Monte Carlo study of multicomponent monolayer adsorption on square lattices
Guillermo D. García, Fabricio O. Sánchez-Varretti, Fernando Bulnes, Antonio J. Ramirez-Pastor
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y Monte Carlo study of multicomponentmonolayer adsorption on square lattices
G. D. Garc´ıa a , b , F. O. S´anchez-Varretti a , b , F. Bulnes a ,A. J. Ramirez-Pastor a , a Dpto. de F´ısica, Instituto de F´ısica Aplicada, Universidad Nacional de San Luis -CONICET, Chacabuco 917, 5700 San Luis, Argentina. b Universidad Tecnol´ogica Nacional, Regional San Rafael, Gral. Urquiza 314, 5600,San Rafael, Mendoza, Argentina.
Abstract
The monolayer adsorption process of interacting binary mixtures of species A and B on square lattices is studied through grand canonical Monte Carlo simulation inthe framework of the lattice-gas model. Four different energies have been consideredin the adsorption process: 1) ǫ , interaction energy between a particle (type A or B ) and a lattice site; 2) w AA , interaction energy between two nearest-neighbor A particles; 3) w BB , interaction energy between two nearest-neighbor B particles; 4) w AB = w BA , interaction energy between two nearest-neighbors being one of type A and the other of type B . The adsorption process has been monitored through totaland partial isotherms and differential heats of adsorption corresponding to bothspecies of the mixture. Our main interest is in the repulsive lateral interactions,where a variety of structural orderings arise in the adlayer, depending on the in-teraction parameters ( w AA , w BB and w AB ). At the end of this work, we determinethe phase diagram characterizing the phase transitions occurring in the system. Anontrivial interdependence between the partial surface coverage of both species isobserved. Key words:
Equilibrium thermodynamics and statistical mechanics, Surfacethermodynamics, Adsorption isotherms, Monte Carlo simulations Corresponding author. Fax +54-2652-430224, E-mail: [email protected]
Preprint submitted to Elsevier 7 December 2018
Introduction
The adsorption process of mixture gases on solid surfaces is a topic of greatinterest not only from an intrinsic but also from a technological point ofview, due to its importance for new developments in fields like gas separa-tion and purification [1,2,3,4]. Although this problem has been theoretically[5,6,7,8,9,10,11,12] and experimentally [11,12,13,14,15] studied for many years,some aspects are still unclear being necessary to reach a better understand-ing about the behavior of the adsorbate during the adsorption process of themixture.As in any adsorption process, a complete analysis of the behavior of gasmolecules under the influence of an adsorbent requires the knowledge of theforces of molecular interactions [1,16,17,18]. In other words, the descriptionof real multicomponent adsorption requires to take into account the effect ofthe lateral interactions between each species of the mixture. An exact treat-ment of this problem, including ad-ad interactions, is unfortunately not yetavailable and, therefore, the theoretical description of adsorption relies on sim-plified models [4]. One way of overcoming this complication is to use MonteCarlo (MC) simulation method [19,20,21,22,23]. MC technique is a valuabletool for studying surface molecular processes, which has been extensively usedto simulate many surface phenomena including adsorption [24], diffusion [25],reactions, phase transitions [26], etc.In this line of work, a previous article was devoted to the study of the adsorp-tion of interacting binary mixtures on triangular lattices [27]. In Ref. [[27]],Rinaldi et al. obtained adsorption isotherms and differential heats of adsorp-tion corresponding to both species of the mixture, for different values of thelateral interactions between the adsorbed species. An unusual feature was ob-served when ( i ) the lateral interaction between A and B particles was differentfrom zero, and ( ii ) the initial concentration of B particles was in the range[0 . , . The adsorptive surface is represented by a two-dimensional square lattice of M = L × L adsorption sites, with periodic boundary conditions. The sub-strate is exposed at a temperature T to an ideal gas phase consisting of abinary mixture of particles A and B with