Adsorption of Linear k-mers on Heterogeneous Surfaces with Simple

Heterogeneous surfaces are represented by two kinds of sites, the so-called bivariate surface. Deep and shallow sites with energies ϵS and ϵD form l...
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Langmuir 2000, 16, 682-689

Adsorption of Linear k-mers on Heterogeneous Surfaces with Simple Topographies A. J. Ramirez-Pastor, J. L. Riccardo, and V. Pereyra* Departamento de Fı´sica and Centro Latinoamericano de Estudios Ilya Prigogine, Universidad Nacional de San Luis, CONICET, Chacabuco 917, 5700 San Luis, Argentina Received May 26, 1999. In Final Form: September 13, 1999 The localized monolayer adsorption of noninteracting homonuclear linear k-mers on heterogeneous surfaces is analyzed by means of a theoretical approach and compared with Monte Carlo simulations. Heterogeneous surfaces are represented by two kinds of sites, the so-called bivariate surface. Deep and shallow sites with energies S and D form l × l patches distributed at random or in a chessboardlikeordered domain on a two-dimensional square lattice. The adsorption process is analyzed following the behavior of different quantities such as coverage versus chemical potential (adsorption isotherms), meansquare fluctuation of the coverage, energy of adsorption, and differential heat of adsorption as a function of the coverage. The isotherms and the other quantities depend on the relation between the sizes of the molecule and the patch as well as on the topological distribution of the patches on the surface.

1. Introduction The adsorption of gases on solid surfaces is a topic of fundamental interest for various applications.1 Most materials possessing complex heterogeneous surface and elementary surface processes such as adsorption, desorption, surface diffusion, and surface reactions are strongly affected by structural and/or energetical disorder.2,5 The adsorption of gases has been used to obtain information about the energetic characteristics of heterogeneous surfaces as well as the adsorption energy distribution. It is well demostrated that, in addition to the adsorption energy distribution function, the surface topography is also a very important factor in the process. The patchwise heterogeneous surface, introduced by Ross and Olivier,2 and the random heterogeneous surface are two examples of disordered surfaces with different topographies, which have been extensively used in the analysis of surface processes.3 When correlation between the energies of the adsorption sites is present, the appropriate description of the heterogeneous surface is by use of intermediate adsorption site topography.3,6-9 Although most of the works are devoted to the adsorption of monatomic gas, there have been very few studies related to multisite occupancy adsorption on heterogeneous substrates. Despite this fact, the experimental evidence has shown that the adsorbed molecules, in many cases, consist of a * To whom all correspondence should be addressed. E-mail: [email protected]. (1) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon: Oxford, U.K., 1974. (2) Ross, S.; Olivier, J. P. On Physical Adsorption; Interscience: New York, 1964. (3) Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, The Netherlands, 1997. (4) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, The Netherlands, 1988. (5) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (6) Ripa, P.; Zgrablich, G. J. Phys. Chem. 1075, 79, 2118. (7) Riccardo, J. L.; Chade, M.; Pereyra, V.; Zgrablich, G. Langmuir 1992, 8, 1518. (8) Riccardo, J. L.; Pereyra, V.; Zgrablich, G.; Rojas, F.; Mayagoitia, V.; Kornhauser, I. Langmuir 1993, 9, 2730. (9) Riccardo, J. L.; Pereyra, V.; Rezzano, J.; Rodriguez Saa, D.; Zgrablich, G. Surf. Sci. 1988, 204, 286.

