Langmuir 2003, 19, 1429-1438
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Monte Carlo Simulations of the Adsorption of Chainlike Molecules on Two-Dimensional Heterogeneous Surfaces Vladimir F. Cabral, Charlles R. A. Abreu, Marcelo Castier, and Frederico W. Tavares* Escola de Quı´mica, Universidade Federal do Rio de Janeiro, Caixa Postal 68542, CEP 21949-900, Rio de Janeiro, RJ, Brazil Received July 10, 2002. In Final Form: October 29, 2002 In this work, the Monte Carlo method of molecular simulation for a grand canonical ensemble (µ, M, and T specified) was used to simulate pure component and binary mixture adsorption of chainlike molecules in homogeneous and heterogeneous surfaces. The generated simulation data were used for the evaluation of an isotherm model. The simulations and the theoretical isotherm are based on the lattice gas model. Each surface site can be occupied by only one molecular segment. The adsorbed segments only interact with their contacting neighbors, through a characteristic potential contact energy. The studied isotherm model was adequate for describing adsorption in homogeneous solids, mainly for mixtures with low interaction energy.
Introduction Separation processes that use solid adsorbents are currently under intense investigation because of the growing need for energy savings. Most of the solids used in adsorption processes are heterogeneous, that is, they present distributions of active sites with different adsorption energies. The existing models for homogeneous solids usually fail in the description of adsorption in these highly nonideal systems. In this way, there is an increasing number of models based on the energy heterogeneity of the active sites. Two heterogeneous surface models are frequently used: the patchwise model and the random distribution model. The first model was suggested by Langmuir and popularized by Ross and Oliver.1 In this model, the adsorption surface is composed of large clusters. The second model assumes that a random distribution of the sites, with different adsorption energies, exists on the solid surface. Despite this, comparative studies of isotherm models for the adsorption of multicomponent mixtures in heterogeneous solids are still scarce. In this context, the molecular simulation technique can be a useful tool in the analysis of the hypotheses and simplifications used in such isotherms. Although the molecular simulation techniques were proposed in the 1950s, their applications to the adsorption phenomenon are more recent. The method commonly used to simulate this phenomenon is the Monte Carlo technique for a grand canonical ensemble, where the chemical potential (µ), the number of sites (M), and the temperature (T) are specified. Examples of the use of the Monte Carlo technique in adsorption are the work of Vlugt et al.2 and of Macedonia and Maginn.3 A common feature of these publications is that they use detailed force fields for the solid and for the fluid. * Corresponding author. E-mail:
[email protected]. Phone: +55-21-25627650. Fax: +55-21-25627631. (1) Ross, S.; Oliver, J. P. On Physical Adsorption; Interscience: New York, 1964. (2) Vlugt, T. J. H.; Smit, B.; Krishna, R. J. Phys. Chem. B 1999, 103, 1102. (3) Macedonia, M. D.; Maginn, E. J. Fluid Phase Equilib. 1999, 160, 19.