chemical potentials µ A and µ B , re-spectively. Particles can be adsorbed on the lattice with the restriction of atmost one adsorbed particle per site and we consider a nearest-neighbor ( N N )interaction energy w XY (X, Y = A, B) among them. The adsorbed phase isthen characterized by the Hamiltonian: H = 12 M X i X l ∈{ NN,i } [ w AA δ c i ,c l , + w BB δ c i ,c l , − + w AB ( δ c i , δ c l , − + δ c i , − δ c l , )] ++ ǫ M X i ( δ c i , + δ c i , − ) − M X i ( µ A δ c i , + µ B δ c i , − ) (1)where c i is the occupation number of site i ( c i = 0 if empty; c i = 1 if occupiedby A and c i = − l ∈ { N N, j } runs on the four N N sitesof site i ; the δ ’s are Kronecker delta functions and ǫ is the interaction energybetween a monomer (type A or B) and a lattice site. In this contribution, thechemical potential of one of the components is fixed throughout the process( µ B = 0), while the other one ( µ A ) is variable, as it is usually assumed instudies of adsorption of gas mixtures [28]. In the actual implementation of themodel ǫ was set equal to zero, without loss of any generality.With respect to the computational simulations, the Monte Carlo procedureused has been discussed in detail in Ref. [[27]] and need not be repeated here.In this case, the first 10 Monte Carlo steps (MCS) were discarded to allowequilibrium, while the next 10 MCS were used to compute averages.3
Results and discussion
The computational simulations have been developed for square L × L lattices,with L = 96, and periodic boundary conditions. With this lattice size weverified that finite-size effects are negligible. Note, however, that the lineardimension L has to be properly chosen such that the adlayer structure is notperturbed.In order to understand the basic phenomenology, we consider in the first placethe case of single-gas adsorption. This was achieved by making µ B → −∞ .Fig. 1 shows the behavior of the adsorption isotherms and the differential heatsof adsorption for different strengths of repulsive interparticle interactions. Asexpected, we obtain the well-known Langmuir isotherm passing through thepoint ( µ A /k B T = 0, θ A = 1 /
2) when w AA /k B T = 0 (being k B the Boltz-mann constant). Two features, which are useful for the analysis of mixed-gasadsorption, are worthy of comment: (a) as the N N repulsive interaction is in-creased, the coverage at zero chemical potential decreases and asymptoticallyapproaches θ A = 0 . N N interaction passes a criticalvalue w c /k B T ≈ .
763 [29,30], a plateau develops in the isotherm at θ A = 1 / c (2 ×
2) ordered phase on the surface. In whatfollows, we consider mixed-gas adsorption but keeping species B at a fixedvalue of the chemical potential µ B /k B T = 0. In addition, we have considered k B T = 1 for simplicity, without any lost of generality.We start with the case of a binary mixture in the presence of repulsive lateralinteractions between the particles. The effect of AA interactions is depictedin Fig. 2, where w AA /k B T = 0, w AB /k B T = 0 and w BB /k B T = 0. We haveplotted the partial (a) and total (b) adsorption isotherms, and the differentialheats of adsorption corresponding to the species A (c) and B (d). It canbe observed in Fig. 2 (a) and (b), that the initial coverage takes the samevalue θ = 0 . µ A /k B T → −∞ the A particle coverage is zero while the B particles arerandomly distributed on the lattice with θ B given by the Langmuir isotherm θ B = exp( µ B /k B T ) / [1+exp( µ A /k B T )+exp( µ B /k B T )], which for µ B /k B T = 0is θ = θ B = 1 / w c /k B T ≈ . w ,determines the character of the phase transition: (i) if w < w > w AA /k B T is increased. Thus, the interaction between Amolecules determines a c (2 ×
2) ordered phase for such particles. Therefore,the A isotherm presents a plateau at half coverage [see Fig. 2(a)]. At equilib-rium, the B particles occupy half of the empty sites, and the correspondingB isotherm presents a plateau at θ B = 0 .
25; this behavior is a consequenceof the excluded volume but is not due to the interactions. The total isotherm[Fig. 2(b)] is the sum of the partial isotherms, then the plateau appears at θ = 0 . q A corresponding to the Aspecies is plotted versus θ A . The behavior of the curves can be explained byanalyzing two different adsorption regimes: (i) for 0 . < θ < .
