 2nd ed.; Wiley: New York, 1971; Vol. II. (14) Gardiner, C. W. Handbook of Stochastic Methods, 2nd ed.; Springer: Berlin, 1985.

Mobile Adsorption on Heterogeneous Surfaces

Langmuir, Vol. 19, No. 17, 2003 6739

Figure 1. Adsorption energy surface (AES) for a crystal of atoms A with 20% impurity of atoms B.

We start by assuming the validity of the statistical homogeneity hypothesis

〈E ˆ (R B 1+a b,...,R B n+a b)〉 ) 〈E ˆ (R B 1,...,R B n)〉

(5)

where 〈...〉 denotes average over an ensemble of many realizations of the surface. The most general statistical information for a continuous stochastic process is given by its generating functional. If we assume that the AES is a Gaussian stochastic process, then its generating functional is given by13,14

∫

B R(R B )[E ˆ (R B) - E h ]}〉 ) F(R) ≡ 〈exp{ d2R 1 exp 2

[

∫∫d RB d RB ′R(RB )H(RB ,RB ′)R(RB ′)] 2

2

(6)

n

δ[E ˆ (R B i) - Ei]〉 ) ∏ i)1

[

[(2π)n detH]-1 exp -

1

n

]

∑ (Ei - Eh )(H-1)i,j(Ej - Eh )

2 i,j)1

(7)

where, by virtue of the condition (5), the covariance matrix

Hi,j ) 〈(Ei - E h )(Ej - E h )〉 ) Ω2C(R Bi - R B j)

[ ( )]

C(r) ) exp -

where H(R B ,R B ′) ) 〈[E ˆ (R B) - E h ][E ˆ (R h ′) - E h ]〉 is the covariance function and E h ) 〈E ˆ (R B )〉. From the generating functional, the multivariate probability density distribution for the adsorptive energies at n points on the surface is obtained as

Φn(E1,...,En) ≡ 〈

statistically isotropic, C is only a function of the distance r between two points. In this model the mean value of any macroscopic quantity of interest depending on the AES could then be evaluated by knowing E h , Ω, and C(r). The correlation function C(r) carries all the useful information about the energetic topography and should, in principle, be determined from the geometric and chemical structure of the adsorbent (even though the methodology to achieve this has not been developed so far). However, we could simplify the model even more by proposing for C(r) a simple Gaussian decay like

(8)

is a function of the relative position vector between two points. Here Ω is the adsorptive energy dispersion and C the correlation function. If furthermore the surface is

1 r 2 r0

2

(9)

where r0 is a correlation length. This expression, which we do not intend to take as a realistic correlation function valid for any surface, simply tells us that the spatial correlation between adsorptive energies at points separated by a distance r < r0 is very high (close to 1) while for r > r0 is very low (close to 0). Thus the present model becomes very attractive in the sense that the energetic topography is characterized by a single parameter, the correlation length, and this opens the possibility for the determination of the three simple parameters of the model (E h , Ω, and r0) by, for example, fitting experimental adsorption isotherms. It is worth to remark that the present model is a continuous one and not a lattice model of adsorption sites. This is an appealing feature, since, as we can see from Figure 1, adsorption sites hardly form a regular lattice and furthermore many of them are so shallow that an adsorbed particle will most probably be quite mobile on appreciably large regions. To obtain a manageable equation for the adsorption isotherm in this model, without losing the generality of a continuous model (in the near future we will develop a lattice formulation of the GGM), we make use of a virial

6740



expansion for the two-dimensional spreading pressure φ of the adsorbed phase15

φ ) kBTF[1 +

∑ ng2

Bn(T)Fn-1]

(10)

p ) K(T)F exp

[∑

ng2

]

B 12) ) F(R B 13 - R

g(B) t )-

(12)

and that the stochastic process E ˆ (R B ) has the distribution given by eq 7, then the coefficients in eq 11 are obtained as12

[

Bn(T) ) -

1 n(n - 2)!

( )]

1 Ω E h kBT 2 kBT

2

R B n)S′1,...,n

∏

[( )

exp

i>j)1

Ω

kBT

]

2

C(R Bi - R B j) (14)

t B ) ∫0∞ dRB F(RB ) eiB‚R 2π 1 ∞ dr F(r)r ∫0 dϑ e-itr cosϑ ) 2π ∫0 1 ∞ dr F(r)r J0(tr) 2π ∫0

B3(T) ) -

2π 3

∫∫∫ dBt 1 dBt 2 dBt 3 g(Bt 1)g(Bt 2)g(Bt 3) ) 2π dBt g3(B) t (20) 3 ∫

For the Lennard-Jones potential, eq 15, and introducing the notation E h ) -kBTa and Ω ) kBTs, we now have

F(r) ) F1(r) - F2(r)

