## Brazilian Journal of Chemical Engineering

##
*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol.18 no.4 São Paulo Dec. 2001

#### http://dx.doi.org/10.1590/S0104-66322001000400004

**MONTE CARLO SIMULATIONS OF THE ADSORPTION OF DIMERS ON STRUCTURED HETEROGENEOUS SURFACES**

C.R.A.Abreu, F.C.Peixoto, R.O.Corrêa, A.S.Telles and F.W.Tavares

Escola de Química, Universidade Federal do Rio de Janeiro

Cx. Postal 68542, CEP 21949-900, Rio de Janeiro - RJ, Brazil

E-mail: tavares@h2o.eq.ufrj.br

(Received: May 24, 2001 ; Accepted: September 14, 2001)

Abstract -The effect of surface topography upon the adsorption of dimer molecules is analyzed by means of grand canonical ensemble Monte Carlo simulations. Heterogeneous surfaces were assumed to consist of a square lattice containing active sites with two different energies. These were distributed in three different configurations: a random distribution of isolated sites; a random distribution of grains with four high-energy sites; and a random distribution of grains with nine high-energy sites.

For the random distribution of isolated sites, the results are in good agreement with the molecular simulations performed by Nitta et al. (1997). In general, the comparison with theoretical models shows that the Nitta et al. (1984) isotherm presents good predictions of dimer adsorption both on homogeneous and heterogeneous surfaces with sites having small differences in characteristic energies. The molecular simulation results also show that the energy topology of the solid surfaces plays an important role in the adsorption of dimers on solids with large differences in site energies. For these cases, the Nitta et al. model does not describe well the data on dimer adsorption on random heterogeneous surfaces (grains with one acid site), but does describe reasonably well the adsorption of dimers on more patchwise heterogeneous surfaces (grains with nine acid sites).: Adsorption isotherms, heterogeneous surfaces, Monte Carlo simulations.

Keywords

INTRODUCTION

The modeling of adsorption phenomena on solid surfaces is an important tool to predict and correlate the physical adsorption involved in the separation or purification processes and to understand the chemical adsorption (adsorption on solids with a broad distribution of energies and irregular surface patterns) inherent in catalytic processes. The diversity of solids used in these processes gives rise to a large number of isotherm models. These are used to describe different observed adsorption characteristics. Generally, the task of choosing the best model to apply is not an easy one. Molecular simulation techniques can be a useful tool in testing hypotheses and simplifications used to develop isotherms. In particular, surface topography is an important parameter in view of the heterogeneity of solids.

The isotherm models presented in the literature fall within one of the following extreme situations: sites distributed in large clusters called patchwise heterogeneous surfaces or randomly distributed sites (Rudzinski and Everett, 1992). Recently, Nitta et al. (1997) used the Monte Carlo simulation method for a grand canonical ensemble to analyze dimer adsorption on heterogeneous solids with randomly distributed sites. This work was based on studies performed by Riccardo et al. (1993) and Ramirez-Pastor et al. (1995) on the adsorption of dimer molecules on heterogeneous surfaces. Ramirez-Pastor et al. (1995) used the molecular simulation technique as an isotherm equation of state to directly predict the number of adsorbed moles for different pressure and temperature conditions. The comparison between the molecular simulation results and experimental data demonstrates the potentiality of this technique to describe real systems.

The objective of the present work is to systematically analyze the effect of solids energy topography on the adsorption of dimers using Monte Carlo simulations. We also explore the conditions under which the Nitta et al. (1984) isotherm model best describes the adsorption behavior of dimers on heterogeneous solids.