number of single k components, the so-called k-mers. Even the simple gases such as oxygen, nitrogen, and carbon monoxide are composed of more than one atom. It is interesting to note that also the multisite adsorption on homogeneous surfaces has been less analyzed compared with monatomic adsorption, and many aspects of the process are recently reported. For instance, Phares et al.10 have analyzed the structural ordering of interacting dimers with repulsive nearest-neighbor interaction. The thermodynamics implication of such structural ordering is demonstrated in the analysis of the collective diffusion coefficient of dimers with nearest-neighbor repulsion.11 The phase diagrams for both attractive and repulsive nearest-neighbor interactions have been reported in ref.12 On the other hand, the exact one-dimensional solution of the thermodynamic functions and chemical diffusion coefficient of linear polymers has been obtained in ref.13 The results were compared with those corresponding to Flory’s approach.14 The works of Nitta et al.15,16 and Marczewski et al.17 were the beginning of a systematic study of adsorption with multisite occupancy on heterogeneous surfaces. Models of dimer physisorption on disordered substrates have also been introduced and analyzed by using theoretical approaches and compared with Monte Carlo simulations and experimental adsorption isotherms.18,19 On the other hand, a very interesting study of adsorption of flexible k-mer molecules on random and patchwise (10) Phares, A. J.; Wunderlich, F. J.; Curley, J. D.; Grumbine, D. W., Jr. J. Phys. A 1993, 26, 6847. (11) Ramirez-Pastor, A. J.; Nazzarro, M. S.; Riccardo, J. L.; Pereyra, V. Surf. Sci. 1997, 391, 267. (12) Ramirez-Pastor, A. J.; Riccardo, J. L.; Pereyra, V. Surf. Sci. 1998, 411, 294. (13) Ramirez-Pastor, A. J.; Eggarter, T. P.; Pereyra, V.; Riccardo, J. L. Phys. Rev. B 1999, 59, 11027. (14) Flory, P. J. Chem. Phys. 1942, 10, 51. (15) Nitta, T.; Kuro-Oka, M.; Katayama, T. J. Chem. Eng. Jpn. 1984, 17, 45. (16) Nitta, T.; Yamaguchi, T. J. Chem. Eng. Jpn. 1992, 25, 420. (17) Marczewski, A. W.; Derylo-Marczewska, A.; Jaroniec, M. J. Colloid Interface Sci. 1986, 109, 310. (18) Ramirez-Pastor, A. J.; Nazzarro, M. S.; Riccardo, J. L.; Zgrablich, G. Surf. Sci. 1995, 341, 249. (19) Nitta, T.; Kirayama, H.; Shigeta, T. Langmuir 1997, 13, 903.

10.1021/la990643w CCC: $19.00 © 2000 American Chemical Society Published on Web 01/18/2000

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heterogeneous surfaces has been done by Rudzinski and co-workers in ref.20 More recently, a modified form of the adsorption isotherm for linear adsorbates on heterogeneous substrates is proposed. The isotherms are calculated by introducing the rigorous analytical solution for the Helmholtz free energy in one dimension, in the framework of a multisite occupancy model proposed by Nitta et al.19 In the present work, we study the effect of topography in the adsorption of k-mers on heterogeneous surfaces. To show the pure heterogeneity effect, we have considered the adsorption of noninteracting k-mers on patchwise and random bivariate heterogeneous surface topographies. On the basis of previous results, we have obtained a closed analytical expression for the local isotherm21 by using the same approach developed in ref19 and introduced in the so-called overall isotherm.3 Comparisons with Monte Carlo simulations show that the model works rather good for a wide range of heterogeneities. The outline of the paper is as follows: in section 2, we present the model of noninteracting k-mer adsorption on bivariate heterogeneous surfaces and the simulation scheme. In the final expression of the adsorption isotherm, we used the exact calculations for the k-mer adsorption isotherms on homogeneous one-dimensional lattices, results which are deduced in Appendix I. In section 3 we discuss our results, and finally in section 4 we give our conclusions. 2. Adsorption of k-mers on Heterogeneous Surfaces 2.1. The Model. The homonuclear linear k-mer molecule adsorbed on a heterogeneous surface is modeled as k interaction centers at a fixed separation, which is equal to the lattice constant a. In the adsorption process we assume that each monomer occupies a single adsorption site. We have not considered here the high-frequency stretching motion along the molecular bond. The k-mer bond length remains constant throughout the treatment. Let us consider a molecule with k identical monomer units. Let us further assume adsorption sites with a discrete energy distribution of W different energies 1, 2, ..., W and frequencies f1, f2, ..., fW, where fi is the frequency of i. In the absence of lateral interaction, the total adsorption energy for a particular k-mer is given by W

˜ j )

mj(i) i ∑ i)1

(1)

with W

mj(i) ) k ∑ i)1

(2)

where j run over MW,k different energy levels and mj(i) is the number of monomers occupying sites with the same energy i in the j level. MW,k is given by the number of k-uples that can be arranged between W different values (20) Rudzinki, W.; Nieszporek, K.; Cases, J. M.; Michot, L. I.; Villeras, F. Langmuir 1996, 12, 170 and references therein. (21) Ramirez-Pastor, A. J.; Pereyra, V.; Riccardo, J. L. Langmuir 1999, 15, 5707. (22) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H. J. Chem. Phys. 1953, 21, 1087.

of energy sites

MW,k )

Wk + (k - 1)W k

(3)

where in the calculation of MW,k we have considered that two configurations with the same values of mj(i) belong to the same energy level (present the same energy) independent of its sequence. The mean surface coverage θ h is then obtained by MW,k

θ h)

