Modeling of Binary Adsorption on Heterogeneous Surfaces

The integral equation (IE) approach coupled with a quasi-Gaussian adsorption energy distribution is used to model the adsorption of single gases and t...
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Langmuir 2005, 21, 7335-7341

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Modeling of Binary Adsorption on Heterogeneous Surfaces Characterized by a Quasi-Gaussian Adsorption Energy Distribution Krzysztof Nieszporek,* Paweł Szabelski, and Mateusz Drach Department of Theoretical Chemistry, Maria Curie-Skłodowska University, Pl. M. C. Skłodowskiej 3, 20-031 Lublin, Poland Received January 24, 2005. In Final Form: June 7, 2005 The integral equation (IE) approach coupled with a quasi-Gaussian adsorption energy distribution is used to model the adsorption of single gases and their binary mixture on a heterogeneous solid surface. The adsorbing surface is assumed to be characterized by two, generally different in width, quasi-Gaussian distribution functions, each of them related to a single component of the mixture. The influence of correlations between the distribution functions associated with different components on the corresponding adsorption isotherms and phase diagrams is discussed. In particular, it is demonstrated that a lack of microscopic correlations between the adsorption energies of the components may lead to the formation of an azeotropic mixture. The predictions of the theory are also compared with the results of the grand canonical Monte Carlo (GCMC) simulations carried out for the system studied.

1. Introduction The surfaces of adsorbents which are commonly used in processes aimed at separation and purification of gases or liquids are often highly disordered.1-3 This effect, caused by, for example, structural defects or impurities built into the surface, is primarily responsible for the observed energetic heterogeneity of these materials. In particular, the structural defects and chemical impurities lead to the formation of a complicated potential field with several minima in the fluid-solid interaction potential. In consequence, interaction of a single molecule with the surface becomes largely diversified, depending on the local environment of the adsorbed molecule (impurities, steps, kinks, etc.). Obviously, the resulting energetic heterogeneity of the surface may largely affect the adsorption process in both single and multicomponent modes. Also, theoretical description of heterogeneous adsorption becomes much more difficult compared to that related to entirely homogeneous surfaces. In this case, to predict accurately the adsorption isotherms or isosteric heats of adsorption in systems in which the heterogeneity effects play a dominant role, it is often necessary to use quite complicated equations. This refers particularly to mixed adsorption, where heterogeneity effects may change drastically the adsorption behavior of the components, for example, by affecting the selectivity in the adsorptive separation of gases.4 Of course, from a practical point of view, these equations should be as simple as possible but, on the other hand, also sufficiently accurate. This is mainly because the computer programs which control industrial installations should respond as fast as possible to the changes in the parameters of a given adsorption/separation system. * Corresponding author. Phone: +48 81 537 5518. Fax: +48 81 537 5685. E-mail: [email protected]. (1) Yang, R. T. Gas Separation by Adsorption Processes; Butterworths: Boston, 1987. (2) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (3) Rudzin˜ski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Solid Surfaces; Academic Press: London, 1992. (4) Sircar, S. Ind. Eng. Chem. Res. 1991, 30, 1032.