75 (0 < θ A < . N N occupancy which produces q A ≈ w AA /k B T = 0), and (ii) for 0 . < θ < . < θ A < cw AA in the energy of the system,where c is the lattice connectivity (in this case, c = 4). The maximum in q A for θ → . − corresponds to the critical coverage at which a dramatic changeof order takes place in the system (the system passes from the disordered tothe ordered phase [31]). A similar situation occurs for the minimum in q A at θ → . + .The adsorption of the A species induces an interesting behavior in the Bisotherm, which also exhibits well-defined steps although the B particles donot interact neither with B particles nor with A particles ( w AB /k B T = 0 and w BB /k B T = 0). This behavior is a consequence of the excluded volume but isnot due to the interactions. Then, as it is expected, q B is strictly zero over allthe coverage range [see Fig. 2 d)].We now continue the study of the effect of AA interactions with w AB /k B T = 0and w BB /k B T = 2 (Fig. 3), therefore introducing BB interactions; w AB /k B T =2 and w BB /k B T = 0 (Fig. 4) introducing interspecies interactions (and remov-ing BB interactions) and, finally, we analyze w AB /k B T = 2 and w BB /k B T = 2(Fig. 5) where all interactions are present.In the case of Fig. 3, the behavior of the partial [Fig. 3 a)] and total [Fig. 3b)] isotherms and the differential heat of adsorption q A [Fig. 3 c)] is similarto those in Fig. 2. The main difference is associated to the value of θ B at lowpressures (which is lower than 0.5), and to the behavior of q B that is not zeroany more.Clearly, for w AA /k B T = 0, A particles are distributed at random and B par-ticles start from a low coverage (close to 0 . q B increases steadily. However, as w AA /k B T becomes sufficiently high so that an5rdered phase is formed, a sudden increase in q B is produced due to a suddenincrease in the screening effect between B particles produced by A particles.An unusual feature is observed in the case of Fig. 4: as µ A /k B T increasesand A particles start to adsorb, B particles are displaced from the surface sothat the total A + B coverage decreases and shows a local minimum [Fig.4 b)]. This effect, which has been previously called mixture effect [27], canbe explained as follows: as the coverage of A particles is sufficiently high sothat AB interactions occur, the repulsive character of w AB leads to more Bparticles being displaced from the surface than A particles being adsorbed onthe surface.The mixture effect is clearly reflected in the behavior of q A [Fig. 4 c)] and q B [Fig. 4 d)]. In fact, for 0 < θ A < /
2, A particles do not interact with other Aor B molecules and, consequently, q A = q B = 0 (see insets). The c (2 ×
2) phaseof A particles starts to develop and is completed at θ A = 1 /
2. For 1 / < θ A , Aparticles fill the vacancies. In this regime, the coverage of B particles tends tozero and does not perturb significantly the adsorption of the A species. Theimportant fluctuations in q B are clear signals that the number of B particlesis practically zero.Our simulations show how the competition between two species in presenceof repulsive mutual interactions reinforces the displacement of one speciesby the other and leads to the presence of the mixture effect. To complete thisanalysis, it is interesting to note that the mixture effect [also called adsorptionpreference reversal (APR) phenomenon] has also been observed for methane-ethane mixtures [7,8,9,10]. The rigorous results presented in Ref. [[10]] showedthat, in the case of methane-ethane mixtures, the APR is not a consequenceof the existence of repulsive interactions between the ad–species, but it is theresult of the difference of size (or number of occupied sites) between methaneand ethane. A similar scenario has been observed for different mixtures oflinear hydrocarbons in silicalite [12], carbon nanotube bundles [33] and metal-organic frameworks [34].In the case of Fig. 5, the presence of repulsive BB interactions results in ainitial coverage of B particles close to 0 . w AA /k B T = w BB /k B T = 0 (Fig.6). Here no ordered phases are formed and for sufficiently high w AB /k B T an6mportant mixture effect appears. Let us choose for our analysis the curvescorresponding to w AB /k B T = 1. B coverage is initially 0.5. As A moleculesare adsorbed B molecules are eliminated from the surface in such a way that θ decreases. At the same time q A starts at a low value and increases rapidly(with a bivaluated behavior) tending to 0 as θ →
1, while q B starts near 0 andtends to 4 w AB as θ →
1. As soon as BB interactions are added, the starting Bcoverage is ∼ .
226 and the mixture effect disappears (Fig. 7). In the presenceof AA and AB interactions (Fig. 8), w AB /k B T = 0 . w AA /k B T = w AB /k B T = 0 (Fig. 10); w AA /k B T = 0and w AB /k B T = 2 (Fig. 11); w AA /k B T = 5 and w AB /k B T = 0 (Fig. 12) and w AA /k B T = 5 and w AB /k B T = 2 (Fig. 13). Given the value of the parametersin Fig. 10, neither the coverage of A [Fig. 10 a)] nor the differential heat of ad-sorption [Fig. 10 c)] are affected by BB interactions. As discussed in Fig. 1, thecoverage of B at low pressure starts at 0 . w BB /k B T increasestowards the limiting value of 0 .