˜fj(f1,f2,...fW) θ(˜ j) ∑ j)1

(4)

where with the symbol ˜ we identify the quantities associated with the k-mers in order to differentiate those associated with the sites. Because we neglect the lateral interaction in the adsorption process, we have considered, in a first approximation as a local isotherm θ(˜ j), the exact solution for the one-dimensional k-mer isotherms on a homogeneous surface given in Appendix I

k exp(µ-˜ j)/kBT )

[ (k -k 1)θ(˜ )]

k-1

θ(˜ j) 1 -

j

[1 - θ(˜ j)]k

(5)

Analytical expressions for ˜fj can be written down by supposing a particular adsorption site topography. Particularly, in the present work we analyze the bivariate heterogeneous surface. Let us consider a square lattice with only two kinds of adsorption sites, namely, shallow and deep adsorption sites, having adsorption energies S and D. On this basis, simple topographies are created by placing square patches of l × l deep or shallow adsorption sites onto the lattice, which is represented by a two-dimensional array of L × L sites with periodic boundary conditions. To create the same number of patches, the ratio L/l ) 48 is kept constant in the present work. Within a given patch, all sites have the same adsorption energy. Two different geometrical structures are easily prepared: (i) randomly located patches of deep adsorption sites and (ii) chessboardlike topographies (see Figure 1). The corresponding fraction of deep/shallow adsorption sites is denoted by fD/fS. To analyze the adsorption of k-mers on the bivariate heterogeneous surfaces, let us define the total adsorption energy for a k-mer occupying j shallow sites and k - j deep sites as

˜ SjDk-j ) jS + (k - j)D

(6)

where j ) 0, 1, ..., k. In the rest of the paper we assume the notation ˜ SkD0 ) ˜ Sk and ˜ S0Dk ) ˜ Dk. Then, let us calculate the isotherms for the two limit cases, the random and the two big patches surfaces. The overall isotherm for a given k-mer on the random topography of patches with l ) 1 is given by k

θ h rand(µ,T) )

( kj )f jSf k-j ˜S D ∑ D θ( j)0 j

k-j

)

(7)

Because in this topography the energies of the sites are completely at random, the factor (kj)f jSf k-j D is the probability to have a k-mer molecule adsorbed on j shallow sites and k - j deep sites. In the case of two big patches, the overall isotherm is given by

θ h patch(µ,T) ) fSθ(˜ Sk) + fDθ(˜ Dk)

(8)

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Figure 1. Patchwise heterogeneous surface formed by only two kinds of square patches with different energies, i.e., deep and shallow patches with adsorption energies S (white sites) and D (black sites), respectively. Both types of sites are present with the same concentration. Two different topographies are shown, namely, (a) the chessboardlike-ordered and (b) the random patches distribution. The lattice size used here is L ) 32, and the patch size corresponds to l ) 4. The unit cell is remarked in the upper part of the figure.

where θ(˜ SjDk-j) is obtained by solving eq 5 for a fixed value of k. Note that in eq 8 we neglected the effect of the border. It seems instructive to show explicitly the calculation for k ) 2. Then, the expression for the overall isotherm on the random topography can be written as

θ h rand(µ,T) ) fD2θ(˜ D2) + 2fDfSθ(˜ DS) + fS2θ(˜ S2) (9) Note that a dimer can be adsorbed by occupying a pair of deep sites with probability fD2, a pair of shallow sites with probability fS2, or a pair of shallow and deep sites with probability 2fSfD. For two big patches the corresponding overall isotherm is given by

θ h patch(µ,T) ) fDθ(˜ D2) + fSθ(˜ S2)

(10)

where θ(˜ SiD2-i) (with i ) 0, 1, 2) is given by

θ(SiD2-i) ) 2 exp(µ-˜ SiD2-i)/kBT + 1 - (2 exp(µ-˜ SiD2-i)/kBT + 1)1/2 2 exp(µ-˜ SiD2-i)/kBT + 1 (11) Other topography can be considered by taking fj(f1,f2,...,fW) adequately.