Other works use less detailed models, which still capture the major aspects of adsorption phenomena. Several recent papers followed this approach. In these publications, the authors supposed that the adsorbent surface is represented by a two-dimensional square lattice of M active sites. Ramirez-Pastor et al.4 studied the adsorption of dimers on heterogeneous surfaces, using experimental adsorption isotherms for O2 and N2 adsorbed on zeolites 5A and 10X to test the reliability of their simulation model. The parameters of the simulation model were adjusted to fit the experimental data. Nitta et al.5 used the Monte Carlo method to simulate the adsorption of dimers on heterogeneous solid surfaces represented by square lattices with two types of sites, each of them characterized by a different adsorption energy. The active sites were randomly distributed. Ramirez-Pastor et al.6 used Monte Carlo simulations to study the adsorption of noninteracting homonuclear linear k-mers on heterogeneous surfaces. The authors modeled the heterogeneous surface with two kinds of sites. These sites formed square patches distributed at random or in a chessboardlike ordered domain on a two-dimensional square lattice. Bulnes et al.7 studied the adsorption of binary mixtures on solid heterogeneous substrates using the Monte Carlo simulation in the framework of the lattice gas model. The objective of the present work is to use the Monte Carlo technique for a grand canonical ensemble (µ, M, and T specified) as a tool in the analysis of the hypotheses and simplifications used to derive isotherm models. For this purpose, the results obtained by molecular simulation are compared to those predicted by the theoretical models. Molecular Simulation In this work, the Monte Carlo technique is used to simulate the adsorption of chainlike molecules on homogeneous and heterogeneous surfaces. The lattice gas model (4) Ramirez-Pastor, A. J.; Nazzaro, M. S.; Riccardo, J. L.; Zgrablich, G. Surf. Sci. 1995, 341, 249. (5) Nitta, T.; Kiryama, H.; Shigeta, T. Langmuir 1997, 13, 903. (6) Ramirez-Pastor, A. J.; Pereyra, V. D.; Riccardo, J. L. Langmuir 2000, 16, 682. (7) Bulnes, F.; Ramirez-Pastor, A. J.; Pereyra, V. D. J. Mol. Catal. A: Chem. 2001, 167, 129.
10.1021/la026217j CCC: $25.00 © 2003 American Chemical Society Published on Web 01/21/2003
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is used. In this way, each molecular segment occupies a specific site of the solid surface. Each site is occupied by only one molecular segment. The most important parameters in a lattice model with chainlike molecules are (1) the total number of sites (M); (2) the number of neighboring sites to each of them, named coordination number (Z); (3) the adsorption energy between a molecular segment and a site of the solid (); (4) the number of segments of each molecule (m); and (5) the interaction energy between two adsorbed segments in neighboring sites (ω). The Monte Carlo method for a grand canonical ensemble (T, M, µ1, µ2, ..., µnc specified), according to the Metropolis algorithm (Allen and Tildesley8), consists of three basic movements: displacement, insertion, and removal of an adsorbed molecule. The transition probability from a configurational state m to a new state n is expressed by
{ }
Pmfn ) min 1,
Fn Fm
(1)
where Fn/Fm is the ratio between the probability densities of the configurational states n and m. The movement is accepted if such probability Pmfn is larger than a number randomly generated between 0 and 1. The following sections detail each movement. Displacement Movement. The adsorbed molecule that will be moved is randomly chosen. One end of the molecule is chosen at random to be the “head”, while the other is the “tail”. The head is moved to a new position on the lattice, all the other segments move one site along the chain, and the tail position becomes vacant. This type of motion is termed “reptation” (Allen and Tildesley8). If the site chosen as the new position of the head is already occupied, the movement is immediately rejected. If the displacement is possible, the ratio between the probability densities of the new state (n) and the old state (m) is calculated by the following expression (Allen and Tildesley8):
[
]
(Un - Um) Fn ) exp Fm KT
(2)
where T is the temperature, K is Boltzmann’s constant, and Um is the configurational energy of the state m, which can be obtained by nc ns
Um ) -
∑ ∑ i)1 j)1
N(m) ij ji
-
1 nc
nc
Nc(m) ∑ ∑ ik ωik 2i)1k)1
(3)
or an occupied site is found. Next, the energy of this new configuration is computed and the ratio between the probability densities of the new (n) and old (m) states (Allen and Tildesley8) is calculated as follows:
[
]
Fn (Un - Um) µi M exp ) Fm Ni + 1 KT KT
(4)
