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Multilayer adsorption of gases in solids
Colloids and Surfaces A: Physicochem. Eng. Aspects 219 (2003) 187 /192 www.elsevier.com/locate/colsurfa
Multilayer adsorption of gases in solids Abderrahim Saber a,b,*, S. Dal Toe` a, S. Lo Russo a, G. Mattei a a
INFM, Dipartimento di Fisica, Universita` di Padova, Via Marzolo 8, 35131 Padua, Italy b The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy Received 26 August 2002; accepted 21 January 2003
Abstract Adsorption phenomena of gas in solids are investigated using the Ono /Kondo lattice model. Layer concentrations rn (n /1, 2, . . . L ) and Gibbs adsorption G are presented versus the bulk density rb and temperature. It is found that the density in core region (inner layers) and the density of the first layers next to the walls exhibit an opposite temperature dependence: while the density near the walls increases with decreasing temperature the density of the core region decreases to reach rb as T increases. The density profile of the layers is also presented. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Lattice model; Gibbs adsorption; Density profile
1. Introduction The study of the adsorption in porous materials has important application in fields such as separation and purification, chemical reactions and catalysis, the storage of gases, as well as many others. One popular approach for the modeling of adsorption phenomena is lattice /gas model, which provides the capability to predict adsorption isotherms and phase behavior in the pore including phase transitions (e.g. wetting, capillary condensation) and hysteresis effects.
* Corresponding author. Permanent address: De´partement de Physique, Faculte´ des Sciences, Universite´ Moulay Ismail, B.P. 4010, Mekne`s Morocco. Tel.: /39-049-827-7039; fax: / 39-049-827-7003. E-mail address: [email protected] (A. Saber).
Ising [1] gave an exact solution to the onedimensional lattice problem in 1925. Onsager [2] in 1944 was able to get an exact solution for behaviour of two-dimensional lattice at the critical density, but to date, no general solution has been obtained. The three-dimensional problem is even more complicated [3 /6]. No three-dimensional problem yet has a solution that is both complete and exact [6]. However, approximate methods, such as cluster expansion [3 /5], low-temperature expansions, and high-temperature expansions [7], have provided important insights. The article is organized as follows. In the next section the basic points of the model used to the present work, is briefly reviewed. This is followed by a presentation of numerical results, and their discussion, in Section 3. A brief conclusion is given in Section 4.
0927-7757/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0927-7757(03)00033-5
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A. Saber et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 219 (2003) 187 /192
2. Model Consider a pure component (A) distributed on a fixed lattice with L layers, such situation is shown in Fig. 1. Each site may either be occupied by a molecule of A or may be empty. An expression of the free energy for the system, F /H/TS , can be derived by determining the Hamiltonian H and the entropy S for the gas confined in the lattice. Following Aranovich and Donohue [8], this is done using the mean field approximation [9] yielding: XX X Ho AA xnr xn?r? o AS xnr ; (1) n;n?
r; r?
nf1;Lg;r
where xnr is the occupation number on the lattice site (n, r ), (n , n ?) are plane indices and (r, r?) are different sites of the planes. xnr /1 if the lattice site is occupied by a molecule A and zero otherwise, oAS and oAA are, respectively, the energies for adsorbate /surface and adsorbate/adsorbate interactions. The gas density in the n -th layer, expressed as the fraction of lattice sites in layer n which are occupied by a molecule of A, is independent of r, so we can write rn x¯ nr which is the mean value of the occupation number. We now seek to find the entropy of the system. Consider taking an adsorbate molecule from the layer n and moving it to an empty site in the bulk (outside the pore). This is equivalent to the exchange of a molecule with a vacancy: An V Vn A
(2)
where A is the adsorbate molecule, and V is the vacancy that it fills (and vice versa). If this exchange occurs at equilibrium, then: DH TDS 0
(3)
where DH and DS are the enthalpy and entropy changes and T is the absolute temperature. The value of DS can be represented in the form: DS kB ln W1 kB ln W2
(4)
Fig. 1. Sketch of the lattice model used here for gas adsorption in solids.
where W1 is the number of configurations where the certain site in the layer n is occupied by an adsorbate molecule and the infinitely distant site is empty, W2 is the number of configurations where the infinitely distant site is occupied by an adsorbate molecule and the certain site in the layer n is empty, and k is the Boltzmann’s constant. If the overall number of configurations for the system is W0, then in the mean field lattice approximation: W1 =W0 rn (1rb )
(5)
and W2 =W0 (1rn )rb
(6)
where rn is the density that the certain site in the n-th layer is occupied by an adsorbate molecule A, and rb is the density of adsorbate in the bulk. Substituting Eqs. (5) and (6) into Eq. (4) we have: r (1 rb ) DS kB T ln n (7) rb (1 rn ) where kB is the Boltzmann constant. Using the Hamiltonian of the system Eq. (1) and in the frame of the mean-field approximation the enthalpy is:
DH
o AS o AA (z2 r1 z1 r2 z0 rb ) for n 1 o AA (z1 rn1 z2 rn z1 rn1 z0 rb ) for 2 5n5L1
(8)
A. Saber et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 219 (2003) 187 /192
where z1 and z2 are, respectively, the interlayer and intralayer coordination numbers (note that z0 / 2z1/z2 is the coordination number for the threedimensional lattice). From Eqs. (3), (7) and (8) it follows that: r1 (1 rb ) ln o AS =kB T rb (1 r1 ) (9) (z2 r1 z1 r2 z0 rb )o AA =kB T 0 for n 1 rn (1 rb ) ln rb (1 rn ) [z1 (rn1 rb )z2 (rn rb )
(10)
z1 (rn1 rb )]o AA =kB T 0 for
25 n5L1
The periodic conditions have to be satisfied, namely: (11)
r1 rL
Eqs. (9) /(11) constitute a nonlinear equations system of L equations in L unknowns. For a given values of the coordination numbers z1 and z2, (z1 /1, z2 /4 in the case of a simple cubic (sc) lattice; z1 /1, z2 /6 for simple hexagonal (sh) lattice; etc. . .), energy parameters EAS /oAS/kBT and EAA /oAA/kBT , and bulk density rb , this equation system can be solved for the density distribution vector r (r1, r2, . . . rL) describing the fractions of A in each of the L layers. The Gibbs adsorption for a lattice is given by: G
L X
(rn rb )
(12)
n1
By numerical treatment, iterative method of successive substitutions is used to solve the set of L nonlinear equations Eqs. (9) /(11) with respect to L unknowns, r1, r2, . . . rL , for a fixed parameters of oAA, oAS and the temperature T . Therefore, using Eq. (12), we can plot the adsorption isotherms. The Ono /Kondo lattice model is able to predict all known types of adsorption behavior, by changing two energetic parameters
189
(energies for adsorbate/adsorbate and adsorbate / adsorbent interactions).