2.2. Simulation Scheme. In the grand canonical ensemble, the k-mer adsorption on heterogeneous surfaces is simulated by putting in contact a heterogeneous square lattice of L × L adsorption sites with an ideal gas phase of linear molecules characterized by chemical potential µ and temperature T, where the surface as well as the adsorbent are inert upon adsorption. The standard Metropolis algorithm18 is used for two steps: adsorption and desorption. In adsorption-desorption equilibrium there are two elementary ways to perform a change of 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). It has also 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 of moving an adsorbed molecule to a new position chosen randomly within the adsorbed phase volume V. The algorithm to carry out an elementary Monte Carlo step (MCS) in simulation is the following: Given a square lattice of L × L adsorption sites with energies already assigned, (i) Set the value of µ and temperature T. (ii) Choose randomly one linear k-uple of sites on the lattice, and generate a random number ξ[0, 1]. (a) If the k sites are empty, then adsorb a molecule if ξ e Pads, where Pads is the transition probability from a state with N k-mers to a new state with N + 1 k-mers. Pads is obtained by following the Metropolis scheme.18,21 (b) If the k sites are occupied by atoms belonging to the same k-mer, then desorb the molecule if ξ e Pdes, where Pdes is the transition probability from a state with N k-mers to a new state with N - 1 k-mers. Pdes is also obtained by following the Metropolis scheme.18,21 (iii) Repeat from step ii L × L times. The equilibrium state, at intermediate temperature, can be reached after discarding the first ≈105 MCS and averaging over the next ≈105 MCS. However, at low temperatures more than ≈107 MCS are necessary to discard, and the average is performed over the next ≈106 MCS in order to have an equilibrium configuration. Apart from the thermal average, each configuration is obtained averaging over 100 different topological distributions of the patches (in the random case). Thermodynamic quantities, such as mean coverage θ and mean adsorption energy U (all normalized per lattice site), are obtained as simple averages.12 Apart from the isotherms and the adsorption energy, other quantities are obtained such as the differential heat of adsorption qd which can be obtained from the simulation23 as

qd ) -(∂U/∂θ)T

(12)

The qd is equivalent to the differential change in the energy of the system per adsorbed particle. The normalized mean-square fluctuations of the coverage can also be obtained as

(

)

∂µ/kBT 〈(δN)2〉 ) ∂ ln θ 〈N〉

-1

(13)

where N is the number of k-mers adsorbed on the lattice and the thermal average 〈...〉 means the time average throughout the Monte Carlo simulation. (23) Razmus, D.; Hall, C. AIChE J. 1991, 37, 5.

Adsorption of Linear k-mers

Figure 2. (a) Adsorption isotherms for dimers on bivariate random topography with equal concentrations of deep and shallow adsorption sites and different temperatures. (b) The energy of adsorption -U/kBT as a function of coverage corresponding to the same values of temperature given in part a. (c) Differential heat of adsorption versus coverage. The lines correspond to the theoretical approach, while the symbols correspond to the Monte Carlo simulations, ∆/kBT ) 0 (squares), 2 (diamonds), 4 (up triangles), and 8 (circles).

3. Results and Discussion Adsorption of noninteracting k-mers on a bivariate heterogeneous surface presents a rich and complex behavior depending on the relation between the number k and the linear size of the patches l, the topological distribution of the patches, and the difference between the energies of the patches. 3.1. Dimers. In Figure 2a, we plot a set of isotherms for dimers adsorbed on a surface formed by a random distribution of l × l patches with fD ) fS ) 0.5 and for different temperatures ∆/kBT (with ∆ ) D - S). In parts b and c of Figure 2, the corresponding adsorption energies U and differential heat of adsorption qd are plotted, as a function of coverage. The analytical curves (lines) show a good agreement with Monte Carlo simulations (symbols) at high temperatures, while at low temperatures the differences are more pronounced. Although at very low temperature (∆/kBT ) 8) there is a strong quantitative difference between the analytical and simulated isotherms, the qualitative behavior seems to be the same. In fact, both isotherms present three different coverage regimes, separated by two well-defined steps. The number of coverage regimes is related to the three different possible energy pairs associated with a given adsorbed dimer, DD, ˜ SD, and ˜ SS. In fact, the filling process is as follows: at low values of chemical potential, the DD pairs are preferentially occupied by the dimers (regime 1); as the chemical potential increases, DS pairs start to be occupied (regime 2); finally, for high values of chemical potential, the SS pairs are filled, (regime 3). These regimes can be confirmed from the curve of the energy of adsorption (Figure 2b) and differential heat of adsorption (Figure 2c), particularly at low temperatures. From the figures, we can calculate the crossover coverages between the three regimes, taking the value of the coverage where the curve

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Figure 3. Same as Figure 2, for two big domains topography.

