Monte Carlo study of multicomponent adsorption on triangular lattices

Surface Science 602 (2008) 1783–1794

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Monte Carlo study of multicomponent adsorption on triangular lattices P. Rinaldi, F. Bulnes, A.J. Ramirez-Pastor *, G. Zgrablich Departamento de Física, Instituto de Física Aplicada, Universidad Nacional de San Luis – CONICET, Chacabuco 917, 5700 San Luis, Argentina

a r t i c l e

i n f o

Article history: Received 20 September 2007 Accepted for publication 17 March 2008 Available online 25 March 2008 Keywords: Gas mixture adsorption Lattice-gas models Monte Carlo simulation

a b s t r a c t The adsorption process of interacting binary gas mixtures containing particles A and B on triangular substrates is studied through grand canonical Monte Carlo simulation in the framework of the lattice-gas model. The energies involved in the adsorption process are four: (1) 0 , interaction energy between a monomer (type A or B) and a lattice site; (2) wAA , nearest-neighbor interaction energy between two A particles; (3) wAB (=wBA ), nearest-neighbor interaction energy between an A particle and a B particle and (4) wBB , nearest-neighbor interaction energy between two B particles. The process is monitored through partial and total isotherms, differential heats of adsorption and energy of the system, which appear as very sensitive to all lateral interactions. We focus on the case of repulsive lateral interactions, where a rich variety of structural orderings are observed in the adlayer, depending on the value of the parameters wAA , wAB and wBB . Results are rationalized through the determination of the phase diagrams characterizing second order phase transitions in the system. A nontrivial interdependence between the partial surface coverage of both species is observed. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction Physical adsorption of gas mixtures is of central importance both from a practical point of view for the design of adsorbents and on a theoretical basis for the understanding of the interactions between gases and solid surfaces [1–7]. For a complete analysis of the behavior of gas molecules under the influence of an adsorbent, the forces of molecular interactions must be quantitatively treated [8–14]. In other words, the description of real multicomponent adsorption requires to take into account the effect of the interactions between adsorbed particles. In the context of the lattice-gas model, this analysis involves a set of energy parameters wAi ;Aj ðrÞ, to describe the lateral interaction between a pair of adparticles Ai and Aj separated by a distance r. An exact statistical mechanical treatment, including lateral interactions, is unfortunately not yet available and, therefore, the theoretical description of adsorption relies on simplified models [1]. One way of overcoming this complication is to use Monte Carlo (MC) simulation method [15–17]. MC technique is a valuable tool for studying surface molecular processes, which has been extensively used to simulate many surface phenomena including adsorption [18], diffusion [19], reactions, phase transitions [20], etc. In previous papers, the effects of lateral interactions and surface heterogeneity on the adsorption of a binary mixture were studied for the case of square substrates [21,22]. In contrast to the statistics * Corresponding author. Tel.: +54 2652 436151; fax: +54 2652 430224. E-mail address: [email protected] (A.J. Ramirez-Pastor). 0039-6028/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2008.03.014

for non-interacting particles, the degeneracy of arrangements of interacting particles is strongly influenced by the structure of the lattice space. Because the structure of lattice space plays such a fundamental role in determining the statistics of interacting species, it is of interest and of value to inquire how a specific lattice structure influences the main thermodynamic properties of adsorbed mixtures. In particular, since the (1 1 1) facet is the most frequently exposed facet in supported metallic nanoparticles (supported catalytic phase) and is also representative of many metallic-oxide surfaces, it is important to consider lattices with triangular geometry. In this sense, the aim of the present work is: (1) to extend previous studies [21,22] to triangular lattices and (2) to discuss the effect of the lattice structure on the adsorption of interacting mixtures. For this purpose, interacting multicomponent gases adsorbed on triangular lattices are studied by using grand canonical ensemble Monte Carlo (GCEMC) simulation [15–17, 21,22]. The process is monitored by following the total and partial isotherms and the differential heats of adsorption corresponding to both species of the mixture. Special interest is devoted to the repulsive case, where different ordered structures in the adsorbate are observed for low temperatures. The model to be considered here is a particular case of the Blume–Emery–Griffiths (BEG) model [23], which is a very general model, used in a variety of phenomena ranging from liquid helium phase separation to phase transitions in adsorbed films [23,24], whose phase diagram has been extensively studied [25–28]. In our case of adsorption of a binary gas-mixture, the detailed behavior of adsorption isotherms and heats of adsorption will be shown to be directly related to the phase diagrams of the system. The

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(iv) Choose randomly one of the M sites, i, and generate a random number n 2 ½0; 1. (a) if the site i is empty, and n 6 W ads , then adsorb an X particle on i, otherwise, the transition is rejected. (b) if the site i is occupied by an X particle, and n 6 W des , then the X molecule is desorbed from i, otherwise, the transition is rejected. (v) Repeat from (iii) M times.