Bi - R B j|)/kBT] - 1 S′1,2 ) f12 ) exp[-Ugg(|R S′1,2,3 ) f12f13f23

{ [( ) ( ) ]} [( )

]

[( )

(23)

12

6

[( ) ( ) ]

Ugg(r) ) 4kBTgg

σ r

12

(15)

where σ is the particle diameter and kBTgg is the depth of the potential. We briefly outline the procedure for the calculation of B ij ) R Bj - R B i and the B3(T). We define relative coordinates R function

Bi - R B j)/kBT) - 1] F(R B ij) ) fij[exp(-Ugg(R

(16)

(15) Hill, T. L. Statistical Mechanics; McGraw-Hill: New York, 1956.

σ r

-

F2(r) ) exp

and so on. It is clear that the calculation of gas-solid virial coefficients is very difficult, so that only the first few of them could be evaluated. This means that the model will be useful only at low values of the adsorbed phase density. But on the other hand, the most important effects of heterogeneity can be seen for the low-pressure part of the adsorption isotherm. To study how the first few virial coefficients depend on the energetic topography, we assume an interparticle interaction given by a Lennard-Jones potential

(21)

where

Tgg σ T r

S′1,2,3,4 ) f12f13f14f23f24f34 + 6f12f13f14f23f34 + 3f12f23f34f14

(19)

where J0 is the zeroth order Bessel function. With this, eq 17 becomes

F1(r) ) exp 4

where

σ r

t exp[iB‚(R t B 13 - R B 12)] ∫ dBt g(B)

1 2π

(13)

∫ ... ∫ dRB 1 ... dRB nδ(RB 1 + ... + n

1 2π

(18)

(11)

1 k (Z - Z0)2 + E ˆ (R B) 2 z

K(T) ) (kBTkz/2π)1/2 exp

∫ ∫ dRB 12 dRB 13F(RB 12)F(RB 13)F(RB 13 - RB 12)

Now we put

where K(T) is an integration constant. By assuming that the potential energy of the system of adsorbed particles Bi - R B j|), and is the sum of the interparticle potential Ugg(|R the gas-solid potential1

B ,Z) ) Ugs(R

1 3

where

n-1

Bn(T)F n-1

B3(T) ) -

(17)

where F is the adsorbate surface density and Bn(T) is the nth two-dimensional virial coefficient. If the adsorbed phase is in equilibrium with an ideal gas phase whose density is F0 and whose pressure is p, then making use of Gibbs equation F0 dφ ) F dp, the adsorption isotherm equation is given by

n

Then, from eq 14 we have

6

exp

Ts 2 -1/2(r/r0)2 e T (22)

Ts 2 -1/2(r/r0)2 e T

]

Therefore, by using a cutoff for the Lennard-Jones potential for r > b, we obtain

B3(T) ) -

(2π)2 [ 3

∫0∞ tg13 dt - 3 ∫0∞ tg12g2 dt + ∞ ∞ 3 ∫0 tg1g22 dt - ∫0 tg23 dt]

(24)

where now gi(t) ) ∫0b dr Fi(r)rJ0(tr). In a similar way, we have for B2(T)

{ [ ( ) ( ( ) )] [ ( ) ]}

B2(T) ) Ψ + r02π Ei -

Ts T

2

exp -

1 b 2 r0

Ei -

2

-

Ts T

2

(25)

where Ψ ) π ∫0b rF1(r) dr and Ei is the exponential-integral function. The integrals involved in B2 and B3 can be evaluated numerically. B4 could also be evaluated numerically within reasonably large computer time, but it would not be worth the much greater effort, since already at very low adsorbed phase density topography effects could be appreciated.


Langmuir, Vol. 19, No. 17, 2003 6741

Figure 3. Adsorption isotherms calculated from the GGM for Lennard-Jones interacting particles, for T/Tgg ) 2.0 and different values of the correlation length.

Figure 2. Normalized gas-solid virial coefficients for a Lennard-Jones potential, as a function of the reduced temperature T/Tgg, for different values of the correlation length r0 and for a given value of the standard deviation of the adsorptive potential kBTs.