changes the slope. At low temperature, the theoretical prediction for the crossover coverages can be obtained from 23 eq 9 given θ12 Th ≈ 0.25 and θTh ≈ 0.75, while for a simulated 12 isotherm we have, θMC ≈ 0.4 and θ23 MC ≈ 0.6, respectively. The difference between the isotherms is directly associated with the topological distribution of the energy sites in the random bivariate surface used to obtain the results in the simulation. This is so because for a random distribution of 1 × 1 patches the mean cluster size lc can be calculated from percolation theory24 to be lc ≈ 5.4 (given in lattice units). Therefore, dimers are adsorbed on irregular patches of approximately 6 × 5 sites. Then, the ratio between DD(SS) and DS pairs increases compared to the theoretical model. In Figure 3a, we have plotted the adsorption isotherms corresponding to two big domains. As is expected, there is a good concordance between simulations and analytical results for all temperatures, which is also observed in the energy of adsorption U (Figure 3b) and differential heat of adsorption qd (Figure 3c). The main reason for this agreement is that the DS pairs with energy ˜ SD do not have a big influence in the process. They eventually occur in the borderline between the patches; hence, the number is negligible compared with the number of SS or DD pairs. Next, we analyze the adsorption on intermediate topographies. In Figure 4, we have plotted the adsorption isotherm (Figure 4a) energy of adsorption (Figure 4b) and differential heat of adsorption (Figure 4c) for dimers on different topographies, at fixed ∆/kBT ) 8 and different values of l. We have omitted here the comparison with the theoretical isotherms. This is mainly because its behaviors are similar to those of the 1 × 1 random case; that is, qualitatively they present a good agreement with the simulation, but there are strong quantitative differences. At this low temperature, two well-defined sets of isotherms can be clearly observed, depending on the values of l. In fact, for odd l the isotherms present a step at half (24) Stauffer, D. Introduction to percolation theory; Taylor & Francis Ltd.: London, 1987. (25) Ramirez-Pastor, A. J. Doctoral Thesis, Universidad Nacional de San Luis, San Luis, Argentina, 1998. (26) Hill, T. An Introduction to Statistical Thermodynamics; AddisonWesley: Reading, MA, 1960.

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Figure 4. (a) Adsorption isotherms for dimers on different topographies with equal concentrations of deep and shallow adsorption sites and fixed temperature ∆/kBT ) 8. (b) The energy of adsorption -U/kBT and (c) differential heat of adsorption versus coverage corresponding to the same values. Open symbols correspond to chessboardlike order with l ) 1 (circles), 2 (up triangles), and 3 (squares). Filled symbols correspond to random topography with l ) 1 (circles), while the results corresponding to two big domains are given by open diamonds.

coverage θ ) 0.5. The behavior is similar to that corresponding to two big patches. In fact, because the patches are formed also by an odd number of sites, the dimers fill completely the deep patches. As the chemical potential increases, the shallow adsorption sites start to be occupied until the full coverage is reached. Then in the equilibrium, the number of DS pairs is negligible. Therefore, in this case the isotherms are almost independent of the topological distribution of the patches and the size l. For even l, the isotherms present three adsorption regimes as in the case previously analyzed in Figure 2 (1 × 1 random case). The filling mechanism is explained based in the same argument. The width in coverage, corresponding to regime 2, increases according to the number of DS pairs. This quantity, which is the controlling parameter in the adsorption process, can be changed because of two factors: (i) For a given topography (random or chessboardlike order), when l increases, the fraction of DS pairs decreases (particularly, for l g 3 the fraction of DS pairs is practically negligible and the isotherms are very close to those corresponding to two big patches). (ii) For fixed l, the chessboardlike topography presents a larger value of the interface between deep and shallow patches compared with the corresponding random topography. Consequently, the number of DS pairs is, for the order topography, bigger than the random surface. A very different shape presents the isotherms for chessboardlike topography for the 1 × 1 patches surface. In fact, dimers are always adsorbed on DS pairs, and regime 2 determines the whole adsorption process. In this case, an analytical expression for the isotherm can be trivially obtained. The three regimes described above are observed as changes in the slope of the energy of adsorption as a

Ramirez-Pastor et al.