Despite the considerable complexity of the problem, the adsorption of mixtures on heterogeneous surfaces has been intensively studied over the past three decades, which resulted in significant progress in understanding of this phenomenon. Different theories of mixed adsorption, based on the properties of the associated single-component systems, have been proposed so far. These include, for example, the ideal adsorbed solution (IAS) theory formulated by Myers and Prausnitz5 and further generalized for the case of heterogeneous surfaces by Myers,6 the vacancy solution theory (VST) developed by Suwanayuen and Danner,7 the potential theory (PT) proposed by Polyanyi and further generalized by Moon,8 Grant,9 and Mehta and Danner,10 or finally the integral equation (IE) approach developed by Rudzinski,3,11 Jaroniec,2,12-15 and Wojciechowski.16 The IE approach coupled with an uniform adsorption energy distribution (AED) has been recently a subject of extensive parametric studies17,18 as well as of computer simulations.19 In the latter case, we proposed the approximate isotherm equations describing the adsorption of a binary mixture on a strongly heterogeneous surface and compared their predictions with the analogous results obtained from the Monte Carlo simulations. Good quantitative matches between the theory and the simulations obtained for the simple uniform AEDs motivated us to verify the proposed approach in the case of a more complicated pattern of the heterogeneity. In particular, (5) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (6) Myers, A. L. Fundam. Adsorpt., Proc. Eng. Found. Conf., 1983 1984, 365. (7) Suwanayuen, S.; Danner, R. P. AIChE J. 1980, 26, 77. (8) Moon, H.; Tien, C. Chem. Eng. Sci. 1988, 43, 2967. (9) Grant, R. J.; Manes, M. Ind. Eng. Chem. Fundam. 1960, 490. (10) Mechta, S. D.; Danner, R. P. Ind. Eng. Chem. Fundam. 1985, 24, 325. (11) Rudzin˜ski, W.; Nieszporek, K.; Moon, H., Rhee, H.-K. Heterog. Chem. Rev. 1994, 1, 275. (12) Jaroniec, M. J. Colloid Interface Sci. 1977, 59, 230. (13) Jaroniec, M. Colloid Polym. Sci. 1977, 255, 32. (14) Jaroniec, M.; Rudzin˜ski, W. Phys. Lett. 1975, 53A, 59. (15) Jaroniec, M. Thin Solid Films 1978, 50, 163. (16) Wojciechowski, B. W.; Hsu, C. C.; Rudzin˜ski, W. Can. J. Chem. Eng. 1985, 63, 789. (17) Ritter, J. A.; Al-Muhtaseb, S. A. Langmuir 1998, 14, 6528. (18) Al-Muhtaseb, S. A.; Ritter, J. A. Langmuir 1999, 15, 7732. (19) Szabelski, P.; Nieszporek, K. J. Phys. Chem. B 2003, 107, 12296.

10.1021/la050192h CCC: $30.25 © 2005 American Chemical Society Published on Web 07/06/2005

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in this article we study the adsorption of a binary mixture of non-interacting molecules on a heterogeneous surface chracterized by a quasi-Gaussian AED. This choice is also dictated by the fact that the AEDs of a Gaussian shape are frequently encountered in adsorption studies of real solid surfaces.2,3 2. Theory The adsorption of a binary mixture of components 1 and 2 on a heterogeneous solid surface can be modeled by using the integral equation (IE) approach2,3 coupled with the Langmuir local model of adsorption, that is,

θi )

∫Ω∫1 + λ

λi exp(i/kT) 1

exp(1/kT) + λ2 exp(2/kT) i ) 1, 2 (1) χ12(1,2) d1 d2

where k and T have their usual meanings and θi, λi, and i are the fractional surface coverage, the absolute activity, and the adsorption energy of component i, respectively. The symbol χ12 in the above equation denotes the twodimensional AED function with the adsorption energy domain Ω. As it may seem, the simple eq 1 should be solvable for an arbitrary function χ12. However, even for single-component systems (i.e., for λ1 ) 0 or λ2 ) 0) analytical solutions of the problem are known only for these continuous AEDs which have a very simple form, such as, for example, the rectangular AED.4,17-19 In most other cases, application of different AEDs such as, for example, the exponential or the Gaussian distribution leads to an integral equation whose solution cannot be obtained in a closed form. Obviously, the situation becomes more complicated when two components adsorb simultaneously. In this case, to obtain the analytical solution of eq 1 it is usually necessary to use approximate methods, such as the condensation approximation (CA),3 and to assume a sufficiently simple functional relationship between the adsorption energies 1 and 2. Let us recall two such functional relationships which are often used in the theory of mixed adsorption on heterogeneous surfaces.2,3 The first one involves a complete lack of correlations between the adsorption energies of both components. In particular, for each adsorption site x, it is assumed that 1x is entirely independent of 2x and, in consequence,

χ12(1,2) ) χ1(1) χ2(2)

(2)

where χ1(1) and χ2(2) are the individual, one-dimensional, AEDs associated with the adsorbing components. Owing to the above assumption, integration of eq 1 becomes, in general, much easier than integration of the original equation with the two-dimensional AED. In the second approach, the adsorption energies 1x and 2x are usually assumed to be highly correlated, being, for example, linearly dependent, that is,

2x ) 1x + ∆12

(3)