Adimensional virial coefficients can be defined as Bn* ) Bn/(π σ2/2)n. The second and third coefficients are shown in Figure 2 as a function of T/Tgg for Ts/Tgg ) 2.0 (which represent a reasonably high heterogeneity with respect to interparticle interactions) and different values of r0. As can be seen, sensitivity of the virial coefficients with respect to the correlation length r0 is very high at low temperature and still appreciable even at relatively high temperature. As Ts/Tgg decreases (figures not shown here), the effect of topography becomes weaker and practically disappears for Ts/Tgg < 0.5. It is interesting to analyze the adsorption process to understand the peculiar behavior of B2 at low temperature. For r0 ) 0 (completely random topography) and r0 f ∞ (macroscopic homogeneous patches) the relative positions of adsorbed particles are not dictated by the adsorption energy topography but rather by the interparticle potential, with prevalence of the attractive region, then making the integrand in eq 14 preferentially positive, and therefore B2 f -∞ as T f 0. For 0 < r0 < 2σ, on the contrary, adsorbed particles are forced by the adsorptive energy topography to be close enough so that the repulsive part of the interparticle potential makes the prevailing contribution to the integrand in eq 14, and B2 f +∞ as T f 0. As it can be easily understood, the virial coefficients for r0 greater than a few particle diameters will behave approximately as for r0 ) ∞. Once the virial coefficients have been evaluated, the adsorption isotherm for low pressure is obtained through

[

p ) K(T)F exp 2B2(T)F +

3 B (T)F2 2 3

]

(26)

1 K(T) ) (kBTkz/2π)1/2 exp -Ta/T - (Ts/T)2 2

[

]

(27)

The constant K(T), known as Henry’s constant, repre-

senting the slope of the adsorption isotherm at very low pressure, depends not only on the mean adsorptive energy, E h ) -kBTa, as classically believed,3 but also on the adsorptive energy dispersion Ω ) kBTs. Adsorption isotherms calculated from the above equations for Ts/Tgg ) 2, T/Tgg ) 2 and different values of the correlation length r0 are shown in Figure 3. The effect of the correlation length can be clearly appreciated as a considerable decrease in adsorption density as r0 increases. Theoretical adsorption isotherms could be fitted to experimental ones obtaining the parameters K(T), Ts, and r0, characterizing the adsorptive energy surface for a given real gas-solid system. In what follows, however, we point to a quite stronger test of the GGM; namely, we produce artificial (computer made) heterogeneous adsorbents with well-controlled energetic topography, determine the AED and the correlation function corresponding to the gassolid system, then simulate the adsorption process in the continuum, and finally compare the observed behavior with the predictions (not data fitting) of the GGM. 4. Simulations on Ideal Heterogeneous Systems A collection of solids is prepared as explained in section 2, corresponding to different concentrations of impurity atoms, and their AESs are generated. We can then study the statistical properties of these AESs, like the AED and the spatial correlation function C(r). These statistical properties for a set of ideally prepared heterogeneous solids are shown in Figures 4 and 5. As the concentration of impurity atoms increases, the mean value of the adsorption energy distribution (Figure 4) shifts toward lower energy values (stronger adsorption) and its dispersion also increases. At the same time, the spatial correlation function (Figure 5) presents an attenuated oscillatory behavior, decaying slower for higher concentrations of impurity atoms. Once the ideal heterogeneous solids are prepared, the adsorption process is simulated through a continuum space Monte Carlo method in the grand-canonical ensemble.16,17 The simulation method can be briefly outlined as follows. (a) Values for the pressure, p, and temperature, T, are fixed. (16) Binder, K. Monte Carlo Methods in Statistical Physics; SpringerVerlag: Berlin, 1986. (17) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982.

6742



(c1) Particle Displacement. A particle is chosen at random and a change in its position by a displacement vector B δ is attempted. The modulus of the displacement vector is fixed, but its direction is randomly chosen. The energy of the final state of the system (if the displacement were accepted), U(SfN), is calculated and the transition is accepted with probability

{ [(

W(SiNfSfN) ) min 1, exp -

)]}

U(SfN) - U(SiN) kBT

(c2) Particle Adsorption. A position on the surface is chosen at random and the adsorption of a new particle at that position is attempted. The transition is accepted with probability

W(SiNfSfN+1) )

{

[(

)]}

U(SfN+1) - U(SiN) pA exp min 1, kBT kBT(N + 1)

(c3) Particle Desorption. An adsorbed particle is randomly chosen and its desorption is attempted. The transition is accepted with probability

W(SiNfSfN-1) )

{

[(

)]}

kBTN U(SfN-1) - U(SiN) exp min 1, pA kBT

Figure 4. Adsorptive energy distributions for ideal heterogeneous solids with different concentrations of impurity atoms.

where A is the area of the solid surface sample in the simulation. Step c is repeated until thermodynamical equilibrium is reached, and then further Monte Carlo steps are executed to obtain the mean value of adsorbed particle density. By changing the value of p, the adsorption isotherm can be obtained. 5. Comparison Test for the GGM

Figure 5. Spatial correlation functions for ideal heterogeneous solids with different concentrations of impurity atoms.