Figure 5. (a) Fraction of DD (filled squares), DS (crossing squares), and SS (dotted center squares) normalized versus the total number of adsorbed dimers, b) differential heat of adsorption, and (c) mean-square fluctuations of the coverage, as a function of coverage for dimers adsorbed on a chessboardlike topography with l ) 3. In the inset, we have shown one configuration corresponding to θ ) 8/18 and 10/18.

function of coverage (Figure 4b) or as steps in the differential heat of adsorption versus coverage (Figure 4c). In Figure 5, we analyzed in detail the case of dimers adsorbed on a patchwise surface of patch size l ) 3. The fractions of occupied DD, SS, and DS pairs (normalized versus the number of adsorbed dimers) are plotted as a function of coverage in Figure 5a. Clearly, we can observe the three adsorption regimes previously described. To follow the adsorption process, let us define the unit cell for one particular chessboardlike-ordered surface as two consecutive patches (see Figure 1). Then, for coverage less than θ < 8/18 (see the inset in Figure 5a), the dimers only occupied the deep sites of the unit cells. For coverage between 8/18 < θ < 10/18, the dimers are adsorbed occupying deep and shallow sites; particularly at θ ) 10/18, there are four dimers adsorbed on deep sites and one dimer occupying a DS pair in the unit cells. Finally, for coverage θ > 10/18 the dimers are adsorbed, occupying the rest of the shallow sites. This adsorption process can also be monitored by following the fluctuation on the coverage (Figure 5b) and the differential heat of adsorption qd (Figure 5c). 3.2. Trimers. The adsorption isotherms for trimers at fixed temperature ∆/kBT ) 8 are plotted in Figure 6 a. The energy of adsorption and the differential heat of adsorption qd versus coverage are plotted in parts b and c of Figure 6, respectively. Different behaviors are also observed depending on the linear size of the patch l and the topography of the surface, as is discussed above for the dimer case. In fact, for those patches which allowed an integer number of trimers, l ) 3n with n ) 1, 2, 3, ..., the isotherms present a unique step at θ ) 0.5; this is also valid for two big patches. The relation between the number of DDS, DSS, and DDD (or SSS) triplets is the reason for such behavior. In fact, the number of DDS and DSS triplets is negligible compared with DDD and SSS triplets. Therefore, two well-defined adsorption regimes can be observed in the isotherms. These regimes are shown in the change

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Figure 6. (a) Adsorption isotherms for trimers on different topographies with equal concentrations of deep and shallow adsorption sites and fixed temperature ∆/kBT ) 8. (b) The energy of adsorption -U/kBT and (c) differential heat of adsorption versus coverage corresponding to the same values. Open symbols correspond to chessboardlike order with l ) 1 (circles), 2 (up triangles), 3 (squares), and 4 (down triangles). Filled symbols correspond to random topography with l ) 1 (circles) while the results corresponding to two big domains are given by open diamonds. In the inset of part a the adsorption isotherms for chessboardlike topography with l ) 1 and 2 are shown. The crossing symbols correspond to ∆/kBT ) 4.

of the slope for the energy of adsorption (Figure 6b) or in the step of the differential heat of adsorption (Figure 6c). For those values of l which allowed the formation of DDS and DSS triplets, the isotherms present a more complicated coverage regime. Let us analyze the adsorption isotherm for a 1 × 1 random distribution of deep and shallow sites (filled circles in Figure 6a). The curve presents a first step at coverage θ ≈ 0.3, a second step at coverage θ ≈ 0.55, and a third step at coverage θ ≈ 0.65. These singularities in the adsorption isotherm are also observed for the energy of adsorption and the differential heat of adsorption defining four adsorption regimes. They correspond to the sequential filling of DDD, DDS, DSS, and SSS triplets in the surface. As is discussed below, the linear size of the average cluster corresponding to this arrangement is lc ≈ 5.4 (in lattice unit). However, the shape of the clusters is completely irregular; therefore, a detailed analysis of the adsorption mechanism, as is given for the other cases, cannot be possible here. For a 1 × 1 chessboard array, the isotherm presents a unique step at coverage θ ) 0.75. This is because the most favorable energetic configurations, at this coverage, occur when trimers are adsorbed, forming DSD triplets. Hence, in the equilibrium the trimers form domains of parallel zigzag rows separated by rows of S empty sites (see Figure 7a). A similar situation occurs for a 2 × 2 chessboard array. As we can observe, the isotherm presents again a step at θ ) 0.75. Now the most favorable configuration corresponding to DDS triplets is showed in Figure 7b. To adsorb new molecules in both cases, it is necessary to remove diagonal row configurations. The Monte Carlo times are extremely large at this temperature. However, this is not

Figure 7. (a) Snapshot of the adsorbed phase, corresponding to the adsorption of trimers on chessboardlike topography with l ) 1 at θ ) 0.75. (b) Same as part a for l ) 2.