where Ni is the total number of molecules of the component i adsorbed in the old state and µi is the chemical potential of such component. The movement is accepted if a generated random number between 0 and 1 is lower than the transition probability (Pmfn) defined by eq 1. Removal Movement. Among the adsorbed molecules, one is randomly chosen and removed. Then, the energy of this new configuration is calculated and the ratio between probability densities of the new (n) and old (m) states (Allen and Tildesley8) is computed as follows:
[
]
(Un - Um) µi F n Ni exp ) Fm M KT KT
(5)
The movement is accepted if a random number generated between 0 and 1 is lower than the transition probability (Pmfn) defined by eq 1. Relationship with Thermodynamics The relationship connecting the fugacity, ˆfi, and the chemical potential of component i in the gaseous phase is given by the classical equation
µi ) µ0i + KT ln(fˆi)
(6)
where µ0i is the reference-state chemical potential for the ideal gas under atmospheric pressure and at the temperature of the system. On the other hand, the Henry constant for the adsorption on a homogeneous solid containing only sites of a given type b is (Hill9)
( ) ( )
Kbi ) ζi exp
µ0i mibi exp KT KT
(7)
where ζi 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 i compared to the same degrees of freedom in the condition of the ideal gas reference state. In simulations, as well as in the models, the value of ζi was incorporated to Kbi (or was set ζi ) 1). Therefore, from eqs 6 and 7, an expression for the chemical potential is obtained:
is the total number of segments of kind i where N(m) ij adsorbed in sites the kind j, ji is the adsorption energy between them, Nc(m) ik is the number of contacts between adsorbed segments of the kinds i and k, ωik is the interaction energy between segments i and k adsorbed in neighboring sites, nc is the number of adsorbed components, and ns is the number of kinds of sites. Insertion Movement. To add a molecule to the lattice, a position is chosen to place the first segment of the molecule and, through another random choice, a neighboring site is selected for adding the second segment. If one of these sites is already occupied, the movement is rejected. To insert the third segment, only (Z - 1) neighboring sites are available. The procedure described above is repeated until the molecule is completely inserted
The solid is modeled as a square lattice of dimension 100 × 100 (M ) 10 000), and its heterogeneity is represented by the existence of two kinds of sites, characterized by the energies a and b. The sites that perform the strongest connection (larger energy, a) are called active sites. The fraction of these sites, which are randomly distributed over the solid matrix, is denominated νa. Figure 1 shows an example of a square lattice with 30% active sites (νa ) 0.3). To minimize the effect of the
(8) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, 1987.
(9) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover: New York, 1986.
mibi µi ) ln(Kbi ˆfi) KT KT
(8)
Solid Model
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The quasi-chemical equations have to be simultaneously solved in order to obtain the isotherms. Such equations are
(
∑R where
)
νRrRiYi
1+
∑j rRjθjYj
-1)0
(i ) 1 ... nc)
θbi 1 - θb
θiYi )
(12)
(13)
and
(
rai ) exp
Figure 1. Illustrative diagram of a 40 × 40 square lattice with 30% randomly distributed active sites.
solid size, periodic boundary conditions are considered both for distributing the active sites on the solid and for moving a molecule in the lattice. Isotherm Model Romanielo et al.10 published a modification of the Nitta et al.11 model for the adsorption of chainlike molecules on homogeneous solids. The isotherm model is based on the lattice gas theory, where the Bragg-Williams approach is used in the determination of the most probable configuration. The configurational entropy is given by the Guggenheim approach for noncyclical molecules. The average configurational energy is calculated by the direct sum of the pair interactions, corresponding to the random distribution of the molecules. Abreu et al.12 extended (see the Appendix) the model for the case of heterogeneous solids, where the heterogeneity was estimated by the quasi-chemical theory as described by Nitta et al.13 In that work, the authors only considered the adsorption of pure components with no contact interaction. The present work considers the case of mixtures of chainlike molecules adsorbed on heterogeneous surfaces with two different kinds of sites. Neighboring segments of components i and k interact with contact energy ωik. The expressions of the extended model are
( )( ) nc
1-
miKbi ˆfi )
∑ βkθk k)1 θi
mi-1
nc
1-
{
exp -
1
nc