χ1(1) ) χ2(1 + ∆12)

which in practice means that the corresponding AEDs have the same shape but they are shifted along the energy axis, one with respect to the other. Interestingly, the converse is not always true. For example, it is possible to distribute 1 and 2 among the adsorption sites in such a way that the associated AEDs satisfy eq 4 while, at the same time, the microscopic condition given by eq 3 may not be fulfilled. We will come back to this problem later in this section. Using the two approaches described above it is possible to derive relatively simple expressions for the adsorption isotherms in qualitatively different two-component systems. Obviously, these expressions correspond to distinct energetic landscapes of the surface but both of them can be obtained only by using further auxiliary assumptions inherent to such methods as the CA. Accordingly, in this study we solved eq 1 using the CA and assuming that each component of the mixture is characterized by the corresponding quasi-Gaussian AED, that is, by

χi(i) )

ηi exp{ηi(i - °i)}

(5)

[1 + exp{ηi(i - °i)}]2

where, for a pure component i, °i is the most probable value of the adsorption energy and ηi ∈ (0,1) is the heterogeneity parameter which is inversely proportional to the distribution width. Note also that the function given by eq 5 is a normalized symmetrical distribution which is defined for i ∈ (-∞,+∞). Let us now proceed with the approximate solutions of eq 1 obtained for the single and mixed adsorption. In the latter case, we discuss the isotherm equations derived for both correlated and uncorrelated adsorption energies 1 and 2. In particular, for the one-component adsorption, the CA approach coupled with the integral eq 1 and the distribution function χi(i) leads to the well-known Langmuir-Freundlich (LF) isotherm,2,3 that is,

[ { }] [ { }]

°i ηi kT θit ) °i 1 + λi exp kT λi exp

(6)

ηi

For the binary mixture, when the adsorption energies of the components 1 and 2 are correlated in a way adopted here (see eq 3), the CA gives the following equations that describe the adsorption of a single component from the mixed bulk phase:11

{ } (∑[ { }]) ∑[ { }] (∑[ { }]) λi exp

θi )

2

λj exp

j)1

where ∆12 is the relative shift on the adsorption energy axis. The above assumption may refer to the adsorption of mixtures on selective adsorbents where specific interactions with the surface (e.g., enantioselective interactions) are much stronger for one component than for the other. In this case, application of eq 3 causes the double integral in eq 1 to become a single integral, which is usually simpler to calculate either analytically or numerically. It is also worth noting that eq 3 implies that

(4)

°i

2

kT

j)1

°j

kT

λj exp

kT

2

1+

λj exp

j)1

ηi

°j

°j

ηi

kT

i ) 1, 2 (7)

where, according to eq 3, °j ) °i + ∆ij. We would like to remind the reader that eq 7 refers to the case where the distribution functions χ1(1) and χ2(2) are exactly the same in shape, that is, η1 ) η2 ) η. However, in this equation we left ηi to indicate that it may be also used with some caution for distributions which are very close in width, that is, for η1 ≈ η2.

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On the other hand, in the case of a complete lack of correlations between the adsorption energies, derivation of the partial adsorption isotherms θi can be additionally facilitated by using the idea proposed by Wojciechowski.16 In particular, Wojciechowski’s approach assumes that the presence of the second component affects the adsorption of the first one only by random blocking of the adsorption sites and that the blocking effect is proportional to the coverage of the second component. Taking into account the above assumption and using the CA, the corresponding adsorption isotherms for the binary system can be expressed as11

[ { }] ∑[ { }] λi exp

θi )

°i

kT

2

1+

λj exp

j)1

ηi

°j

ηj

i ) 1, 2

(8)

kT

Note that eq 8 has a more general character than eq 7 because it can be used also in the case when η1 ) η2. Furthermore, it has also another interesting property which is worth mentioning here. Namely, eq 8 allows formation of an azeotropic mixture in our system. To demonstrate this, let us define the mole fraction of component i in the adsorbed phase as

Xi ) θi/(θ1 + θ2) ) θi/θT

(9)

and assume that the difference between the standard chemical potentials of the components is negligible, that is, the components have very similar physical properties like, for example, a pair of enantiomers. Although, in general, not necessary, the above assumptions allow the adsorption isotherms to be treated as functions of the total absolute activity λ, which may be identified with the total pressure of the mixture. In consequence λi (i ) 1, 2) can be substituted by λYi, where Yi denotes the mole fraction of component i in the bulk phase and, obviously, Y1 + Y2 ) 1. Taking into account the above definitions, the azeotropy in the system means that Xi ) Yi or, in other words, that at a given λ the composition of the adsorbed phase is identical with that of the bulk phase. According to eqs 8 and 9 this occurs when Yi ) 0 or Yi ) 1 (trivial solutions) but also when Yi fulfills the following condition, which we provide here for the first component of the mixture, that is,