a jamming state as in the irreversible adsorption.27,28 As soon as the temperature increases, the simulation is more efficient and the full coverage is reached, as is shown in the inset of Figure 6a. The energy of adsorption (Figure 6b) and the differential heat of adsorption (Figure 6c) for both cases (1 × 1 and 2 × 2) present the same behavior for the values of the coverage informed in this work. More complicated shapes present the isotherms for 3 < l < 6. In fact, for a 4 × 4 chessboard array, the isotherm shows a first step at θ ≈ 0.465 and a less pronounced step at θ ≈ 0.565. To explain the isotherm, let us observe the behavior of the fraction of occupied DDD, SSS, SDD, and SSD triplets (normalized versus the number of adsorbed trimers) which are plotted as a function of coverage in Figure 8 a. Here, we observe three adsorption regimes: (i) for coverage less than θ < 15/32, the trimers only occupied the deep sites of the unit cells. For coverage between 15/32 < θ < 18/32, the trimers are adsorbed occupying deep and shallow sites; particularly at θ ) 18/32, there are five trimers adsorbed on deep sites and one trimer occupying a DSS triplet in the unit cells. Finally, for coverage θ > 18/32 the trimers are adsorbed occupying the rest of the shallow sites. The fluctuation on the coverage and the differential heat of adsorption qd present singularities at these coverages, confirming our explanation. (27) Stacchiola, D.; Eggarter, T. P.; Zgrablich, G. J. Phys. A 1998, 31, 185. (28) Stacchiola, D.; Ciacera, M.; Zuppa, C.; Eggarter, T. P.; Zgrablich, G. J. Chem. Phys. 1998, 108, 1730.

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Figure 8. (a) Fraction of DDD (filled circles), DSS (crossing circles), and SSS (dotted center circles) normalized versus the total number of adsorbed trimers, (b) differential heat of adsorption, and (c) mean-square fluctuations of the coverage, as a function of coverage for trimers adsorbed on a chessboardlike topography with l ) 4. In the inset, we have shown one configuration corresponding to θ ) 15/32 and 18/32.

Finally, for l g 5 the isotherms are very close to those corresponding to two big patches. This is because the fractions of DDS and DSS are negligible compared with those corresponding to DDD and SSS triplets. 4. Conclusions In this paper we study the adsorption of linear rigid homonuclear molecules, k-mers, on heterogeneous surfaces. To analyze the effect of topography on the adsorption process, we have considered the noninteracting adsorption of molecules on a heterogeneous bivariate surface. Deep and shallow sites, with energies S and D, form l × l patches distributed at random or chessboardlikeordered domains on a simple square lattice. A modified theoretical expression for the overall isotherms is obtained by using as a local isotherm the one-dimensional exact solution for the isotherm of linear k-mers on a homogeneous substrate. In the present work we have analyzed in detail the dimers k ) 2 and trimers k ) 3 adsorption. The resulting adsorption isotherm shows significant qualitative agreement with the Monte Carlo simulation. The main difference appears at low temperatures. Because no adsorbate-adsorbate interactions are included, all of the differences may be attributed to entropic contributions which have appreciable effects in the adsorption isotherm and remaining thermodynamic functions. In fact, both numerical and theoretical adsorption isotherms present different coverage regimes determined mainly by three factors (i) the relation between the size of the patches l and the number of k units; (ii) the topological distribution of the patches, and (iii) the adsorption energies. These factors determine the relation between the number of D and S sites occupied by a given k-mer. For instance, for random topography the dimer isotherms present three regimes, which correspond to three possible configurations for an adsorbed molecule, DD, SS, and SD, while for these patchwise topographies, where the size of the patches l is an odd number, the

Ramirez-Pastor et al.