∑ θbk k)1
[ ( )] } qk
∑ qi + mi m 2KTk)1
Zqi ) mi(Z - 2) + 2 θi )
miNi M
Zωkiθk
(9)
k
(10)
nc
θ)
θi ∑ i)1
(14)
In these equations, ˆfi is the fugacity for component i in the gas phase, Kbi is the Henry constant of component i on a homogeneous surface composed only of sites b, Ri is the adsorption energy between a site R and a segment of component i, qi is a parameter related to the molecular area (calculated using eq 10), Z is the coordination number of the lattice (in this study, Z ) 4), T is the absolute temperature of the system, βi ) (mi - qi)/mi is a constant that relates the structural parameters mi and qi of the molecule i, ωik is the interaction energy parameter between the adsorbed segments of the molecules i and k in neighboring sites, and θi and θbi are the coverage fractions of component i on the whole lattice and the coverage fraction of component i on nonactive sites, respectively. Using eq 13, eq 12 can be rewritten as
[ ] ( ) ∑ ( )
νRrRi
θi )
∑R
1+
1 - θb
rRj
j
where
θbi
θbj
(i ) 1 ... nc) (15)
1 - θb
nc
θb )
θbi ∑ i)1
(16)
Then, by substitution of eq 15 into eq 9, nc nonlinear equations for θbi (i ) 1 ... nc) are generated. Here, this system is solved using a Newton-Raphson algorithm with finite-difference approximation to the Jacobian. Results and Discussion Pure Component. Covering fraction (θ) results were obtained in different pressures for systems of molecules with 2 and 8 segments. The results were compared to those calculated using the model presented in the preceding section.
mi
θbi
)
ai - bi KT
(11)
(10) Romanielo, L. R.; Tavares, F. W.; Rajagopal, K. Tercer Simposio Latino Americano de Propriedades de Fluidos y Equilibrio de Fases para el Disen˜o de Procesos Quimicos, Oaxaca, 1992. (11) Nitta, T.; Shigetomi, T.; Kuro-Oka, M.; Katayama, T. J. Chem. Eng. Jpn. 1984, 17, 39. (12) Abreu, C. R. A.; Telles, A. S.; Tavares, F. W. V Conferencia IberoAmericana Sobre Equilibrio entre Fases para el disen˜o de Procesos, Vigo, 1999. (13) Nitta, T.; Kuro-Oka, M.; Katayama, T. J. Chem. Eng. Jpn. 1984, 17, 45.
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Figure 2. Adsorption of molecules with 2 and 8 segments with adsorbate-adsorbate interactions in homogeneous solids.
Figure 3. Adsorption of molecules with 2 and 8 segments with adsorbate-adsorbate interactions in heterogeneous solids.
In all cases, the sampling was made in a lattice composed by 10 000 sites (100 × 100). To reach equilibrium, 2 × 106 Monte Carlo steps were performed when m ) 2 and 4 × 106 steps when m ) 8, in each simulated point. The average properties of the system were computed at each 105 steps. This procedure was repeated 10 times, for different randomly generated solids with the same fraction of active sites. The global average and the standard deviation of the properties, at each point, were calculated with the 100 obtained average values. In the case of pure component, the covering fraction results are shown as functions of the product between the Henry constant in nonactive sites (Kb) and the fugacity (f). The variable Kbf can be defined as a dimensionless fugacity that is directly related to the system’s pressure. Another analyzed parameter is the interaction energy (ω) between two adsorbed segments in neighboring sites. The isotherm model is evaluated in the description of systems where adsorbate-adsorbate interactions take
place. The solid surface heterogeneity is characterized by the existence of two types of sites with different adsorption energies. Figure 2 shows the covering fraction results for homogeneous solids as a function of Kbf. It can be seen that the model presents larger deviations in the simulations of molecules with 8 segments. These deviations increase with the interaction energy between the adsorbed segments. For the molecules with m ) 8 and ω/KT ) 1, both the simulations and the model predict the coexistence of two adsorbed phases at the same pressure, that is, at the same value of Kbf (approximately 1 × 10-3). This means that two possible covering fractions exist in equilibrium with the gaseous phase at the specified temperature and pressure. Figures 3 and 4 show the results of adsorption on heterogeneous solids. The parameter ra is set equal to 10 (Figure 3) and 100 (Figure 4). In the both cases, the active site fraction was imposed to be 30% (νa ) 0.3). In the case where ra ) 10 (Figure 3), the model presented better results
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Figure 5. Adsorption of a binary mixture in a homogeneous solid in the pressure of 1 atm, with m1 ) 2 and m2 ) 8, without energy interaction between the adsorbed segments. Figure 4. Adsorption of molecules with 2 and 8 segments with adsorbate-adsorbate interactions in heterogeneous solids.