work is organized as follows: In Section 2, the basic definitions are given along with the general basis of the MC simulation. Results and discussions are presented in Section 3. Finally, the conclusions are given in Section 4. 2. Model and Monte Carlo simulation In the framework of the lattice-gas approximation, we assume that the homogeneous surface is represented by a two-dimensional triangular lattice of M ¼ L  L adsorption sites, with periodic boundary conditions. The substrate is exposed to an ideal A–B mixed-gas phase, at temperature T and chemical potentials lA and lB . Particles can be adsorbed on the substrate with the restriction of at most one adsorbed particle per site and we only consider nearest-neighbor (NN) interaction energies. Let us introduce the occupancy variable ci which can take the values ci ¼ 0 if the site i is empty, and ci ¼ 1 [ ci ¼ 1] if the site i is occupied by an A-atom [ B-atom]. The energy parameters of the model are: (a) 0 , interaction energy between a monomer (type A or B) and a lattice site. (b) wAA , lateral energy interaction between a NN pair A–A. (c) wBB , lateral energy interaction between a NN pair B–B. (d) wAB (=wBA ), lateral energy interaction between a NN pair A–B. Under these considerations, the adsorbed phase is characterized by the Hamiltonian: H¼

M X 1X ½wAA dci ;cl ;1 þ wBB dci ;cl ;1 þ wAB ðdci ;1 dcl ;1 þ dci ;1 dcl ;1 Þ 2 i l2fNN;ig M M X X þ 0 ðdci ;1 þ dci ;1 Þ  ðlA dci ;1 þ lB dci ;1 Þ i

W ads and W des are the transition probabilities from a state with N particles to a new state with N þ 1 and N  1 particles, respectively. By following the Metropolis scheme, these probabilities are given by W i!f ¼ ½1; expðbDHÞ

where DH ¼ Hf  Hi is the difference between the Hamiltonians of the final and initial states, and b ¼ 1=kB T, with kB the Boltzmann constant. It has been supposed that there is no diffusion in the adsorbed phase. However, diffusion could be qualitatively taken into account by introducing an additional elementary step consisting in moving an adsorbed molecule to a new position chosen randomly within the adsorbed phase volume V. The approximation to thermodynamical equilibrium is monitored through the fluctuations in the number N of adsorbed particles. The first m0 MCS are discarded in order to reach equilibrium; after that, mean values of thermodynamic quantities, like total and partial isotherms are obtained as simple averages over m successive uncorrelated configurations: hðlA ; lB Þ ¼

ð1Þ

i

where d is the Kronecker delta and ‘‘l 2 fNN;ig” means that for a given site i, the sum runs over its six nearest-neighbor sites. In this contribution, the chemical potential of one of the components is fixed throughout the process (lB ¼ 0), while the other one is variable, as it is usually assumed in studies of adsorption of gas mixtures [29]. In addition, we focus on the case of repulsive lateral interactions among adsorbed particles (wij > 0) since, as we shall see, different structures appear in the adsorbed phase in correspondence with the antiferromagnetic Blume–Emery–Griffiths model. Finally, 0 is set equal zero, without any loss of generality. The adsorption process of an ideal binary gas-mixture on a substrate is simulated through a GCEMC method, by following the Metropolis algorithm [30]. In adsorption–desorption equilibrium there are two elementary ways to produce a change in the system state, namely, adsorbing one molecule onto the surface (adding one molecule into the adsorbed phase volume V), and desorbing one molecule from the adsorbed phase (removing one molecule from the volume V). For a given value of the temperature T and chemical potentials lA and lB , an initial configuration of adparticles is generated. Then an adsorption–desorption process is started, where a site is chosen at random and an attempt is made to change its occupancy state according to the Metropolis scheme of probabilities. A Monte Carlo Step (MCS) is achieved when M sites have been tested to change its occupancy state. The basic algorithm to carry out an elementary MCS during the simulation can be summarized as follows: (i) Set the value of lA , lB and temperature T. (ii) Set an initial state xN by placing randomly N molecules onto the lattice. (iii) Choose randomly one of the components of the mixture ! X ( X  A or B).

ð2Þ

hNi

hA ðlA ; lB Þ ¼

L2

hN A i L2

hB ðlA ; lB Þ ¼

hN B i L2

ð3Þ

where h, hA and hB are the total and partial surface coverages, defined, respectively as h ¼ N=M, hA ¼ NA =M, hB ¼ NB =M; the brackets h. . .i denote averages over statistically uncorrelated configurations. The differential heat of adsorption qi for the i-specie is defined as [31–33]   oU qi ¼  oN i T;Nj6¼i

ð4Þ

where U is the energy of the adsorbed phase. The RHS part of the last equation can be written as 

oU oN i

 ¼ T;Nj6¼i

( X  k

)    oU oðlk =kB T Þ oðlk =kB T Þ T;lj6¼i oN i T;N j6¼i

ð5Þ

Expressing both derivatives in the RHS of this equation as fluctuations in the grand canonical ensemble we finally obtain the following forms for the differential heats of adsorption [21,22] h i UðAÞ hN B i  hN B i2  UðBÞ½hN A N B i  hN A ihN B i ih i qA ¼  h ð6Þ hN 2A i  hN A i2 hN 2B i  hN B i2  ½hN A N B i  hN A ihN B i2 h i UðBÞ hN A i  hN A i2  UðAÞ½hN A N B i  hN A ihN B i ih i qB ¼  h ð7Þ hN 2A i  hN A i2 hN 2B i  hN B i2  ½hN A N B i  hN A ihN B i2 where UðaÞ ¼ hUNa i  hUihN a i

ð8Þ

During the simulations, the process is monitored through the total and partial isotherms, hðlA Þ, hA ðlA Þ and hB ðlA Þ, respectively, and the differential heats of adsorption qA ðhA Þ and qB ðhA Þ.