MOLECULAR SIMULATION

The Monte Carlo method for a grand canonical ensemble was used to simulate the adsorption of dimers (with two identical segments) on heterogeneous solids with two kinds of sites, characterized by energies e_{a} and e_{b}, distributed in a square lattice. The dimer molecule is adsorbed on two adjacent sites, interacting with the adsorbent solid by one of three energy values, e_{aa} = 2e_{a,} e_{bb}.= 2e_{b, }OG e_{ab }= e_{a }+ e_{b. }For the simulations and models studied here, adsorbate-adsorbate interactions were neglected. Thus, the configurational energy (U) of the adsorbed phase results from the configuration of occupied sites,

where N_{ab} is the number of dimers adsorbed on pairs of distinct sites (i.e., one of its segments on a site with energy e_{a} and the other on a site with energy e_{b}) and N_{aa} and N_{bb} are the number of pairs of site a and pairs of site b occupied, respectively. The solid lattice, a 100´ 100 (M = 10000) square matrix, was filled with acid sites in the proportions of 10%, 30%, and 50%. To each fraction of acid sites, three topologies, characterized by the random distribution of square grains with one, four and nine sites (clusters of sites with g = 1, g = 4, and g = 9 ) were used. To visualize this statement, Figure 1 shows two solid square-lattice samples with a fraction of 10% acid sites and clusters (grains) of sites with g = 1 and g = 9 . In order to minimize the size effect of the solids, the periodic boundary conditions were used.

Once the surface structure was defined, the surface coverage due to dimer adsorption was calculated by the Monte Carlo method, given the chemical potential (m), temperature ( T ), and number of specified sites ( M ) in the lattice. The method is based on three fundamental movements, according to the Metropolis algorithm (Allen and Tildesley, 1987 and Nitta et al., 1997): creation, destruction, and rotation of an adsorbate molecule. The transition probability from a configurational state (m) to another configurational state (n) is given as

where r_{n }/ r_{m }is the ratio of probability densities of the two configurational states (n) and (m). Details of each movement are presented below:

Rotation Movement

Among the adsorbed molecules, one is randomly chosen to rotate around one of its segments. This segment is fixed as a pivot and another random number is generated to choose a new position for the second segment. If this position is already occupied, the movement is rejected (the configuration is thermodynamically impossible). If not, the energy of this new configuration is computed and the new ratio of the probability density of the new configurational state (n) to that of the old state (m) is given by (Allen and Tildesley, 1987)

The movement is accepted if the random number, generated in the interval 0-1, is lower than the transition probability (R_{m®n}) defined by equation (2).

Creation Movement (Adsorption)

In order to add one molecule, a position in the lattice is chosen to place the first segment of the dimmer, and using another random number, a neighboring site is chosen to complete the addition. If one of these sites is already occupied, the movement is rejected, as its configuration is thermodynamically impossible. If not, the energy of this new configuration is computed and the ratio of the probability density of the new (n) state to that of the old (m) (Allen and Tildesley, 1987) is calculated as follows:

where N is the total number of dimers previously adsorbed and M is the number of sites in the lattice. The movement is accepted if the generated random number between 0 and 1 is lower than the transition probability (R_{m®n}) defined by equation (2).

Destruction Movement (Desorption)

One of the adsorbed molecules is randomly removed and the energy of this new configuration is calculated and the ratio of the probability density of the new (n) state to that of the old (m) (Allen and Tildesley, 1987) is computed by

The movement is accepted if the generated random number between 0 and 1 is lower than the transition probability (R_{m®n}) defined by equation (2).

The relationship connecting fugacity, f, and chemical potential is given by the classic equation (Smith and van Ness, 1987):

where m^{o} is the chemical potential of the reference state for the ideal gas under atmospheric pressure and at the temperature of the system. On the other hand, the Henry constant for dimer adsorption on a homogeneous solid exclusively constituted by sites of the b kind is given by (Hill, 1960)

where x represents a correction term for the partition function corresponding to the internal degrees of freedom (such as vibrational, electronic, rotational, etc.) of the adsorbed molecule compared to the same degrees of freedom under the condition of the ideal-gas reference state. In simulations as well as in the models, the value of x was incorporated into K_{b} (or was set at x = 1). Therefore, the chemical potential is given by

Periodic boundary conditions were considered for both distributing the acid grains on the solid and moving a molecule in the lattice. The Monte Carlo method is based on a large number of configurations involving these three basic movements.