for the adsorption of dimers. The model deviations increase with the interaction energy between the adsorbed segments. In the situation in which ra ) 100 (Figure 4), the model also showed better results for dimers. For the adsorption on heterogeneous solids in systems with larger energy differences between the sites and also with larger interaction energy, the model presented less accurate results. In this point, our study shows a similarity with the results of Nitta et al.:5 the isotherms studied in this work and by Nitta presented larger deviations for ra > 50. The present work also shows that for ω/KT < 0.5, the model predicts the phenomenon of the adsorption very well. In general, when compared to Nitta’s isotherm, the proposed model presents similar results for systems composed by molecules with 1 or 2 segments. However, for cases of molecules with 4 or more segments, the proposed model presents better agreement with the simulation results. This fact reinforces the conclusion that Guggenheim’s approach becomes more appropriate than
Flory’s for accounting for the entropy of the lattice, filled with chainlike molecules, as the number of segments of the molecules increases. Binary Mixture. For the case of binary mixtures, 4 × 106 Monte Carlo steps were performed at each simulated point in order to equilibrate the system. For the calculation of the average values of the thermodynamic properties, 106 steps were used. Each average value was calculated using 105 steps. In this study, the following combining rule was used for evaluating the cross contact energy (ωij):
ωij ) φxωiiωjj
(17)
where φ is a binary interaction parameter for correlating equilibrium data in binary systems. Figures 5 and 6 show the results for the adsorption of binary mixtures with m ) 2 (component 1) and m ) 8 (component 2), with no contact energy, under the pressures of 1 and 10 atm. The predicted behaviors for the phase diagrams and values of total covering fraction (θ) are
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Figure 6. Adsorption of a binary mixture in a homogeneous solid in the pressure of 10 atm, with m1 ) 2 and m2 ) 8, without energy interaction between the adsorbed segments.
shown in these illustrations. The Henry constants of components with m ) 2 and m ) 8 were set equal to 1 and 2 atm-1, respectively. The model presented good results in the prediction of adsorption in both situations. Component 1 adsorbes preferentially on the surface, even though component 2 has a larger value of the Henry constant. This can be explained by means of the entropic effects. This occurs because the molecules with 2 segments are easier to place in the lattice than those with 8 segments. Figures 7-9 show results for systems in which the adsorbed molecular segments present interactions with their first neighbors. All the simulations were performed at the pressure of 1 atm. The Henry constants of components with m ) 2 and m ) 8 were set equal to 1 atm-1. The value of the parameter ω/KT was fixed at 0.2 for the molecule with 2 segments and at 0.6 for the molecule with 8 segments. The parameter value for the cross interaction was calculated through the combining rule defined in eq 12. In this way, we obtained the diagrams of phase equilibrium and of total covering fraction for different values of the parameter φ (1, 0.9, and 0.4). When parameter φ is close to unity (Figures 7 and 8), the simulations and the model do not present inversions
Cabral et al.
Figure 7. Adsorption of a binary mixture in a homogeneous solid in the pressure of 1 atm, with m1 ) 2 and m2 ) 8 and with energy interaction between the adsorbed segments (φ ) 1).