226 [see Fig. 10 a)]. However, the coverage of Bdoes not present any special features. Such a special feature does indeed ap-pear in the behavior of q B [Fig. 10 d)], which decreases steadily as w BB /k B T increases below a certain critical value, w cBB /k B T (being w cBB /k B T ≈ w cBB /k B T , thereby allowing some BB interac-tions which contribute to the decrease in the differential heat of adsorption.Above w cBB /k B T , B particles adsorb forming an ordered structure so that BBinteractions stop contributing and q B increases. The condition w AB /k B T = 0restricts the possibility of mixture effect.In Fig. 11, the presence of AB interactions favors the displacement of B parti-cles and, consequently, the slopes of the B partial isotherms are increased. Onthe other hand, as the initial fraction of B particles is high (0 . ≤ θ iB ≤ . w BB /k B T (0 ≤ w BB /k B T ≤ . . ≤ θ ≤
1, thecurves in Fig. 11 are very similar between them (this is clearly visualized inthe case q A and q B ), which is indicative of the rapid decreasing of the numberof B particles on the surface.In Fig. 12, the adsorption of A particles [Fig. 12 a)] follows a unique isothermand is independent of the strength of BB interactions. Adsorption of B parti-cles decreases at low A coverage with increasing values of w BB /k B T and tendsto the limiting value θ B = 0 . w BB /k B T = 0 because there are no N N vacant7ites in that range available for the adsorption of B particles. This determinesthe total coverage behavior shown in Fig. 12 b). The curves of q A , shown inFig. 12 c), present a very similar behavior between them. This is, one markedjump appears, corresponding to the plateau in A isotherms. In contrast, q B presents two types of behaviors [Fig. 12 d)]. Thus, at very low w BB /k B T val-ues (negligible interactions) and very high w BB /k B T values (above the criticalvalue for the formation of the ordered phase), it remains practically constant,while at intermediate values it increases in line with the total coverage. In thelast case (Fig. 13), the adsorption process can be easily understood: the Bparticles disappear for low values of µ A /k B T . Then, for higher µ A /k B T ’s, allcurves collapse on a unique curve. In order to rationalize the results presented in previous section, we will deter-mine the temperature-coverage phase diagram characterizing our system in therange of the parameters studied. The curves will be obtained as a generaliza-tion of the well-known phase diagram for a lattice-gas of repulsive monomersadsorbed on a homogeneous square lattice, which is shown in Fig. 14.Some of the exact properties of this system have been found, especially byOnsager [35]. These are confined mostly to the special condition θ = 1 /
2, butby symmetry, it can be deduced that θ c = 1 /
2, if a critical point exists, sothis is the most interesting value of θ . Thus, the maximum of the coexistencecurve (occurring to θ = 1 /
2) corresponds to a critical value k B T c /w AA = h | √ − | i − ≈ .
567 [29,30]. On the other hand, zones I, II and IIIcorrespond to a disordered lattice-gas state, a ordered state [ c (2 ×
2) phase],and a disordered lattice-liquid state, respectively.We start with the analysis of Fig. 13, where our model is almost identical to alattice-gas of one species. As indicated in Fig. 13, the existence of repulsive ABinteractions favors the displacement of B particles. Once θ B ≈
0, which occursat µ A /k B T ≈
5, the binary mixture is equivalent to the square lattice-gas ofone species. To corroborate this affirmation, Fig. 15 shows the total isothermsof Fig. 13 b) (symbols), in comparison with the adsorption isotherm corre-sponding to a square lattice-gas of one species and w AA /k B T = 5 (solid line).The analysis can be separated in two parts: for θ < /
2, there exist B parti-cles on the lattice and, consequently, the curves of the mixture deviate fromthat corresponding to one species. For θ > /