P. Rinaldi et al. / Surface Science 602 (2008) 1783–1794

3. Results and discussion 3.1. Adsorption properties The computational simulations have been developed for triangular L  L lattices, with L ¼ 96, and periodic boundary conditions. With this lattice size we verified that finite-size effects are negligible. Note, however, that the linear dimension L has to be properly chosen such that the adlayer structure is not perturbed. As a basis for the analysis of the behavior of the system, we begin by considering the case of single-gas adsorption on a homogeneous surface. This can be achieved, for example, by making lB ! 1. Fig. 1 shows the behavior of the adsorption isotherms and the differential heats of adsorption for different strengths of repulsive interparticle interactions. As expected, we obtain the well-known Langmuir isotherm passing through the point (lA ¼ 0; hA ¼ 1=2) when wAA ¼ 0. Two features which are useful for the analysis of mixed-gas adsorption are worthy of comment: (a) as the repulsive NN interaction increases, the coverage at zero chemical potential decreases and approaches hA ¼ 0:162 asymptotically; (b) as the repulsive NN interaction passes a critical value kB T c =w ¼ 0:3354ð1Þ [34], two plateaus develop in the isotherm at hp ¼ 1=3 A ffiffiffi pffiffiffi and 2=3, indicating the appearance pffiffiffi pof ffiffiffi an ordered ð 3  3Þ phase and its complementary ð 3  3Þ . This phase transition is also announced by characteristic signals in the differential heat of adsorption. In fact, a plateau in the isotherm appears

A

1.0

0.5

wAA / kBT

0 3 5 10

0.0 0

20

μA

40

60

0

A

-20

-40

-60 0.0

0.2

0.4

0.6

0.8

1.0

θA Fig. 1. (a) Adsorption isotherms and (b) differential heats of adsorption for the single-gas adsorption of A particles onto a homogeneous surface showing the effect of lateral A—A interactions.

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as a step in the differential heat of adsorption. This behavior is shown in Fig. 1b. In what follows, we consider mixed-gas adsorption but keeping species B at a fixed value of the chemical potential lB ¼ 0. We start with the analysis of the adsorption of a mixture of molecules A and B on a homogeneous surface and focus on the effects of lateral interactions among the different adsorbed species. The effect of AA interactions is depicted in Fig. 2, where wAB ¼ wBB ¼ 0. We have plotted the partial (a) and total (b) adsorption isotherms, and the differential heats of adsorption corresponding to the species A (c) and B (d). For lA ! 1 the state of the system is the following: hA ¼ 0 and hB is the equilibrium coverage given by the Langmuir isotherm, hB ¼ expðlB =kB TÞ=½1 þ expðlA =kB TÞ þ expðlB =kB TÞ, being hB ¼ 1=2 for lB ¼ 0. As lA is increased, the A particles adsorb on the surface and the B particles reach its equilibrium coverage in the rest of the lattice. This results in a decreasing (increasing) of the B(A) isotherm. As the interaction wAA increases, two well-defined and pronounced steps appear in the partial isotherms. In the case the pffiffiffi of p ffiffiffi A species, pffiffiffi the pffiffiffi plateaus indicate the formation of the ð 3  3Þ and ð 3  3Þ structures at h ¼ 1=3 and 2=3, respectively. The behavior of qA reflects clearly the three regimes of adsorption described above. Namely, in the first regime ( 0 6 h < 1=3), the A particles do not interact with each other and the adsorption– desorption p offfiffiffi anpA ffiffiffi molecule involves qA ¼ 0 (see Fig. 2c). At h ¼ 1=3, a ð 3  3Þ structure of the A species is formed inpthe ffiffiffi adlayer; for p1=3 h < 2=3 the system changes from the ð 3 pffiffiffi ffiffiffi 6 pffiffiffi  3Þ to the ð 3  3Þ . Each incoming A particle can be allocated on the lattice at an additional energy cost of 3wAA . Finally, for 2=3 6 h 6 1, the energy cost upon adsorption of an A particle is equal to 6wAA . The adsorption of the A species induces an interesting behavior in the B isotherm, which also exhibits well-defined steps although the B particles do not interact neither with B particles nor with A particles (wAB ¼ wBB ¼ 0). This behavior is a consequence of the excluded volume but is not due to the interactions. As it is expected, qB is strictly zero over all the coverage range (Fig. 2d). The total isotherms follow the behavior of the sum of A and B isotherms. We now study the effect of AA interactions with wAB ¼ 0 and wBB ¼ 2kB T (Fig. 3); wAB ¼ 2kB T and wBB ¼ 0 (Fig. 4) and wAB ¼ 2kB T and wBB ¼ 2kB T (Fig. 5). In the first case, the behavior of the curves is similar to those in Fig. 2. The main difference is associated to the value of coverage of B at low pressures, which is lower than 0:5, and to the behavior of qB . This behavior can be understood as follows. For wAA ¼ 0, A particles are distributed at random and B particles start from a low coverage (0.2) which rapidly decreases as more A particles are adsorbed, therefore as the total coverage increases B particles see each other always less frequently and qB increases steadily. However, pffiffiffi as pffiffiffi wAA becomes pffiffiffi pffiffiffi sufficiently high so that the ordered ð 3  3Þ and ð 3  3Þ phases are formed, a sudden increase in qB is produced due to a sudden increase in the screening effect between B particles produced by A particles in each ordered phase. With respect to Fig. 4, new interesting features appear. As lA increases and A particles start to adsorb, B particles are displaced from the surface where they occupy 50% of the free sites left by A particles (Fig. 4a). The coverage of A particles is sufficiently high so that A–B interactions occur, repulsive interactions lead to more B particles being displaced from the surface than A particles are adsorbed so that the total A þ B coverage decreases and shows a local minimum (Fig. 4b). The position of the minimum varies between lA ¼ 0 for wAA ¼ 0 and lA  2:7 for wAA ! 1. Under these circumstances, there exists a wide range of h (between 0.3 and 0.5), where