Ya1 ) (1 - Ya1)pλq exp[r(°2η2 - °1η1)]

(10)

where r ) 1/(η1 - 1), q ) (η2 - η1)r and p ) (η2 - 1)r. Without a detailed analysis of eq 10 its two important implications can be indicated immediately. The first is that Ya1 becomes totally independent of λ when η1 ) η2 ) η. This means that, for the distribution functions of the same shape, the position of the azeotropic point in the X1 - Y1 phase diagram should not change with the total pressure of the mixture. In this case, the composition of the bulk phase at which the azeotropy appears is given by

Ya1 )

1 1 + exp[-ηr(°2 - °1)]

we are dealing with the situation in which the mean energies of adsorption of both components are equal, but in spite of that the azeotropy is still possible in the system. The effects discussed above refer exclusively to the case of the uncorrelated adsorption energies, and according to eqs 7 and 9, they are impossible to observe in the systems in which the energies are microscopically correlated. This is because, for the correlated AEDs the condition X1 ) Y1 is fulfilled only for Y1 equal to 0 or 1. The adsorption azeotropy predicted in the former case is particularly interesting because, for example, for two ADSs of the same shape, it originates only from a different spatial distribution of the adsorption energies. This result is interesting also because in this study we assumed no lateral interactions in the adsorbed phasesan important factor which is often treated as the major source of the azeotropic behavior of binary mixtures. However, it is worth noting here that the presence of interactions between molecules adsorbed on heterogeneous surfaces has been shown long ago to be an unnecessary condition for the azeotropy.4 For these reasons it seems useful to assess the validity of the approximate expressions (eqs 6-8) by comparing their predictions with analogous results obtained by using different approaches. In particular, it is worthwhile to check whether the azeotropy discussed above is just an artifact originating form the approximate character of the proposed equations or it can really occur in an adsorption system following the assumptions made here. The objective formulated above can be reached in two ways. Namely, the first way involves numerical integration of eq 1 while the second one relies on computer simulations performed using grand canonical Monte Carlo technique. In this study we use both methods and compare their predictions with the approximate theory described in this section. 3. Monte Carlo Simulation The simulations described below were carried out on a square L × L lattice using standard the grand canonical Monte Carlo (GCMC) technique.20,21 Here we briefly outline the applied algorithm; a more detailed discussion can be found elsewhere.19 Before we proceed with the MC simulations let us first focus on a few questions related to the energetic properties of the adsorbing surface. According to the theory presented in the preceding section, the quasi-Gaussian distribution function given by eq 5 and used for the derivation of eqs 6-8 is defined inside an infinite interval of the adsorption energy. While, on one hand, this assumption largely simplifies derivation of the associated analytical expressions, on the other hand, it is completely impractical with respect to the simulations. Obviously, this is because of a finite range of numbers generated by a computer. For that reason, it is necessary to define a sufficiently wide finite interval from which the adsorption energy is sampled. This procedure allows the minimization of errors associated with the resulting distribution cutoff. It is also worth noting that such errors may greatly affect the simulated results making, them hardly comparable with the predictions of the theory, especially in the case of broad distributions of the adsorption energy.22 To avoid this, in the present study, we always used the interval centered at °i whose radius was equal to 25, regardless of the assumed parameters of

(11)

The second consequence of eq 10 is that the adsorption azeotropy occurs regardless of the values of both °1 and °2, in particular, also for °1 ) °2. In this particular case,

(20) Landau, D. P.; Binder, K. A Guide to Monte Carlo Simulations in Statistical Physics; Cambridge University Press: Cambridge, 2000. (21) Nicholson, D.; Parsonage, G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (22) Szabelski, P.; Zarzycki, P.; Charmas, R. Langmuir 2004, 20, 997.