2, the coverage of B particles isnegligible and all curves collapse on a unique curve. Then, the phase diagramcharacterizing a binary mixture with the set of parameters of Fig. 13 is identi-cal to that shown in Fig. 14. The unique difference with the one-species phasediagram is that the zone I now corresponds to an A-B mixture with different8roportions according to the value of w BB /k B T .As a basis for the analysis of the behavior of the system for variable w AB /k B T ,we begin by considering one of the cases of Fig. 13 (that corresponding to w BB /k B T = 0), which is characterized by a phase diagram as shown in Fig.14. This behavior is representative of systems with high values of w AB /k B T (in the range w AB /k B T ≥ . w AB /k B T = 2 . w AB /k B T is decreased, the maxima ofthe coexistence curves in zone II shift to higher values of coverage . This canbe better visualized in Fig. 17 (solid circles, bottom axis), where we plot thedensities corresponding to the maxima of the coexistence curves in zone II, θ c ’s, as a function of w AB /k B T . The figure allows to analyze the behavior ofa binary mixture with w BB /k B T = 0, variable w AB /k B T and w AA /k B T inthe critical regime. The curves can be understood according to the followingreasoning. As it was explained for high values of the AB lateral interaction,the zone II corresponds to a phase of A particles. As w AB /k B T decreases, thephase is complemented with B particles, which are randomly distributed inthe empty sites of the structure. As a typical example, we will analyze thecase of w BB /k B T = w AB /k B T = 0. In this case, the B particles, which are atchemical potential µ B /k B T = 0, occupy at random 1 / . Then, the phase corresponding to zone II is formed by a struc-ture of A particles (which occupy 1 / / θ c is 3 /
4. The rest of the points in Fig. 17 can be explained by similararguments.We now turn to the effect of BB interactions (see Fig. 17, open circles, upperaxis). For this purpose, we set w AB /k B T = 0 and w AA /k B T in the criticalregime. The adsorption properties corresponding to this case were discussedin Fig. 12. We start the analysis with the case w BB /k B T = w AB /k B T = 0,where θ C = 3 /
4. As w BB /k B T is increased, θ C remains constant. In this case,the phase corresponds to a c (2 ×
2) phase of A particles complemented witha partial coverage of B particles equal to 1 / • In Fig. 2, the total isotherms have a pronounced plateau at θ = 3 / w AA /k B T < The figure shows the upper part of each coexistence curve. A complete analy-sis of the curve should require a more complex study (percolation of phase, zero–temperature calculations, etc.), which are out of the scope of the present paper. Note that 1 / c (2 ×
2) represents 1 /
9. This result is reflected in Fig. 17. • Figs. 3 and 4 correspond to particular cases in Fig. 17. • Figs. 5 and 9 can be explained by combining the results in Fig. 17. • Figs. 8, 12 and 13 were discussed in details above. • Finally, no phase transition develops in the system when k B T /w AA > . Using Monte Carlo simulations, we have studied the adsorption of a gas mix-ture of interacting particles A and B on homogeneous square surfaces. A va-riety of behaviors arise due to the formation of different ordered structuresin the adlayer for different values of the lateral interactions among adsorbedparticles. The analysis of partial and total adsorption isotherms and differen-tial heats of adsorption provides a detailed understanding of the adsorptionprocess. This study yields to the construction of a phase-diagram which allowsto understand the critical behavior of the system.An interesting feature of this work is the occurrence of a minimum in theglobal adsorption isotherm. This singularity appears as the initial fraction ofB particles on the surface is high (0 . ≤ θ iB ≤ .
5) and w AB /k B T >
0. Thiseffect might be interesting due to the fact that seems counterintuitive to see anegative slope (therefore a minimum) in the total coverage. Nevertheless, thephenomenon is because the A particles adsorbing in the lattice expel the Bparticles at a higher ratio; then the partial A [B] coverage increases [decreases](partial isotherms must cross). In the above regime, the desorbed B particlesare more than the adsorbed A particles, therefore we see a total coverage ( θ )with a negative slope followed by a minimum.The computational technique used here has proven to be very powerful toolfor these kind of lattice-gas models and many other systems in a much widerscope of sciences, allowing the interpretation of many experimental resultswithout heavy or time-consuming calculations. This work was supported in part by CONICET (Argentina) under projectnumber PIP 112-200801-01332; Universidad Nacional de San Luis (Argentina)under project 322000; Universidad Tecnol´ogica Nacional, Facultad RegionalSan Rafael (Argentina) under projects PQPRSR 858 and PQCOSR 526 and10he National Agency of Scientific and Technological Promotion (Argentina)under project 33328 PICT 2005. 11 igure Captions