(b) An arbitrary initial state with N adsorbed particles, SiN, is established (for example, by adsorbing N particles at randomly chosen positions on the solid surface) and its energy is calculated as N

U(SiN) )

1

Ugg(R B j,R B k) ∑ E(RB k) + 2 ∑ k)1 j,k

(c) One of the following three processes is randomly chosen with equal probabilities:

We now compare the predictions of the GGM with the behavior observed through simulations for the ideal heterogeneous systems. As we can see from Figure 4, the adsorptive energy distribution could be qualitatively described by a Gaussian distribution, as assumed by the GGM, whose dispersion increases as the concentration of impurity atoms increases. The case corresponding to 0% concentration of impurity atoms is the less favorable, but it is also true that a distortion of the AED in the high-energy region (weak adsorption energy) is not important for adsorption at low pressure, where the deeper adsorptive energy regions are preferentially occupied by adsorbed particles. It is to be expected that more general heterogeneous solids, where heterogeneity could be due not only to impurity atoms but also to a number of defects, or even to the presence of amorphous structures, the AED would be even more similar to a Gaussian distribution. On the other hand, for the spatial correlation function, the Gaussian decay assumed by the GGM is also qualitatively acceptable, as can be seen from Figure 6, where the solid symbols represent the Gaussian decay for different correlation length and the open symbols represent the spatial correlation function obtained from the AESs for different concentrations of impurity atoms. In fact, even if the “real” correlation function presents the oscillatory structure induced by the periodic character of the solid lattice, these oscillations are not relevant to the adsorption of molecules, whose size is usually larger than


Langmuir, Vol. 19, No. 17, 2003 6743

the parameters used in Table 1, and compare adsorption isotherms obtained by the GGM with simulated isotherms for those samples. This comparison is shown in Figure 7, where solid symbols represent simulated isotherms, while full curves represent GGM predictions. As we already mentioned, the comparison can only have significance at low pressure, given that we only use the virial expansion up to the third coefficient. In this region, and considering that this is not the result of a parameter fitting procedure, we may say that the predictions of the model are satisfactory.

Figure 6. Comparison between “real” spatial correlation functions and those assumed by the GGM.

Figure 7. Comparison between adsorption isotherms simulated on ideal heterogeneous solids (black symbols) and those predicted by the GGM (full lines), for three different samples. Table 1. Parameter Values Used in the GGM To Obtain the Adsorption Isotherms of Figure 7 concn of impurities

r0

kBTa

kBTs

Ts/Tgg

1% 7% 30%

2.0 2.5 3.5

2.76 2.85 3.49

0.10 0.16 0.32

0.20 0.32 0.64

that of the solid lattice spacing. What is important is the attenuation of the oscillations. Visual inspection of Figure 1 suggests the importance of the size of dark and bright regions, rather than the small grains within these regions. The important fact then is that the GGM provides a simple correlation function, which takes into account such a decay with only one parameter, the correlation length r0. We now choose more or less appropriate (by visual comparison) AED and correlation length values for different samples of heterogeneous solids, as indicated by

6. Conclusions We have addressed the mobile adsorption (i.e., more suited to physical adsorption) of gases on heterogeneous surfaces at low pressure. We have stressed the importance of the adsorptive energy topography, which can be taken into account by a theoretical model like the GGM, and we have extended such a model by calculating the second and third gas-solid virial coefficients for particles interacting through a Lennard-Jones potential. The GGM turns out to be quite an attractive model due to its simplicity; in fact in this model the adsorptive energy surface is statistically described by only three parameters: the mean value of the adsorptive energy distribution, kBTa, and its dispersion, kBTs, and the correlation length, r0. This last parameter is the most relevant one describing the topography. The gas-solid virial coefficients were shown to depend strongly on the topography and, consequently, so does the adsorption isotherm at low pressure. The only way to test the validity of such a model is to compare its predictions with the behavior of a system whose adsorptive energy surface properties are wellspecified, and this is the case when adsorption is simulated on ideally constructed heterogeneous solids, as done here. The test turned out to be satisfactory for adsorption at low pressure. From the above, we may say that the present form of the GGM can be used to fit experimental adsorption isotherms of physically adsorbed gases on heterogeneous solids at low pressure, obtaining in this way the parameters characterizing the heterogeneity. We may expect that the model would work better with substrates presenting a rough AES, due either to chemical impurities or to roughness in the physical surface. Finally, since virial coefficients are found to be more sensitive to the correlation length at lower temperature, the appropriate adsorbates should be selected in such a way as to obtain experimental low-density adsorption isotherms at the lowest possible temperatures to ensure good sensitivity in the fitting parameters. Acknowledgment. We gratefully acknowledge financial support from CONICET of Argentina and CONACYT of Mexico, which made possible the development of the present research. One of us (G.Z.) gratefully acknowledges a Fellowship from the John Simon Guggenheim Memorial Foundation. LA034227U

Energetic Topography Effects on Mobile Adsorption on

Recommend Documents