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the distribution function χi(i). As we examined, further increase of the interval length does not lead to any visible change in the obtained results, even for the widest distributions considered (ηi ) 0.2). Another important question related to the mixed adsorption is the way in which the energies 1 and 2 are distributed among the adsorption sites. This refers specifically to the case when χ1(1) and χ2(2) are of the same shape (i.e., η1 ) η2) but they are shifted on the energy axis, one with respect to the other. Here, although the distribution functions are correlated in a macroscopic sense (see eq 4), they do not have to be so taking into account microscopic properties of the surface. As we indicated in the previous section, it is possible to assume two entirely different spatial distributions of 1 and 2 which give exactly the same pairs of χ1(1) and χ2(2). To explore this possibility we prepared the lattices of adsorption sites for which the adsorption energy was distributed in two ways. In the first case, which we call uncorrelated distributions, for each site x two totally independent values 1x and 2x were sampled at random from the corresponding distributions. Obviously, this method refers also to the distribution functions of different widths (η1 * η2). On the other hand, in the case of the microscopically correlated distributions, for each site we sampled 1x and next calculated 2x according to eq 3. In the following, we call briefly these functions correlated distributions. Let us now come back to the actual simulation of the adsorption process. The method sketched below refers to the binary adsorption, but it can be easily adapted to a single-component system by setting Yi ) 1. Regardless of the assumed energetic landscape of the surface, the probability of adsorption of a molecule of type i on a vacant site x is given by

paix ) min[1,λYi exp(ix)]

i ) 1, 2

Figure 1. The quasi-Gaussian adsorption energy distributions used for the calculation of the single and mixed adsorption data.

(12)

Obviously, the associated probability of desorption of an already adsorbed molecule from site x is equal to pdix ) 1/pix. In the course of the simulation one of these two probabilities is always calculated in each adsorption or desorption attempt, depending whether the site is empty or occupied, respectively. The calculated probability is next compared with a uniformly distributed number R. When it is less than R the corresponding attempt is successful, otherwise the trial ends. The above procedure is repeated for sufficiently long time, typically 104 × L2 times, until the equilibrium state is reached. Then, the surface coverage of each component at a given λ or at a given gas phase composition is calculated. All of the simulations described in this work were performed for L ) 100. The results presented in the following section are averages over 20 independent runs. 4. Results and Discussion As we mentioned in the theoretical section, the isotherm eqs 6-8 were obtained by using the CA method, which may introduce some error to the proposed theory. For that reason, to examine what influence the CA may have on the accuracy of the isotherm equations, we decided to study the single-component systems first. To that purpose, four typical examples of the AED given by eq 5 were considered. For the sake of simplicity, in further discussion we express all of the adsorption energies in kT units. Accordingly, the calculations were performed for the surfaces characterized by the following distributions of the adsorption energy: (A) °i ) 0, ηi ) 0.4; (B) °i ) 2, ηi ) 0.4; (C) °i )

Figure 2. The single-component adsorption isotherms obtained for the distribution functions shown in Figure 1. The symbols correspond to the MC simulations while the solid lines are the isotherms calculated by using eq 4; the dashed lines denote analogous results obtained by using eq 1 (numerical integration).

0, ηi ) 0.2; and (D) °i ) 0, ηi ) 0.8. Figure 1 shows the AEDs listed above. We would like to emphasize that the mean values assumed for the distributions from Figure 1 were chosen just for illustrative purposes and they are meant to be representative of a wide range of adsorption systems with quasi-Gaussian AED. In particular, for the sake of convenience, for distributions A, C, and D we set their mean values to 0. Of course, a desirable shift of the AEDs on the energy axis would allow the use of the proposed theory also in the case of quantitatively different systems. In Figure 2 we displayed the single-gas adsorption isotherms calculated for the distribution functions shown in Figure 1. As it follows from Figure 2, the theoretical adsorption isotherm obtained for distribution D (solid line) differs markedly from the simulated data (filled triangles) as well as from the numerical solution of eq 1 (dashed lines). On the other hand, for the three remaining functions, the agreement between the theory and the simulations is, in general, much better, being almost complete for distribution C (plus signs). This indicates and confirms again3 that the CA coupled with the IE approach gives reliable predictions of the adsorption isotherms only in the case of strongly heterogeneous surfaces, that is, for relatively

Binary Adsorption on Heterogeneous Surfaces

Figure 3. The adsorption isotherms obtained for the equimolar mixture of components 1 and 2 that are characterized by the distribution functions A and B, respectively. The left part corresponds to the correlated adsorption energies of both components while the right part corresponds to case when the energies are totally uncorrelated. The solid lines are the isotherms obtained with the help of eqs 7 and 9 while the dashed lines are the data calculated numerically using eq 6. From top to bottom: θT, θ2. and θ1.