Fig. 1: Adsorption isotherms for the single-gas adsorption of A particles ontothe surface showing the effect of lateral AA interactions.Fig. 2: Mixed-gas adsorption on a square lattice: (a) adsorption isothermsfor A and B particles; (b) total adsorption isotherms; (c) differential heat ofadsorption for A particles and (d) differential heat of adsorption for B particles.Effect of AA interactions: w AA /k B T ≥ w BB /k B T = 0 and w AB /k B T = 0.Fig. 3: As Fig. 2 for w AA /k B T ≥ w BB /k B T = 2 and w AB /k B T = 0.Fig. 4: As Fig. 2 for w AA /k B T ≥ w BB /k B T = 0 and w AB /k B T = 2.Fig. 5: As Fig. 2 for w AA /k B T ≥ w BB /k B T = 2 and w AB /k B T = 2.Fig. 6: Mixed-gas adsorption on a square lattice: (a) partial adsorption isothermsfor A and B particles; (b) total adsorption isotherms; (c) differential heat of ad-sorption for A particles and (d) differential heat of adsorption for B particles.Effect of AB interactions: w AA /k B T = 0, w BB /k B T = 0 and w AB /k B T ≥ w AA /k B T = 0, w BB /k B T = 2 and w AB /k B T ≥ w AA /k B T = 5, w BB /k B T = 0 and w AB /k B T ≥ w AA /k B T = 5, w BB /k B T = 2 and w AB /k B T ≥ w AA /k B T = 0, w BB /k B T ≥ w AB /k B T = 0.Fig. 11: As Fig. 10 for w AA /k B T = 0, w BB /k B T ≥ w AB /k B T = 2.Fig. 12: As Fig. 10 for w AA /k B T = 5, w BB /k B T ≥ w AB /k B T = 0.Fig. 13: As Fig. 10 for w AA /k B T = 5, w BB /k B T ≥ w AB /k B T = 2.Fig. 14: Temperature-coverage phase diagram corresponding to a lattice-gasof repulsive monomers ( w AA /k B T >
0) adsorbed on a homogeneous squarelattice.Fig. 15: Comparison between the total isotherms in Fig. 13 b) (symbols) andthe one corresponding to a square lattice-gas of one species with w AA /k B T = 5(solid line). The inset shows a snapshot of the c (2 ×
2) phase.12ig. 16: Effect of the lateral interactions between A-B particles, w AB /k B T , onthe temperature-coverage phase diagram corresponding to a binary mixturewith w BB /k B T = 0 and w AA /k B T in the critical regime.Fig. 17: Densities corresponding to the maxima of the coexistence curves inzone II as a function of ( i ) w AB /k B T (solid circles, bottom axis) and ( ii ) w BB (open circles, upper axis). In case ( i ) [( ii )], w BB /k B T = 0 [ w AB /k B T = 0] and w AA /k B T is chosen in the critical regime.13 eferences [1] D. M. Ruthven, Principles of Adsorption and Adsorption Processes, Wiley, NewYork, 1984.[2] R. T. Yang, Gas Separation by Adsorption Processes, Butterworth, London,1987.[3] L. K. Doraiswamy, Prog. Surf. Sci. 37 (1990) 1.[4] W. Rudzinski, W. A. Steele, G. Zgrablich, Equilibria and Dynamics ofGas Adsorption on Heterogeneous Solid Surfaces, Elsevier, Amsterdam, TheNetherlands, 1997.[5] S. Sircar, Langmuir 7 (1991) 3065.[6] M. Heuchel, R. Q. Snurr, E. Buss, Langmuir 13 (1997) 6795.[7] K. Ayache, S. E. Jalili, L.J. Dunne, G. Manos, Z. Du, Chem. Phys. Lett. 362(2002) 414.[8] L. J. Dunne, G. Manos, Z. Du, Chem. Phys. Lett. 377 (2003) 551.[9] M. D´avila, J. L. Riccardo, A. J. Ramirez-Pastor, J. Chem. Phys. 130 (2009)174715.[10] M. D´avila, J. L. Riccardo, A. J. Ramirez-Pastor, Chem. Phys. Lett. 477 (2009)402.[11] A.L. Myers, Molecular thermodynamics of adsorption of gas and liquidmixtures, in : A. I. Liapis (Ed.), Fundamental of Adsorption, EngineeringFoundation, 1987.[12] B. Smit, T. L. M. Maesen, Chem. Rev. 108 (2008) 4125.[13] F. Gonzalez-Cavallero, M.L. Kerkeb, Langmuir 10 (1994) 1268.[14] J. A. Dunne, R. Mariwala, M. Rao, S. Sircar, R. J. Gorte, A. L. Myers, Langmuir12 (1996) 5888.[15] J. A.Dunne, M. Rao, S. Sircar, R. J. Gorte, A. L. Myers, Langmuir 12 (1996)5896.[16] W. A. Steele, The Interaction of Gases with Solid Surfaces, Pergamon Press,New York, 1974.[17] E. Alison Flood, The Solid-Gas Interface, M. Dekker Inc., New York, 1967.[18] V. P. Zhdanov, Elementary Physicochemical Processes on Solid Surfaces,Plenum Press, New York/London, 1991.[19] K. Binder (Ed.), Monte Carlo Methods in Statistical Physics. Topics in CurrentPhysics, vol. 7, Springer, Berlin, 1978.