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P. Rinaldi et al. / Surface Science 602 (2008) 1783–1794

1.0 0

a

-12

0.5

qA

θAB

A

c

-24

B -36

b

d 0 1 2 3 4 5 6

θ

wΑΑ /kΒ T

wΑΒ = 0 0.5

wΑΒ = 0

qB

0.0 1.0

0

-1 0

10

20

30

40

0.5

0.6

μΑ

0.7

0.8

0.9

1.0

θ

Fig. 2. Mixed-gas adsorption on a homogeneous triangular surface: (a) adsorption isotherms for A and B particles; (b) total adsorption isotherms; (c) differential heat of adsorption for A particles and (d) differential heat of adsorption for B particles. Effect of A–A interactions: wAA P 0, wBB ¼ 0 and wAB ¼ 0.

1.0 0

a

c -12

0.5

qA

θAB

A

-24

B 0.0 1.0

-36 1

θ

wΑΑ /kΒ T 0.5

wΑΒ = 0 wΒΒ = 2kΒT 0

20

0 1 2 3 4 5 6 40

d

qB

b

0

-1 0.2

0.4

μΑ

0.6

0.8

1.0

θ

Fig. 3. As Fig. 2 for wAA P 0, wBB ¼ 2kB T and wAB ¼ 0.

the total isotherm is a bi-valuated function. This effect (which we call mixture effect) is clearly reflected in the behavior of qA (Fig. 4c) and qB (Fig. 4d). In order to clarify the analysis, qA and qB are also plotted as a function of hA (see insets in Fig. 4c and d). The behavior of the differential heats allows us to complete the analysis. For 0 6 hA 6 1=3, A particles do not interact with pffiffiffi other pffiffiffi A or B molecules and, consequently, qA ¼ qB ¼ 0. The ð 3  3Þ phase of A particles starts to

develop and is completed at hA ¼ 1=3. For 1=3 6 hA 6 2=3, A particles fill the central vacancies in the triangles formed by adsorbed A particles, so that each newly adsorbed pffiffiffi pffiffiffi particle has three NN occupied sites and qA ¼ 3wAA , the ð 3  3Þ phase of A particles starts to develop and is completed at hA ¼ 2=3. In this regime, the coverage of B particles tends to zero and does not perturb significantly the adsorption of the A species (note that qA does not depend on wAB ). However, each B particle in the lattice is surrounded by three

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P. Rinaldi et al. / Surface Science 602 (2008) 1783–1794

1.0 0

a A

-15

θAB

0.5

0 1 2 3 4 5 6

wΑΒ = 2 kΒ T wΒΒ = 0

B

-15 -30

-45

0.0

θΑ

c

θΑ

d

0.5

1.0

0.5

1.0

0

b 0

θ

-30

2

μA

qB

0.0 1.0

qA 0

qA

wΑΑ /kΒ T

4

0.5

0

-15 0.45

qB -10

θ 0.30

0.0 0

10

μΑ

20

-30 30

-20 0.0

40

0.4

0.6

0.8

1.0

θ Fig. 4. As Fig. 2 for wAA P 0, wBB ¼ 0 and wAB ¼ 2kB T.

1.0

wΑΑ /kΒT

0.5

wΑΒ =2kΒT 0.0 1.0

0

A

B

wΒΒ =2kΒ T

0 1 2 3 4 5 6

c -12

qA

AB

a

-24

-36 0

b

d qB

-6 0.5 -12

-18 0

10

20

30

40

0.2

0.4

0.6

0.8

1.0

μΑ Fig. 5. As Fig. 2 for wAA P 0, wBB ¼ 2kB T and wAB ¼ 2kB T.