isotherms present only two regimes; in fact, the dimers fill completely the deep patches DD and then fill the shallow patches SS. This is so because in the equilibrium the DS pairs are negligible and do not contribute to the process. This scheme is also valid for a surface composed of two big patches. For even l, the isotherms present three adsorption regimes as in the 1 × 1 random topography; also in these cases the reason for such behavior is mainly because the number of DS pairs is not negligible. In particular, we can analyze in detail the filling process and even estimate the crossover coverage for a given value of l. The other thermodynamic quantities present a behavior which can be explained with the same arguments. For trimer, the behavior of the isotherms as well as the heat of adsorption can be explained by following the same scheme. In fact, for patchwise topographies where the patches allowed an integer number of trimers, the isotherms present a unique step at θ ) 0.5; this is also valid for two big patches. The number of DDS and DSS triplets which are negligible compared with DDD and SSS triplets is the reason for such behavior. For random 1 × 1 topography, the isotherms present four regimes which are separated by steps. These steps are determined by the filling of the DDD, DDS, DSS, and SSS triplets. However, a more complicated adsorbed structure results in the case of a 1 × 1 chessboard array. In this case, the trimers are adsorbed, forming DSD triplets; therefore, domains of zigzag rows separated by rows of S empty sites appear. Then, the isotherm presents a unique step at θ ) 0.75. In a similar way we can explain the 2 × 2 chessboard topography. Following this scheme, we can also explain more complicated adsorbed processes such as the 4 × 4 and 5 × 5 chessboard topographies. This argument used for the adsorption of dimer and trimer can be generalized also for large molecules and different l × m ordered and disordered bivariate topographies. Finally, we can conclude that knowledge of the exact coverage and temperature dependence of the adsorption isotherms of linear adsorbates in a homogeneous onedimensional lattice allows the development of a more accurate description of the adsorption isotherm in higher dimensions, even for heterogeneous surfaces. In this case, the observed differences with respect to former approaches can be attributed to the configurational entropy that is more properly taken into account in the present case. Acknowledgment. This work is partially supported by the CONICET (Argentina) and Fundacio´n Antorchas (Argentina). Appendix I Let us assume a one-dimensional lattice of M sites with lattice constant a (M f ∞) and periodic boundary conditions. Under this condition all lattice sites are equivalent; hence, border effects will not enter our derivation. N linear k-mers are adsorbed on the lattice in such a way that each mer occupies one lattice site and doublesite occupancy is not allowed to represent the monolayer regime. Because different k-mers do not interact with each other through their ends, all configurations of N k-mers on M sites are equally probable; henceforth, the canonical partition function Q(M,N,T) results

(

Q(M,N,T) ) Ω(M,N) exp -

)

Nk kBT

(14)

where Ω(M,N) is the total number of configurations and

Adsorption of Linear k-mers

Langmuir, Vol. 16, No. 2, 2000 689

 is the interaction energy between an individual unit of the k-mer and the lattice site. Ω(M,N) can be readily calculated as the total number of permutations of the N indistinguishable k-mers out of ne entities, with ne being25

βF(M,N,T) ) -[M - (k - 1)N] ln[M - (k - 1)N] + [M - (k - 1)N] + [N ln N - N] + [(M - kN) ln(M kN) - (M - kN)] + βNk ) -[M - (k - 1)N] ln[M - (k - 1)N] + N ln N + (M - kN) ln(M kN) + βNk (21)

ne ) number of k-mers + number of empty sites ) N + M - kN ) M - (k - 1)N

(15)

S(M,N) ) [M - (k - 1)N] ln[M - (k - 1)N] kB N ln N - (M - kN) ln(M - kN) (22)

Accordingly,

Ω(M,N) )

( )

ne [M - (k - 1)N]! ) N N![M - kN]!

(16)

In the canonical ensemble framework, the Helmholtz free energy F(M,N,T) relates to Ω(M,N) through

βF(M,N,T) ) -ln Q(M,N,T) ) -ln Ω(M,N) + βNk (17) where β ) 1/kBT. The remaining thermodynamic functions can be obtained from the general differential form26

dF ) -S dT - Π dM + µ dN

∂F (∂T )

M,N

∂F (∂M )

Π)-

T,N

µ)

βΠ ) ln[M - (k - 1)N] - ln[M - kN] βµ ) ln

∂F (∂N )

T,M

(19)

[

which can be written in terms of the Stirling approximation as follows:

]

[

]

Then, when the lattice coverage is defined as θ ) kN/M, the free energy per site as f ) F/M, and the entropy per site as s ) S/M, eqs 21-24 can be rewritten in terms of the intensive variables θ and T.

{[

] [

[

]

(k - 1) (k - 1) θ ln 1 θ k k θ θ ln - (1 - θ) ln(1 - θ) + βθk (25) k k

βf(θ,T) ) - 1 -

] [

}

]

(k - 1) (k - 1) s(θ) θ θ θ ln 1 θ - ln - (1 ) 1kB k k k k θ) ln(1 - θ) (26)

[

]

(k - 1) θ k 1-θ

1-

exp(βΠ) )

Thus, from eqs 16 and 17

βF(M,N,T) ) -{ln[M - (k - 1)N]! - ln N! - ln[M kN]!} + βNk (20)

(23)

N kN + (k - 1) ln 1 - (k - 1) M M kN + βk (24) k ln 1 M

(18)

where S, Π, and µ designate the entropy, spreading pressure, and chemical potential, respectively, which, by definition, are

S)-

Henceforth, from eq 19

[

k-1 θ k (1 - θ)k

θ1Ck exp(β(µ - k)) )

where Ck ) k. LA990643W

(27)

]

k-1

(28)