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Fig. 1: Garcia et al. A = 0.5 A = 0.226 A A / k B T .5 0.6 0.7 0.8 0.9 1.0-0.4-0.20.00.20.4 Fig. 2d: Garcia et al. (d) q B w AA / k B T = 0 w AA / k B T = 1 w AA / k B T = 2 w AA / k B T = 3 w AA / k B T = 4 w AA / k B T = 5 w AA / k B T = 6 Fig. 2c: Garcia et al. (c) q A -5 0 5 10 15 20 250.50.60.70.80.91.0 (b) A / k B T Fig. 2b: Garcia et al. -10 -5 0 5 10 15 20 250.00.20.40.60.81.0 BA w BB / k B T = 0 w AB / k B T = 0 A / k B T AB (a) Fig. 2a: Garcia et al. .2 0.4 0.6 0.8 1.0-0.4-0.20.00.20.4
Fig. 3d: Garcia et al. (d) q B w AA / k B T = 0 w AA / k B T = 1 w AA / k B T = 2 w AA / k B T = 3 w AA / k B T = 4 w AA / k B T = 5 w AA / k B T = 6 (c) Fig. 3c: Garcia et al. q A -10 -5 0 5 10 15 20 250.20.40.60.81.0 Fig. 3b: Garcia et al. (b) A / k B T -10 -5 0 5 10 15 20 250.00.20.40.60.81.0 (a) Fig. 3a: Garcia et al. BA w BB / k B T = 2 w AB / k B T = 0 A / k B T AB .4 0.5 0.6 0.7 0.8 0.9 1.0-24-16-80 Fig. 4d: Garcia et al. (d) q B A q B Fig. 4c: Garcia et al. (c) q A -10 -5 0 5 10 15 20 250.00.20.40.60.81.0 -3 0 30.400.440.48 Fig. 4b: Garcia et al. A / k B T (b) -10 -5 0 5 10 15 20 250.00.20.40.60.81.0 (a) Fig. 4a: Garcia et al.
B A w BB / k B T = 0 w AB / k B T = 2 A / k B T AB .2 0.4 0.6 0.8 1.0-10-8-6-4-202 Fig. 5d: Garcia et al. (d) q B (c) Fig. 5c: Garcia et al. q A w BB / k B T = 2 w AB / k B T = 2 -10 -5 0 5 10 15 20 250.20.30.40.50.60.70.80.91.01.1 (b) Fig. 5b: Garcia et al. A / k B T w BB / k B T = 2 w AB / k B T = 2 -10 -5 0 5 10 15 20 250.00.20.40.60.81.0 Fig. 5a: Garcia et al.
B A w BB / k B T = 2 w AB / k B T = 2 A / k B T AB (a) .4 0.6 0.8 1.0-12-8-40 Fig. 6d: Garcia et al. (d) q B q B Fig. 6c: Garcia et al. w AA / k B T =0 w BB / k B T =0 (c) q A -5 -4 -3 -2 -1 0 1 2 3 4 50.40.50.60.70.80.91.0 A / k B T Fig. 6b: Garcia et al. (b) w AA / k B T = 0 w BB / k B T = 0 -10 -5 0 5 10 15 20 250.00.20.40.60.81.0 (a) Fig. 6a: Garcia et al.
AB A / k B T BA w AA / k B T = 0 w BB / k B T = 0 .2 0.4 0.6 0.8 1.0-12-8-40 q B Fig. 7d: Garcia et al. (d)
Fig. 7c: Garcia et al. (c) q A w AA / k B T = 0 w BB / k B T = 2 -5 -4 -3 -2 -1 0 1 2 3 4 50.20.30.40.50.60.70.80.91.0 Fig. 7b: Garcia et al. A / k B T (b) w AA / k B T = 0 w BB / k B T = 2 -10 -5 0 5 100.00.20.40.60.81.0 (a) w AA / k B T = 0 w BB / k B T = 2 A Fig. 7a: Garcia et al.