A particles and, consequently, qB ¼ 3wAB . Finally, for 2=3 6 hA 6 1, the remaining vacant sites are filled by A particles, so that each newly adsorbed particle has six NN occupied sites and qA ¼ 6wAA . The important fluctuations in qB are clear signals that the number of B particles is practically zero. To complete the analysis of Fig. 4, it is interesting to note that the mixture effect has been observed experimentally for methane–ethane mixtures [35]. Our simulations represent a contribution to the understanding of the phenomenon, showing how the

competition between two species in presence of repulsive mutual interactions reinforces the displacement of one species by the other and leads to a decreasing in the total coverage. In the case of Fig. 5, the presence of repulsive B–B interactions results in a initial coverage of B particles close to 0:18. This small fraction of B molecules does not perturb significantly the adsorption of the A species and the mixture effect disappears. This finding indicates that, in addition to the requirements of repulsive lateral interactions between the two species, the amount of particles on

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P. Rinaldi et al. / Surface Science 602 (2008) 1783–1794

the surface is essential for the existence of the mixture effect. The rest of the figure can be understood following the arguments given above. The effects of variable A–B interactions are now shown in Figs. 6–9. We start with the case where wAA ¼ wBB ¼ 0 (Fig. 6). Here no ordered phases are formed and for sufficiently high wAB an important mixture effect appears. Let us choose for our analysis the curves corresponding to wAB ¼ 1. B coverage is initially 0.5, as A molecules are adsorbed B molecules are eliminated from the surface in such a way that h decreases. At the same time qA starts at a low value and increases rapidly (also with a bi-valuated behavior) tending to 0 as h ! 1, while qB starts near 0 and tends to 6wAB as h ! 1. As soon as B–B interactions are added, the starting B coverage is 0.2 and the mixture effect disappears (Fig. 7). In the presence of A–A and A–B interactions, Fig. 8, wAB ¼ 0:5 is sufficiently high for the appearance of ordered phases for A molecules, so that the resulting behavior is similar to that of Fig. 4. The inclusion of the three interactions (Fig. 9), only produces the elimination of the mixture effect, compared to Fig. 8, due to the low initial B coverage. The effect of variable B–B interactions is discussed in Figs. 10– 13. As in previous study, different cases were considered: wAA ¼ wAB ¼ 0 (Fig. 10); wAA ¼ 0 and wAB ¼ 2kB T (Fig. 11); wAA ¼ 5kB T and wAB ¼ 0 (Fig. 12) and wAA ¼ 5kB T and wAB ¼ 2kB T (Fig. 13). Given the value of the parameters in Fig. 10, neither the coverage of A (Fig. 10a) nor the differential heat of adsorption (Fig. 10c) are affected by B–B interactions. As discussed in Fig. 1, the coverage of B at low pressure starts at 0.5 and decreases as wBB increases towards the limiting value of 0:162 (see Fig. 10a). However, the coverage of B does not present any special features. Such a special feature does indeed appear in the behavior of qB (Fig. 10d), which decreases steadily as wBB increases below a certain critical value, wcBB (being wcBB  1:5), and increases above it. The explanation for this behavior is that B particles adsorb more or less at random below wcBB , thereby allowing some B–B interactions which contribute to the decrease in the differential heat of adsorption. Above wcBB , B particles adsorb forming an ordered structure so that B–B interac-

1.0

tions stop contributing and qB increases. The condition wAB ¼ 0 restricts the possibility of mixture effect. In Fig. 11, the presence of A–B interactions favors the displacement of B particles and, consequently, the slopes of the B partial isotherms are increased. On the other hand, as the initial fraction of B particles is high (0:3 6 hiB 6 0:5), which occurs for small values of wBB =kB T ( 0 6 wBB =kB T 6 0:5), A–B interactions lead to mixture effect. In a wide range of coverage, 0:4 6 h 6 1, the curves in Fig. 11 are very similar between them (this is clearly visualized in the case qA and qB ), which is indicative of the rapid decreasing of the number of B particles on the surface. In Fig. 12, the adsorption of A particles (Fig. 12a) follows a unique isotherm and is independent of the strength of B–B interactions. Adsorption of B particles decreases at low A coverage with increasing values of wBB and tends to the limiting value hB ¼ 0:162. However, at high A coverage all isotherms tend to that corresponding to wBB ¼ 0 because there are no NN vacant sites in that range available for the adsorption of B particles. This determines the total coverage behavior shown in Fig. 12 b. The curves of qA , shown in Fig. 12c, present a very similar behavior between them. This is, three marked jumps appear, corresponding to the plateaus in A isotherms. As wBB =kB T is increased, the number of B particles on the lattice diminishes, and consequently, the width of the first plateau decreases. In contrast, qB presents two types of behaviors (Fig. 12d). Thus, at very low wBB values (negligible interactions) and very high wBB values (above the critical value for the formation of the ordered phase), it remains practically constant, while at intermediate values it increases in line with the total coverage. In the last case (Fig. 13), the adsorption process can be easily understood: the B particles disappear for low values of lA . Then, for higher lA ’s, all curves collapse on a unique curve. Finally, it is interesting to compare the results presented here with those previously obtained for square lattices [22]. In Ref. [22], the adsorption of a gaseous mixture containing particles A and B was studied via lattice-gas simulations on homogeneous and heterogeneous square surfaces. In the homogeneous case, three sets of lateral interactions were considered: (i) wAA > 0,

a

0 0.45

A

θ

0.50

0.5

-1

qΑ

qA

θAB

0

-1

B

c

0.0 1.0

-2 0

wΑΒ /kΒ T 0.25 0.5 2 4 6

qB

θ

0.8

0; 0.37; 1; 3; 5;

wΑΑ= 0

0.6

-4

θ

0.45

0.50

0.0

qB

-8

-0.5

wΒΒ = 0

b 0.4 0

μΑ

2

4

d

-1.0

-12 -2

0.55

0.4

0.6

0.8

1.0

θ

Fig. 6. Mixed-gas adsorption on a homogeneous triangular surface: (a) adsorption isotherms for A and B particles; (b) total adsorption isotherms; (c) differential heat of adsorption for A particles and (d) differential heat of adsorption for B particles. Effect of A–B interactions: wAA ¼ 0, wBB ¼ 0 and wAB P 0.