broad AEDs (A, B, C). On the other hand, the CA introduces a serious error to the theory when the surface is nearly homogeneous (D). It is also worth noting that the singlecomponent adsorption isotherms obtained by numerical integration of eq 1 (dashed lines) are in full agreement with the simulations. With these conclusions in mind, let us now proceed with the mixed adsorption. To facilitate comparison of the results obtained for different systems in all of the following figures, we always use the same notation as that used in Figure 2. That is: the theory proposed here, solid lines; exact numerical solutions of eq 1, dashed lines; simulated data, symbols. Figure 3 presents the adsorption isotherms obtained in the case of the equimolar (Y1 ) Y2 ) 0.5) mixture of components 1 and 2, which are characterized by the distribution functions A and B, respectively. We call this system briefly A+B and use this notation constantly in the further text. The left part of Figure 3 shows the results corresponding to the correlated distributions while the right part presents the isotherms obtained for the uncorrelated distributions. As it follows from the left part, the theory overestimates somewhat the partial adsorption isotherm of the more strongly adsorbed component 2, and thus it overestimates also the total adsorption isotherm. This is particularly visible for the values of λ larger than ∼0.5. On the other hand, the isotherm corresponding to the weakly adsorbed component 1 seems to be almost accurately predicted by the theory, being, however, slightly underestimated compared to the simulations. As we can also observe, the isotherms calculated numerically are very close to their counterparts obtained from the simulations. Quite a different situation can be encountered in the case of the uncorrelated distributions. As it can be seen in the right part of Figure 3, the lack of microscopic correlations between the adsorption energies of both components leads to qualitatively different results. In particular, the relative position of the partial adsorption isotherms is markedly different from that observed for the correlated distributions. For example, the simulated θ1 taken at λ ) 3 is about three times larger for the uncorrelated than for the correlated distributions. On the other hand, θ2 (symbols) taken at this λ decreases from ∼0.6 to ∼0.5 when the correlations are switched off. In consequence, because the growth of θ1 is not compensated by the decrease in θ2, the total coverage becomes higher compared to θT from the left panel of Figure 3. Regarding the reliability of the predictions of the theory, we may

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Figure 4. The adsorption isotherms obtained for Y1 ) 0.3, for the mixture of components 1 and 2 that are characterized by the distribution functions A and B, respectively. The notation used here is the same as that used in Figure 3.

Figure 5. The adsorption isotherms obtained for Y1 ) 0.7, for the mixture of components 1 and 2 that are characterized by the distribution functions A and B, respectively. The notation used here is the same as that used in Figure 3.

conclude that eq 8 is, in general, much less accurate than eq 7 associated with the correlated distributions. In particular, for the uncorrelated distributions A and B, the theory predicts almost correctly θ2 while it largely underestimates θ1. Obviously, this leads to a serious discrepancy between the total adsorption isotherm resulting from the theory and that obtained from the simulations. On the other hand, the numerical results agree completely with the simulations. To examine the influence of the gas-phase composition on the theoretical adsorption isotherms in the system A+B, in Figures 4 and 5 we plotted the results obtained for Y1 ) 0.3 and Y1 ) 0.7, respectively. As it can be seen in both figures, changes in the gasphase composition do not lead to a noticeable reduction of the relative error arising from the application of the theory. Although, obviously, the relative position of the isotherms shown in Figures 4 and 5 changes compared to that observed for Y1 ) 0.5, the gap between the corresponding theoretical and simulated curves seems to be nearly the same for each Y1. In particular, as we may deduce from Figures 3-5, the theory referring to the correlated distributions (the left panels) systematically overestimates θ2. On the other hand, in the case of the correlated distributions (the right panels) the theory seriously underestimates θ1, regardless of the gas-phase composition. It is also worth noting that, as for Y1 ) 0.5, we can observe a very good match between the isotherms calculated using eq 1 and the results of the simulations. Let us now focus on the results obtained for the system C+D (see Figure 1). This combination was chosen as an example of two AEDs of substantially different widths. In this case, to calculate the theoretical adsorption isotherms we used eq 8, which refers to the uncorrelated distributions. The results of the calculations performed for Y1 equal to 0.3, 0.5, and 0.7 are shown in Figure 6.