in selectivity or maxima in the covering fractions, respectively. In both cases, the model represents well the simulation data. Figure 9 presents the case in which parameter φ is equal to 0.4. In this situation, the simulations and the model show the existence of an azeotrope. However, the values of the azeotrope composition obtained by the simulations and by the model do not agree. The simulations predict an azeotrope at a composition of approximately 0.3 (molar fraction of component 1), while the model predicts an azeotrope of composition 0.5. To verify only the effect of surface heterogeneity in the phenomenon of azeotropy, the following test was performed: the simulation of mixture adsorption (m1 ) 2 and m2 ) 8) with no contact energy (ω1/KT ) 0 and ω2/KT ) 0). The component 1 had its parameter ra1 equal to 1 × 101 and the parameter ra2 of component 2 was set equal to 1 × 101, 1 × 102, and 1 × 103. Both components had their Henry’s constants set equal to 1 atm-1. The simulations were performed at a pressure of 1 atm. The surface was modeled with a fraction of active sites equal to νa )
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Figure 8. Adsorption of a binary mixture in a homogeneous solid in the pressure of 1 atm, with m1 ) 2 and m2 ) 8 and with energy interaction between the adsorbed segments (φ ) 0.9).
Figure 9. Adsorption of a binary mixture in a homogeneous solid in the pressure of 1 atm, with m1 ) 2 and m2 ) 8 and with energy interaction between the adsorbed segments (φ ) 0.4).
0.25. The results of this test are shown in Figure 10. In the case where the components had the same value of the parameter ra (ra1 ) 10 and ra2 ) 10), the simulations and the model predict that component 1 (m1 ) 2) adsorbs preferentially. The covering fraction (θ) is almost constant in all ranges of adsorbate fraction of component 1, but one maximum exists when the adsorbate fraction of component 1 is close to 1. In the other cases, the simulations and the model predict the phenomenon of azeotropy, where, initially, the preferential adsorption of component 1 exists. The model deviations increase with the difference between the parameters ra of the components.
The studied model demonstrated good capacity to represent the adsorption of chainlike molecules in homogeneous solids, in which the adsorbed segments interact with their neighbors with a potential interaction energy (ω). In the case of heterogeneous solids, the model showed larger deviations for systems with molecules of 8 segments than for those with 2 segments and larger values of the interaction parameter between adsorbed segments (ω). In this way, the model deviations increase in comparison to the simulated results when the number of segments, the energy difference between the sites, and/or the potential interaction energy between the adsorbed segments increase. In the evaluation of adsorption of binary mixtures in homogeneous solids, the model demonstrated a good capacity to represent the behavior of systems in which the adsorbed segments can interact or not with their neighbors. Therefore, the model is adequate for representing the adsorption of mixtures of chainlike molecules with no
Conclusions In this work, the adsorption of chainlike molecules on homogeneous and heterogeneous surfaces was simulated. The Monte Carlo simulation for a grand canonical ensemble was extended for binary mixture adsorption. The simulated systems presented different behaviors, ranging from ideal to strongly nonideal (azeotropy).
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Figure 10. Adsorption of a binary mixture in a heterogeneous solid in the pressure of 1 atm, with m1 ) 2 and m2 ) 8 and no energy interaction between the adsorbed segments.
interaction between neighboring segments (ω11 f 0 and ω22 f 0). When interactions exist, the results of the simulations and the model agreed for systems with φ close to unity. For the system with φ ) 0.4, the model presented larger deviations. Acknowledgment. This work was supported by CNPq, FAPERJ, ANP/PRH-13, and by PRONEX Grant No. 124/96 (Brazilian Federal Government).
Appendix Romanielo14
presents a modification of Nitta’s model (Nitta et al.11). The differences between these two formulations are in the approaches used for evaluating the number of configurational states of a system and its average configurational energy. (14) Romanielo, L. R. Adsorc¸ a˜o de Gases Multicomponentes. Msc Thesis, PEQ/COPPE-UFRJ, Rio de Janeiro, 1991.
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The canonical partition function (Q) of a system composed by N molecules adsorbed on M sites is given by
Q ) jNg(N,M) exp(E h c/KT)
(A.1)
where jN is the partition function for an isolated molecule, g is the number of ways to distribute N molecules over M sites, and E h c is the configurational energy of the system. When the surface is homogeneous, Guggenheim’s approach can be used for evaluating the number of possible configurations of the system (g′(N,M)):
[
]
(qN + M - mN)! M! g′(N,M) ) ζN M! N!(M - mN)!