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P. Rinaldi et al. / Surface Science 602 (2008) 1783–1794

1.0

0.0

a

-0.4 0.5

qA

θAB

A

-0.8

B 0

b wAB /kBT

wAA= 0 θ

c

0 0.25 0.5 1 2 3 6

wBB= 2 k T

0.5

-6

wAA /kBT

qB

0.0 1.0

4 5

-12

d -18

0.0 -2

0

μΑ

2

0.2

4

0.4

0.6

0.8

θ

1.0

Fig. 7. As Fig. 6 for wAA ¼ 0, wBB ¼ 2kB T and wAB P 0.

1.0

0.5

0

θAB 0.5

0

μΑ

-10

5

qA

θAB

A 0.0

-20

a

B

c -30

0.0 1.0

0

wA = 5kBT

θ

qB

wBB= 0 wAB /kBT 2 3 4 5 6

0 0.25 0.37 0.5 1

0.5

b 0

10

μA

20

30

-20

d

-40 0.0

0.2

0.4

θ

0.6

0.8

1.0

Fig. 8. As Fig. 6 for wAA ¼ 5kB T, wBB ¼ 0 and wAB P 0.

wAB ¼ 0 and wBB ¼ 0; (ii) wAA ¼ 0, wAB ¼ 0 and wBB > 0; and (iii) wAA ¼ 0, wAB > 0 and wBB ¼ 0. The results obtained in (i), (ii) and (iii) are in qualitative agreement with the corresponding data shown in Figs. 2, 10 and 6, respectively. Of special interest is the case (iii), where the mixture effect was also observed for square lattices. The present study shows that this phenomenon is more marked for triangular lattices. The explanation for this behavior is simple: the number of B particles displaced from the surface by A particles increases as the connectivity is increased.

3.2. Phase diagrams In order to rationalize the results presented in previous section, we will determine the temperature-coverage phase diagram characterizing our system in the range of the parameters studied. The curves will be obtained as a generalization of the well-known phase diagram for a lattice-gas of repulsive monomers adsorbed on a homogeneous triangular lattice, which is shown in Fig. 14 [36–39].

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P. Rinaldi et al. / Surface Science 602 (2008) 1783–1794

1.0

0

a

c

A 0.5

qA

θAB

-10

wAB /kBT

-20

B

0.37 0.5 -30

0.0

0

wAA= 5kBT -20

0.5

wAB /kBT

qB

θ

wBB= 2kBT

0 0.25 0.5 1

b

2 3 4 6

-40

d

0.0

-60 0

10

μA

20

30

0.2

0.4

0.6

0.8

1.0

θ

Fig. 9. As Fig. 6 for wAA ¼ 5kB T, wBB ¼ 2kB T and wAB P 0.

1.0

a

0.5

c

A θAB

qA

0.5

B

-0.5

0.0

0.0

b

wAA= 0 wBB /kBT 0 0.25 0.5 1 2 3 6

0.5

0.0

d

-0.4

qB

wAB= 0 θ

0.0

-0.8

-1.2 -4

-2

0

2

4

μA

0.2

0.4

0.6

0.8

1.0

θ

Fig. 10. Mixed-gas adsorption on a homogeneous triangular surface: (a) adsorption isotherms for A and B particles; (b) total adsorption isotherms; (c) differential heat of adsorption for A particles and (d) differential heat of adsorption for B particles. Effect of B–B interactions: wAA ¼ 0, wBB P 0 and wAB ¼ 0.

As it was discussed in Ref. [34], the maxima of the coexistence curves (occurring to h ¼ 1=3 and h ¼ 2=3) correspond to a critical value kB T c =wAA ¼ 0:3354ð1Þ. On the other hand, zones II, III, pffiffiffi I,pffiffiffi and IV correspond to a disordered lattice-gas state, a ð 3  3 Þ orpffiffiffi pffiffiffi dered structure, its complementary ð 3  3Þ phase, and a disordered lattice-liquid state, respectively. We start with the analysis of Fig. 13, where our model is almost identical to a lattice-gas of one species. As indicated in Fig. 13, the existence of repulsive A–B interactions favors the displacement of

B particles. Once hB  0, which occurs at lA  4, the binary mixture is equivalent to the triangular lattice-gas of one species. To corroborate this affirmation, Fig. 15 shows the total isotherms of Fig. 13b (symbols), in comparison with the adsorption isotherm corresponding to a triangular lattice-gas of one species and wAA ¼ 5 (solid line). The analysis can be separated in two parts: for h < 1=3, there exist B particles on the lattice and, consequently, the curves of the mixture deviate from that corresponding to one species. For h > 1=3, the coverage of B particles is negligible and all curves col-

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P. Rinaldi et al. / Surface Science 602 (2008) 1783–1794

1.0

0

A

wAA = 0

c

wBB /kB T

wAB= 2kBT

0 0.25 0.5 1 2 3 6

θAB

0.5

B

qA

a

0.0

-1

-2

b

0

d

qB

θ

-4 0.5

-8

0.0

-12 -4

-2

0

2

4

0.2

0.4

μA

0.6

0.8

1.0

θ

Fig. 11. As Fig. 10 for wAA ¼ 0, wBB P 0 and wAB ¼ 2kB T.