Each simulation cycle starts with the establishment of an initial configuration by randomly placing a specific quantity of dimers in the lattice. With a given initial configuration, the system was first relaxed to equilibrium throughout 10^{6} Monte Carlo steps (rotation, creation, and destruction). After equilibration, another 10^{6} Monte Carlo steps were performed to compute the thermodynamic properties, such as the coverage fraction and the configurational energy. The global means and the standard deviation of the thermodynamic properties were calculated using 10 cycles of simulation, each one marked by a different random distribution of acid grains. The scheme of the Monte Carlo method used in this work is summarized in Figure 2. Typical standard deviation of the thermodynamic properties obtained here was, for example, 10^{-3} for the coverage fraction of dimers.

MODELS

Heterogeneous Langmuir Model

In the Langmuir isotherm model, it is assumed that every molecule adsorbs on only one effective site, without lateral interactions. In particular, for the case of dimer adsorption, the number of effective sites should be equal to M/2, where M is the total number of sites in the original lattice. A second important hypothesis is that the M/2 effective sites are independent; therefore the total amount adsorbed is simply a summation of the parts. In this way, the expression of surface coverage ( q = 2N/M )**, **for the dual-site Langmuir model is (Mathias et al., 1996)

This equation can be rewritten as

where

In these equations, K_{b }is the adsorption equilibrium constant for the dimer molecules adsorbed on the surface composed only of sites b and r_{a} is the relative strength of adsorption on sites a compared to that on sites b.

Nitta et al. Model

In the Nitta et al. (1984) model the probability of a site being occupied is computed using the quasi-chemical theory (Hill, 1960), where the sites are considered to be randomly distributed (as shown in Figure 1a for g= 1 ). On the other hand, the Flory-Huggins equation is used to estimate the entropic effect due to the adsorption of dimer molecules in the lattice. In this way, the equation is (Nitta et al., 1984 and 1997)

where

The quasi-chemical equation should be solved simultaneously to obtain q and q _{b} by

and Y is defined by

In these equations, f is the fugacity for a dimer in the gas phase, K_{b} is the equilibrium constant of the adsorption on the hypothetical homogeneous surface composed only of sites b, and q and q _{b} are fractions of the total coverage and the coverage of nonacid sites, respectively.

**RESULTS AND DISCUSSION**

The Monte Carlo simulation method for a grand canonical ensemble was used to calculate the coverage fraction and configurational energies for adsorption of dimer molecules on heterogeneous surfaces. Special attention was given to analyzing different topographies of the energy surface heterogeneity. The results obtained with Monte Carlo simulations were compared to those obtained with the theoretical dual-site Langmuir (Mathias et al., 1996) and Nitta et al. (1984) models for a large range of fugacity values.

Coverage fractions for adsorption of dimers on completely random heterogeneous solid surfaces ( g = 1 ) obtained by Monte Carlo simulations are presented in Figure 3. The simulation results are in good agreement with those of Nitta et al. (1997). Figure 3a shows the coverage fractions for dimer adsorption on homogeneous surfaces ( r_{a }= 1_{ }) versus dimensionless gas fugacity ( K_{b}f ). In this case, there is no topological effect owing to the differences in grain size. One can observe the significant agreement between the coverage fractions obtained with the Nitta et al. model and those obtained with simulations, mainly in the low pressure regions (which also correspond to low coverage fractions). This agreement is consistent with the inclusion of the entropic effect of dimer adsorption in a crystalline framework according to the Flory-Huggins theory. Besides, under higher pressures, the Nitta et al. model slightly underestimates the dimer coverage fractions. This result is also consistent with the Flory-Huggins theory which, without taking explicitly into account the lattice coordination number, introduces an error in the counting of the number of possible configurations at higher dimer concentrations. Figure 3b shows the adsorption of dimers on heterogeneous solids ( G_{a} = 1000, v_{a }= 0,3_{ }) with a random distribution of acid sites ( g = 1 ). A comparison between the results of simulations and those obtained with both the Langmuir and the Nitta et al. models shows that these two theoretical isotherms cannot describe the adsorption of dimers on random heterogeneous surfaces for large r_{a} values. Additional Monte Carlo simulations were performed for adsorption of dimers on these random heterogeneous surfaces for smaller values of r_{a} Figure 4 shows the influence of r_{a} on the mean absolute deviation between the values of coverage fraction obtained by the Nitta et al. model and those obtained by Monte Carlo simulations (for k_{b}f ranging from 10^{-5} to 10^{5}). In agreement with the results of Nitta et al. (1997), it is found that the Nitta et al. model (Nitta et al., 1984) yields good results when r_{a} is smaller than 100.