AB A / k B T B .3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-25-20-15-10-50 Fig. 8d: Garcia et al. (d) q B w AA / k B T = 5 w BB / k B T = 0 q A Fig. 8c: Garcia et al. (c) w AA / k B T = 0 w BB / k B T = 5 -10 -5 0 5 10 15 20 250.30.40.50.60.70.80.91.0 Fig. 8b: Garcia et al. A / k B T (b) w AA / k B T = 5 w BB / k B T = 0 -10 -5 0 5 10 15 20 250.00.20.40.60.81.0 -4 0 40.00.20.4 (a) A w AA / k B T = 5 w BB / k B T = 0 B Fig. 8a: Garcia et al. AB A / k B T .4 0.6 0.8 1.0-24-16-80 q B Fig. 9d: Garcia et al. (d) q A Fig. 9c: Garcia et al. (c) w AA / k B T = 5 w BB / k B T = 2 -10 -5 0 5 10 15 20 250.20.30.40.50.60.70.80.91.01.1 Fig. 9b: Garcia et al. A / k B T (b) w AA / k B T = 5 w BB / k B T = 2 -10 -5 0 5 10 15 20 250.00.20.40.60.81.0 (a) A / k B T w AA / k B T = 5 w BB / k B T = 2 Fig. 9a: Garcia et al. AB BA .2 0.4 0.6 0.8 1.0-0.8-0.6-0.4-0.20.0 Fig. 10d: Garcia et al. (d) q B Fig. 10c: Garcia et al. q A (c) w AA / k B T = 0 w AB / k B T = 0 -10 -5 0 5 100.20.30.40.50.60.70.80.91.01.1 Fig. 10b: Garcia et al. A / k B T (b) w AA / k B T = 0 w AB / k B T = 0 -10 -5 0 5 100.00.20.40.60.81.0 (a) Fig. 10a: Garcia et al. w AA / k B T = 0 w AB / k B T = 0 A AB B A / k B T .2 0.4 0.6 0.8 1.0-8-6-4-20 q B Fig. 11d: Garcia et al. (d)
Fig.11c: Garcia et al. (c) q A w AA / k B T = 0 w AB / k B T = 2 -5 -4 -3 -2 -1 0 1 2 3 4 50.20.30.40.50.60.70.80.91.01.1 (b) A / k B T Fig. 11b: Garcia et al. w AA / k B T = 0 w AB / k B T = 2 -5 -4 -3 -2 -1 0 1 2 3 4 50.00.20.40.60.81.0 (a) w AA / k B T = 0 w AB / k B T = 2 AB B A A / k B T Fig. 11a: Garcia et al.
10 -5 0 5 10 15 20 25 300.00.20.40.60.81.0
Fig. 12b: Garcia et al. A / k B T (b) w BB / k B T = 5 w AB / k B T = 0 Fig. 12d: Garcia et al. (d) q B Fig. 12c: Garcia et al. (c) w AA / k B T = 5 w AB / k B T = 0 q A -10 -5 0 5 10 15 20 250.00.20.40.60.81.0 Fig. 12a: Garcia et al. (a) AB BA w BB / k B T = 5 w AB / k B T = 0 A / k B T .2 0.4 0.6 0.8 1.0-10-8-6-4-20 Fig. 13d: Garcia et al. (d) q B Fig. 13c: Garcia et al. q A (c) w AA / k B T = 5 w AB / k B T = 2 -15 -10 -5 0 5 10 15 20 25 300.20.40.60.81.0 (b) Fig. 13b: Garcia et al. A / k B T w AA / k B T =5 w AB / k B T =2 -15 -10 -5 0 5 10 15 20 25 300.00.20.40.60.81.0 (a) w AA / k B T = 5 w AB / k B T = 2 AB B A A / k B T Fig. 13a: Garcia et al. .0 0.2 0.4 0.6 0.8 1.00.00.20.40.6
Fig. 14: Garcia et al. k B T C / w AA IIIIII θ θ µ A / k B T w AA / k B T = 5 w AB / k B T = 2 c(2 x 2) Fig. 15: Garcia et al. .0 0.5 1.0 1.5 2.00.10.20.30.4 1.0 0.8 0.6 0.4 0.2 0.0
Fig. 16: Garcia et al. k B T C / w AA θ w AB / k B T w AA / k B T = w BB / k B T = Fig. 17: Garcia et al. w BB / k B T C w AB / k B T k B T C / w AA <<