1.0

0

a A qA

θAB

-10 0.5

-20

B -30

0.0

c

b 0.0

θ

0 0.25 0.5 1 2 3 6

0.5

wAA= 5kBT wAB= 0

qB

wBB /k BT -0.4

-0.8

b -1.2

0.0 0

10

μA

20

30

0.2

0.4

θ

0.6

0.8

1.0

Fig. 12. As Fig. 10 for wAA ¼ 5kB T, wBB P 0 and wAB ¼ 0.

lapse on a unique curve. Then, the phase diagram characterizing a binary mixture with the set of parameters of Fig. 13 is identical to that shown in Fig. 14. The unique difference with the one-species phase diagram is that the zone I now corresponds to an A–B mixture with different proportions according to the value of wBB . As a basis for the analysis of the behavior of the system for variable wAB , we begin by considering one of the cases of Fig. 13 (that corresponding to wBB ¼ 0), which is characterized by a phase diagram as shown in Fig. 14. This behavior is representative of systems with high values of wAB (in the range wAB P 1:5), as is

indicated in Fig. 16 (see the plane corresponding to wAB ¼ 1:5). As wAB is decreased, the maxima of the coexistence curves in zones II and III shift to higher values of coverage.1 This can be better visualized in Fig. 17, where we plot the densities corresponding to the maxima of the coexistence curves in zones II and III, hC ’s, as a function of wAB . The figure allows to analyze the behavior of a binary 1 The figure shows the upper part of each coexistence curve. A complete analysis of the curve should require a more complex study (percolation of phase, zerotemperature calculations, etc.), which are out of the scope of the present paper.

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P. Rinaldi et al. / Surface Science 602 (2008) 1783–1794

1.0

a

c

0

wAA= 5kB T

A

wAB = 2kBT

B

0.5

wBB /kBT

qA

θAB

-10

1 2 3 6

0 0.25 0.5

-20

-30

0.0

b

d

0

qB

θ

-5 0.5

-10

0.0

-15 0

10

20

μA

30

0.2

0.4

0.6

0.8

1.0

θ

Fig. 13. As Fig. 10 for wAA ¼ 5kB T, wBB P 0 and wAB ¼ 2kB T.

0.4 0.3354(1), Ref. [34]

w

AA

0.3

w

B

/k

B

T=

T=

5

0.4

0

I

0.3

IV

0.1

0.2

III

II

0.1

0.0 0.0

0.2

0.4

θ

0.6

0.8

0.0

1.0

0.0 0.5

0.8

T /kB w AB

Fig. 14. Temperature-coverage phase diagram corresponding to a lattice-gas of repulsive monomers ( wAA > 0) adsorbed on a homogeneous triangular lattice.

1.0

AA

0.2

kB T / w C

kB TC /wAA

BB

/k

0.2 0.4 1.0

0.6 0.8 1.5

wAA= 5kBT wAB= 2kBT

θ

1.0

Fig. 16. Effect of the lateral interactions between A–B particles, wAB , on the temperature-coverage phase diagram corresponding to a binary mixture with wBB ¼ 0 and wAA in the critical regime.

θ

0.6

Fig. 15. Comparison between the total isotherms in Fig. 13b (symbols) and the one corresponding to a triangular lattice-gas of one pffiffiffispecies pffiffiffi with pffiffiffi wAA pffiffiffi¼ 5 (solid line). The left [right] inset shows a snapshot of the ð 3  3Þ [ð 3  3Þ ] phase.

As it was explained for high pffiffiffivalues pffiffiffi of the A–B lateral interaction, the zone II corresponds to a ð 3  3Þ phase of A particles. As wAB decreases, the phase is complemented with B particles, pffiffiffi pffiffiffi which are randomly distributed in the empty sites of the ð 3  3Þ structure. As a typical example, we will analyze the case of wBB ¼ wAB ¼ 0. In this case, the B particles, which are at chemical lB ¼ 0, occupy pffiffiffipotential pffiffiffi 2 at random 1=2 of the empty sites of the ð 3  pffiffiffi 3Þ phase. Then, the pffiffiffi phase corresponding to zone II is formed by a ð 3  3Þ structure of A particles complemented with B particles, which occupy at random 1=3 of the total sites. The resulting value of hC is 2=3. The rest of the points in Fig. 17 can be explained by similar arguments.

mixture with wBB ¼ 0, variable wAB and wAA in the critical regime. The curves can be understood according to the following reasoning.

pffiffiffi pffiffiffi Note that 1=2 of the empty sites of the ð 3  3Þ phase represents 1=3 of the total sites.