Molecular simulations of dimer adsorption on heterogeneous solid surfaces with different topographies are shown in Figure 5. Simulations were carried out for 18 cases. The parameter related to the energy difference between acid and nonacid sites ( r_{a }) assumed values equal to r_{a }=_{ } 1000 and r_{a }= 10000. Three values for the fraction of acid sites in the lattice (V_{a} = 0.1, 0.3, and 0.5) were used in the simulations. Additionally, the effect of the different energy topologies of the solids upon the adsorption isotherms was analyzed with the use of different grain sizes. Three values for randomly distributed grains for sites a, with sizes g = 1, 4 or 9, were used. It can be observed that the effect of the energy topology of the solids is more prominent in the lower coverage fraction regions (corresponding to lower pressure regions) and for higher fractions of acid sites. These results are expected since an increase in the energy difference between sites enhances the local concentration effect, which depends upon the ratio of grain size to adsorbed molecule size. On the other hand, under higher pressures, corresponding to the high coverage fractions, the isotherms are almost independent of the energy values, number fraction, and size of the acid-grain sites. This occurs because, under these pressures, almost all sites are occupied and the unique important contribution is the entropic configuration of the dimer molecules in the square lattice. This entropic contribution is well described by the Flory-Huggins equation.

Although the Nitta et al. model was developed to describe the adsorption of molecules on random heterogeneous surfaces (described here for grains with one site, g = 1), Figure 5 shows that this isotherm model overestimates or underestimates the coverage fractions, depending on pressure. However, for larger grains, i.e., for g = 9, the Nitta et al. model describes well the adsorption phenomena in a wide range of pressures. To better understand this behavior, a comparison of the coverage fractions of sites a and b ( q_{a} and q_{b }), obtained with the Nitta et al. model, with those obtained with simulations is presented in Figure 6. We have used the same parameter values as in the results shown in Figures 5c and 5f. It is possible to observe (Figures 6a and 6c) that the Nitta et al. model overestimates the coverage fraction of acid sites ( q_{a }) and underestimates the coverage fraction of nonacid sites ( q_{b }) for adsorption of dimers on random heterogeneous surfaces (grains with g = 1). Figures 6a and 6c also show that these results become more apparent with an increasing in the difference of in energy between acid and nonacid sites ( r_{a }= 1000_{ } and r_{a }= 10000 ). The calculated coverage fractions make use of quasi-chemical theory (equation (7)), which implicitly assumes the independence of the molecular segments. However, adsorption of one segment of a dimer molecule in a lattice requires that the second segment be adsorbed on a neighboring site, and consequently the independence of the molecular segments is broken. Therefore, the quasi-chemical theory is more adequate to describe the adsorption of dimers on patchwise heterogeneous surfaces because it has a higher probability of finding adjacent acid sites. We can observe in Figures 6b and 6d that the quasi-chemical theory predicts well the coverage fractions of acid ( q_{a} ) and nonacid ( q_{b }) sites for adsorption of dimers on patchwise heterogeneous surfaces (grains with g = 9).

In Figure 7, the results for the adsorption internal configurational energy per molecule, expressed in a nondimensional form defined by equation (16), obtained with simulations and with the Nitta et al. model are compared.

The right-hand side of the above equation was obtained from equation (1). In the equation (16), one can note that the constant line equal to zero represents adsorption on homogeneous solids (r_{a} = 1). Consistent with the inherent limitation of the quasi-chemical theory, Figure 7 shows that the internal configurational energy of dimer adsorption on patchwise heterogeneous surfaces (grains with g = 9) is better described by the Nitta et al. model.

CONCLUSIONS

The Monte Carlo method for a grand canonical ensemble was used to estimate adsorption of dimers on heterogeneous solids with surfaces characterized by two energetically distinct sites and with a random spatial distribution in grains with one, four and nine acid sites. The comparison between the adsorption of dimers estimated with simulations and that estimated with the Nitta et al. (1984) model demonstrates the predictive capability of the theoretical model for conditions under which sites have similar characteristic energies.