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Figure 7. The phase diagrams obtained for λ ) 0.1, for the mixture of components 1 and 2 that are characterized by the distribution functions A and B, respectively. The left part corresponds to the model assuming the existence of correlations between the adsorption energies of both components whereas the right part presents the results obtained for the case of uncorrelated energies. The symbols denote the MC results, the solid lines are the predictions of eq 8, and the dashed lines are the results obtained by numerical integration of eq 1.

Figure 6. The adsorption isotherms obtained for the mixture of components 1 and 2 that are characterized by the distribution functions C and D, respectively (lack of correlations). The symbols denote the MC results, the solid lines are the predictions of eq 8, and the dashed lines are the isotherms calculated numerically using eq 1. The gas-phase composition for which the calculations were performed is displayed in each part of the figure. From top to bottom: θT, θ1, and θ2.

As it follows from Figure 6, the predictions of the theory are totally incorrect, except from the range of very low pressures (i.e., small values of λ). This refers to all of the gas-phase compositions considered. Note that, for all of the cases shown in Figure 6, θ1 is greatly underestimated by the theory, while, at the same time, the theoretical θ2 is much higher than the corresponding coverage obtained from the simulations. Another discrepancy which can be noticed in Figure 6 is the fact that the theoretical partial adsorption isotherms cross each other. This effect, observed for Y1 ) 0.5, implies clearly the existence of the azeotropic point in the phase diagram calculated for λ ) 2 (see the middle part of Figure 6). As we mentioned before, the azeotropy in the sytem C+D is a direct consequence of the intrinsic properties of eqs 8-10. However, as seen in Figure 6, this result is not confirmed either by the simulations or by the exact numerical solution of eq 1. The above findings may suggest that the adsorption azeotropy for the uncorrelated distributions is only an artificial effect originating from the approximate character of the theory. As we show later, this hypothesis is, in general, not acceptable. Taking into account the conclusions drawn for the singlecomponent systems, the discrepancies seen in Figure 6 are, however, not surprising. As we mentioned at the beginning of this section, the CA leads to a serious error in θit when the surface is nearly homogeneous, that is, when χi(i) is narrow (see Figure 2). Note that, here we are

dealing with the adsorption of two components, one of which (D, ηi ) 0.8) is characterized by such a narrow AED. For that additional reason, the theory fails even in a qualitative prediction of the partial adsorption isotherms, except from the range of very low pressures. On the other hand, as we already observed for the system A+B, the isotherms obtained from numerical integration of eq 1 are in agreement with the simulated results. In the theoretical, and especially in the experimental, studies of mixed-gas adsorption equilibria it is often useful to plot the adsorption data in the phase diagram. To that purpose, in Figure 7 we display the phase diagrams obtained at λ ) 0.1 for the system A+B (i.e., for the range of low pressures). The left part of Figure 7 shows the results corresponding to the correlated distributions while the right part shows analogous data obtained in the case when the distributions are totally uncorrelated. As it can be seen in the left part of this figure, the theory predicts correctly the phase diagram for the correlated distributions. Note that the theory agrees fully with the simulations as well as with the numerical solution of eq 1. In the latter case, the solid and the dashed lines (not seen) overlap. As we mentioned before, eq 7, which was used for the calculation of the curves shown in the left part, does not allow the azeotropy to appear in the system A+B. This theoretical prediction is also confirmed by both simulations and numerical results. On the other hand, as seen in the right part of Figure 7, the same system displays entirely different behavior when the associated distribution functions are uncorrelated. In particular, in the right panel we can observe the curve of a completely different shape than that obtained for the correlated distributions. Moreover, in the case of a lack of correlations the azeotropy does appear in our system. Taking into account the absence of lateral interactions in the adsorbed phase this is a somewhat surprising effect. However, we would like to emphasize again that the major source of the observed effect is the surface heterogeneity which has been found to be responsible for the azeotropy also in other models of heterogeneous mixed adsorption.4,23 As seen in the right part of Figure 7, the theoretical phase diagram deviates considerably from the simulated data for values of Y1 less than ∼0.6. Nevertheless, the azeotropy predicted by the model is also confirmed by the simulations as well as by the exact numerical results. (23) Bai, R.; Yang, R. T. J. Colloid Interface Sci. 2002, 253, 16.

Binary Adsorption on Heterogeneous Surfaces

Figure 8. The phase diagrams obtained for λ ) 5, for the mixture of components 1 and 2 that are characterized by distribution functions A and B, respectively. The notation used here is the same as that used in Figure 7.