0.4 0.2 0.0 0

10

20

30

μA / kBT

2

P. Rinaldi et al. / Surface Science 602 (2008) 1783–1794

1.00

kB TC / wAA < 0.3354 ; wBB / kBT = 0

θC

0.75

0.50

0.25

Phase II Phase III 0.00 0

1

2

1793

pffiffiffi pffiffiffi the phase corresponds to a ð 3  3Þ phase of A particles complemented with a partial coverage of B particles equal to 0.162. Figs. 14–18 allow to characterize the critical behavior of a binary mixture adsorbed in a triangular lattice in the region of the parameters studied in the present contribution. Namely:

3

wAB / kBT Fig. 17. Densities corresponding to the maxima of the coexistence curve in zones II and III as a function of wAB . In all cases, wBB ¼ 0 and wAA is chosen in the critical regime.

(i) In Fig. 2, the total isotherms have pronounced plateaus at h ¼ 2=3 and h ¼ 5=6 for strongly repulsive A–A interactions, which smoothes out already for wAA < 3. This result is reflected in Figs. 17 and 18. (ii) Figs. 3 and 4 correspond to particular cases in Figs. 18 and 17, respectively. (iii) Figs. 5 and 9 can be explained by combining the results in Figs. 17 and 18. (iv) Figs. 8, 12 and 13 were discussed in details above. (v) Finally, no phase transition develops in the system when kB T=wAA > 0:3354. This is clearly seen in Figs. 6, 7, 10 and 11, where a smooth behavior in the adsorption properties is obtained.

4. Conclusions With respect to the zone III, the behavior of the adlayer is as follows. For pffiffiffi p ffiffiffi high values of wAB , this region corresponds to a ð 3  3Þ phase of A particles. As wAB decreases, pffiffiffi pand ffiffiffi as it was discussed for zone II, the empty sites of the ð 3  3Þ phase are filled by B particles. The total coverage depends on the value of wBB . In the case of wBBp¼ ffiffiffi wAB pffiffiffi¼ 0, the phase corresponding to zone III is formed by a ð 3  3Þ structure of A particles complemented with B particles, which occupy at random 1=6 of the total sites. The resulting value of hC is 5=6. We now turn to the effect of B–B interactions (see Fig. 18). For this purpose, we set wAB ¼ 0 and wAA in the critical regime. The adsorption properties corresponding to this case were discussed in Fig. 12. We start the analysis with the case wBB ¼ wAB ¼ 0, where hC ¼ 2=3 in zone II and hC ¼ 5=6 in zone II. As wBB is increased, the number of B particles in the zone II diminishes and, consequently, hC decreases. The limit value of hC , pwhich ffiffiffi pffiffiffiis reached for wBB > 3, is hC  0:5 and corresponds to a ð 3  3Þ phase of A particles complemented with a partial coverage of B particles equal to 0:162 (see discussion concerning Fig. 1). With respect to zone III, hC remains constant around 5=6. In this case,

1.00

kB TC / wAA < 0.3354 ; wAB / kBT = 0

C

0.75

0.50

In this work we have used Monte Carlo simulation in grand canonical ensemble to study the adsorption of repulsive interacting binary mixtures on triangular surfaces. The adsorption process has been monitored by following total and partial isotherms and differential heats of adsorption corresponding to both species of the mixture, for different values of the lateral interactions between the adsorbed species. An interesting feature, which we called mixture effect, is the appearance of a minimum in the global adsorption isotherm. This novel phenomenon occurs when (i) there exist lateral interactions between A and B particles and (ii) the initial concentration of B particles is in the range 0:3 6 hiB 6 0:5. In these conditions, the A particles adsorbing on the lattice expel the B adsorbed particles; then, the partial A [ B] coverage increases [decreases]. During this regime the number of desorbed particles is greater than the number of adsorbed particles which results in a decreasing of the total coverage h that occurs for a wide range in values of lA where the slope of the curve is negative. The behavior of the system has been fully explained through the analysis of the phase diagrams for order-disorder transitions occurring in the adsorbed layer. These phase diagrams are similar to those obtained by mean-field approximations for the BEG model. With respect to the Monte Carlo simulation, this technique have proven to be an adequate and powerful tool to study multicomponent adsorbates with strong repulsive lateral interactions. In addition, this simulation model can be applied in principle to any kind of topography and seems to be very useful to interpret experimental data without any special requirement and time consuming computation.

Acknowledgements 0.25

Phase II Phase III 0.00 0

1

2

3

4

5

6

wBB / kBT Fig. 18. Densities corresponding to the maxima of the coexistence curve in zones II and III as a function of wBB . In all cases, wAB ¼ 0 and wAA is chosen in the critical regime.

This work was supported in part by CONICET (Argentina) under project PIP 6294; Universidad Nacional de San Luis (Argentina) under projects 328501 and 322000 and the National Agency of Scientific and Technological Promotion (Argentina) under project 33328 PICT 2005. The numerical work were done using the BACO parallel cluster (composed by 60 PCs each with a 3.0 GHz Pentium-4 processor) located at Instituto de Fı´sica Aplicada, Universidad Nacional de San Luis - CONICET, San Luis, Argentina.

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