The molecular simulation results also show that the energy topography of the solids, with higher energy differences between sites, plays an important role in adsorption in low coverage fraction regions (low pressures). For these cases, the Nitta et al. model does not describe well the data on dimer adsorption on random heterogeneous surfaces (grains with one acid site), but describes reasonably well the adsorption of dimers on patchwise heterogeneous surfaces.

**NOMENCLATURE**

f | fugacity |

g | number of sites in the square clusters (grains) |

K | the Boltzmann constant |

K_{b} | the Henry constant for adsorption on homogeneous solid |

M | total number of sites in the lattice |

N | total number of dimers adsorbed in the lattice |

N_{aa} | number of dimers adsorbed on pairs of site a |

N_{ab} | number of dimers adsorbed on pairs of distinct sites |

N_{bb} | number of dimers adsorbed on pairs of site b |

P_{m} _{®n} | transition probability in going from one configurational state (m) to another state (n) |

r_{a} | relative strength of adsorption on site a when compared to that on site b |

T | temperature, K |

U | average configurational energy, J |

U_{n} | energy of the configurational state n, J |

Greek Letters | |

e_{a} | characteristic energy of sites a, J |

e_{b} | characteristic energy of sites b, J |

e_{aa} | energy of a dimer adsorbed on a pair of sites a, J |

e_{ab} | energy of a dimer adsorbed on a pair of distinct sites, J |

e_{bb} | energy of a dimer adsorbed on a pair of sites b, J |

F | nondimensional energy |

m | chemical potential, J/mol |

r_{n} | probability density of configuration n |

q | total coverage fraction |

q_{a} | coverage fraction of site a |

q_{b} | coverage fraction of site b |

n_{a} | number fraction of site a |

n_{b} | number fraction of site b |

**ACKNOWLEDGEMENTS**

This work was supported by CNPq, CAPES, PRH-ANP/MCT and PRONEX grant no. 124/96 (Brazilian Federal Government).

REFERENCES

Allen, M. P. and Tildesley D. J., Computer Simulation of Liquids. Oxford University Press, New York, 1987. [ Links ]

Hill, T. L., An Introduction to Statistical Thermodynamics. Dover Publications, Inc., New York, 1960. [ Links ]

Mathias, P. M., R. Kumar, J. D. Moyer, J. M. Schork, S. R. Srinivasan, S. R. Auviland, and O. Talu, Correlation of Multicomponent Gas Adsorption by the Dual-Site Langmuir Model. Application to Nitrogen/Oxygen Adsorption on 5A-Zeolite, Ind. Eng. Chem. Res., 35, 2477 (1996). [ Links ]

Nitta, T., M. Kuro-Oka, and T. Katayama, An Adsorption Isotherm of Multi-Site Occupance Model for Heterogeneous Surface, J. Chem. Eng. Japan, 17, 45 (1984). [ Links ]

Nitta, T., H. Kiryama, and T. Shigeta, Monte Carlo Simulation Study for Adsorption of Dimers on Random Heterogeneous Surfaces, Langmuir, 13, 903 (1997). [ Links ]

Ramirez-Pastor, A. J., M. S. Nazzaro, J. L. Riccardo, and G. Zgrablich, Dimer Physisorption on Heterogeneous Substrates, Surface Science, 341, 249 (1995). [ Links ]

Riccardo, J. L., V. Pereyra, G. Zgrablich, F. Rojas, V. Mayagoitia, and I. Kornhauser, Characterization of Energetic Surface Heterogeneity by a Dual Site-Bond Model, Langmuir, 9, 2730 (1993). [ Links ]

Rudzinski, W. and D. H. Everett, Adsorption of Gases on Heterogeneous Surfaces, Academic Press: London, 1992. [ Links ]

Smith, J. M. and C. H. van Ness, Introduction to Chemical Engineering Thermodynamics, Mc Graw-Hill Book Company, New York, 1987. [ Links ]