[
]
(qN + M - mN)! M! ζN M! N!(M - mN)!
Z/2
N/R!
where NR denotes the number of occupied sites of kind R and N/R is the number of occupied sites R given by a random distribution in the homogeneous surface. For a surface with two kinds of sites (a and b), NR is subject to the following constraint:
Na + Nb ) mN
mN M
(A.5)
θa )
Na Mνa
(A.6)
Nb
(A.7)
M(1 - νa)
The constraint condition of eq A.4 can be rewritten by use of surface covering as follows:
raY Y + (1 - νa) )1 νa 1 + raθY 1 + θY
θY )
θb 1 - θb
(
ra ) exp
m-1
θb 1 - θb
m
)
ω exp -Zqθ KT
(m - 1) β)2 Zm
(A.12)
(A.13)
The site coverages θ and θb are calculated by solving simultaneously the eqs A.8 and A.12. The above procedure can be easily applied for the case of multicomponent adsorption. In this case, the partition function (Q), the number of configurations (g(N,M,NRi)), and the configurational energy (E h c) are written as
( ) E hc
nc
jiN ) ∑ g(N,M,NRi) exp ∏ KT i)1
Q)(
i
(NRi)
nc
g(N,M,NRi) )
[
(ζi)N ∏ i)1
(M -
∑j
nc
(Ni)! ∏ i)1
mjNj)!
] ()
∑j qjNi + M - ∑j mjNj)! M!
E hc ) -
1
(A.14)
M!
i
(
∑∑ 2 i j
Z/2
∏ ∏i R
NRi!
/ NRi !
ZqjmjNiNjωij M
(A.15)
(A.16)
where N1, N2, ..., Nnc are the number of adsorbed molecules of the components 1, 2, ..., nc, respectively, and NRi is the number of adsorption pairs of sites R and a component i. The adsorption energy per site for the pair Ri is named as Ri. In this case, the variable NRi is subject to the following constraint equation:
(A.8)
where Y and ra are defined as
) ( ) (
1 - βθ θ
(A.4)
θ)
(A.11)
where f is the fugacity of a molecule in the gas phase, Kb is the Henry constant for a hypothetical homogeneous surface composed only by sites b, and
Three surface coverings, θ, θa, and θb, are defined as
θb )
(
mKbf )
(A.3)
-ZqmN2 ω 2M
where ω is the interaction parameter between segments adsorbed on neighboring sites. The equation for the pure component isotherm is obtained by minimizing the Helmholtz free energy subject to the constraint of eq A.4. In this way, the isotherm equation is given by
(A.2)
NR!
∏ R
E hc )
Z/2
where ζ is the constant relating to flexibility and symmetry of the molecule; Zq is the external area of the molecule, where Zq ) m(Z - 2) + 2, for noncyclical molecules; m is the number of sites occupied by molecules; and Z is the coordination number of the lattice. When the surface is heterogeneous, the variable g must be modified to take into account the entropic effect due to the preferential adsorption on more energetic sites. One approach for this problem is to use the quasi-chemical approximation. In this case, g is given by
g(N,M,NR) )
The average configurational energy given by random distribution is obtained by the following equation:
Nai + Nbi ) miNi
(i ) 1, ..., nc)
(A.17)
The above equation can be rewritten as
(A.9)
)
a - b KT
(A.10)
νa 1+
raiYi
Yi
∑j
∑j θjYj
+ (1 - νa) rajθjYj 1+
)1
(i ) 1, ..., nc) (A.18)
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given by
where
θbi
θiYi )
1-
∑j θbj
1-
miKbi ˆfi )
)
ai - bi rai ) exp KT
∑
(A.20)
Finally, the adsorption isotherm for a component i is
LA026217J
nc
1-
θi
exp -
mi
θbi
βkθk
k)1
{
and
(
( )( ) mi-1
nc
(A.19)
∑ θbk
[ ( )] } k)1
1
nc
qk
∑ qi + mi m 2KTk)1
k
Zωkiθk
(A.21)