However, the position of the azeotropic point is different for the theoretical results and for the simulated data. Namely, in the former case, from eq 11 we have Ya1 ) 0.209 while in the latter case Ya1 is located at about 0.29. To test further the behavior of the phase diagram in the system A+B we performed also calculations corresponding to high pressure of the mixture. Figure 8 presents results analogous to those from Figure 7 but obtained for λ ) 5. As it can be seen in the left part of Figure 8, increasing the total pressure does not lead to significant changes in the shape of the phase diagram obtained for the correlated AEDs (compare with Figure 7). As for λ ) 0.1, the theoretical diagram is also consistent with the results of both simulations and numerical calculations. The only discrepancy in this case is a slight underestimation of X1 observed for 0.3 < Y1 < 0.9, compared to the simulated data. On the other hand, a much larger discrepancy between the theory and the simulations can be observed in the case of the uncorrelated distributions. This can be as seen in the right part of Figure 8. Here, the effect of the total pressure on the shape of the phase diagram is, in general, also marginal. However, two quantitative differences between the data shown in the right panels of Figures 7 and 8 can be indicated. Namely, the phase diagram obtained for λ ) 5 is more underestimated by the theory than the phase diagram calculated for λ ) 0.1. This refers especially to Y1 varying from 0 to 0.8. The second difference between these diagrams is a slight shift of the azeotropic point obtained from simulations, from 0.29 to 0.31, that is induced by the increased λ. We would like to remind the reader that, according to eq 11, the position of the azeotropic point for the system A+B should be totally independent of λ. Indeed, as it follows from the comparison of the solid lines from the right parts of Figures 7 and 8 for both λ ) 0.1 and λ ) 5, we have Ya1 ) 0.209. However, as we found for the correlated distributions, the independence of Ya1 of λ seems not to be confirmed either by the simulations or by the exact solution of the integral eq 1. For these two methods we can also observe that the corresponding phase diagrams obtained with their help are identical. Let us now proceed with the system C+D, the phase diagrams for which we present in Figure 9. The left part of this figure shows the results obtained for λ ) 0.1 while the right part corresponds to λ ) 5. In this case, we can observe that the position of the azeotropic point is largely affected by the total pressure of the mixture. In particular, for low pressure, Ya1 obtained from the simulations is very

Langmuir, Vol. 21, No. 16, 2005 7341

Figure 9. The phase diagrams obtained for different values of λ, for the mixture of components 1 and 2 that are characterized by distribution functions C and D, respectively. The notation used here is the same as that used in Figure 7.

close to 1, while for high pressure, it is equal to ∼0.6. The corresponding values of Ya1 calculated by means of eq 10 are equal to 0.999 and 0.276, respectively, which suggests that the theory works much better for the low pressures. Unfortunately, this conclusion is not valid, taking into account the whole phase diagram. Namely, as seen in the right part of Figure 9 the mole fraction of component 1 in the adsorbed phase is largely overestimated within the whole range of Y1. The situation is even worse in the case of the results obtained for λ ) 5. In this case, increasing the total pressure leads to a dramatic deterioration of the agreement between the theory and the simulations. The origin of the observed effects can be, however, easily explained taking into account the intrinsic properties of the CA which were discussed before. Summary The results of this work demonstrate that the energetic heterogeneity of the adsorbing surface is an important factor which can drastically change adsorption behavior of binary mixtures. In particular, it was shown that the heterogeneity may be the source of the adsorption azeotropy when the adsorption energies of the components are uncorrelated in a microscopic sense. The most interesting finding is the fact that for the components characterized by AEDs of the same Gaussian shape the partial adsorption isotherms and phase diagrams can be qualitatively different, depending on whether the adsorption energies are correlated or not. The existence of the adsorption azeotropy is particularly interesting because the theory proposed in this work does not assume lateral interactions in the adsorbed phase. The obtained results indicate also that the predictions of the theory are much more accurate in the case of the correlated adsorption energy distributions. Unfortunately, quite simple analytical expressions derived for the systems with the uncorrelated distributions are unable to provide a reliable description of the mixed adsorption. Also, the quality of the theoretical predictions deteriorates when the adsorbing surface is weakly heterogeneous, that is, when the associated distribution functions are narrow. Acknowledgment. One of the authors (P.S.) is grateful to the Foundation for Polish Science (FNP) for the Award of a Stipend for Young Scientists. LA050192H