3. Results and discussion In this paper, we fix the energies for adsorbate / adsorbate and adsorbate/adsorbent at oAA/kB / /430 K, oAS/kB /40 K, respectively, which are arbitrarily chosen and in the most of the study the number of layers is L /4. Let us begin with the evaluation of the layer concentration as a function of the bulk one for the case of four-layer problem and for a simple cubic and simple hexagonal lattices at two different temperatures 300 and 77 K. Fig. 2(a /b) show the plot of the layer density rn (with r1 (/r4) and r2 (/r3) for the reasons of symmetry) of lattice sites occupied by molecules of A as a function of the bulk mole fraction. We see that, the layer concentration is always increasing with the increase of the bulk concentration for a fixed temperature and it decreases with increasing the coordination number and the temperature for a fixed rb (layers (1 and 4) (see Fig. 2(a))), however, in the core region (layers (2 and 3)) Fig. 2(b), we have just the opposite dependence of coordination number and temperature. We can see clearly this behavior in Fig. 3, where the layer concentrations r1(4)(T ) and r2(3)(T) are depicted as function of temperature for (sc) and (sh) lattices and bulk mole fraction of rb /0.2. As T is lowered, the density increases in the surface layers. The increase is caused by stronger adsorption at the walls due to lower (average) kinetic energy of gas molecules while the core region the density remains constant and never exceeds the bulk density rb /0.2. However, as T is lowered further, r2(T ) eventually decreases because of low kinetic energy which prevents the depletion of the gas. Notice that the layer concentration decreases with the coordination number in the surfaces and we have the opposite tendency in the inner layers. Fig. 4 shows typical adsorption isotherms behaviour at 77 and 300 K and for sc and sh structures. These isotherms show standard type I behaviour of micropores materials [10]. At 300 K the gasadsorption at pores of both structures is mono-
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Fig. 3. Layer concentrations as a function of temperature is shown for a simple cubic (sc) and simple hexagonal (sh) lattices for rb /0.2, oAA/kB //430 K, oAS/kB /40 K.
Fig. 2. (a /b) Layer concentrations as a function of the bulk concentration for L /4 and oAA/kB //430 K, oAS/kB /40 K at two different temperatures T/77 and 300 K and for two crystalline lattices simple cubic (sc) and simple hexagonal (sh).
tonically increasing, while lowering the temperature to 77 K, the adsorption behaviour is characterized by a steep increase in the adsorption-
Fig. 4. Adsorption isotherms as a function of bulk density rb for L/4, oAA/kB //430 K, oAS/kB /40 K at two different temperatures T /300 and 77 K and for the simple cubic (sc) and simple hexagonal (sh).
A. Saber et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 219 (2003) 187 /192
191
isotherms at low bulk densities, a maximum appeared at rb /0.1051 and 0.1391 for sc and sh, respectively, when the saturation of gas-molecules in the first adsorbed layers is reached and a monotonic decrease for higher bulk densities. rb is fixed, the adsorption isotherm increases with the decrease of the coordination number (a dense adsorbent is likely to have less sites of adsorption, since the process of densification reduces the number of sites available by blocking the pores) and temperature, respectively, this behaviour is clearly seen in Fig. 5 for the adsorption along isochores as function of temperature for rb /0.2. The temperature dependence of G (T ) is identical to that of the layer density r1(T ). Qualitative results has been found using the crossover theory [11]. Finally, in Fig. 6(a /b) we show the density profiles of the gas A in different layers for L /40. These is drawn for a two selected values of the bulk density rb /0.05 and 0.2 and at T /77 and
Fig. 6. (a /b) Layer concentration profiles for L /40 layers for simple cubic lattice, (a) T /77 K and for two values of the bulk density rb /0.2 and 0.05 (b) rb /0.2, at T /77 and 300 K. oAA/kB //430 K, oAS/kB /40 K. Fig. 5. Adsorption isochores as a function of temperature for L /4, rb /0.2 , oAA/kB //430 K, oAS/kB /40 K and for the simple cubic (sc) and simple hexagonal (sh).
300 K. rn exhibits an oscillatory structure in the vicinity of the walls that reflects layering of gas molecules. Layering is a direct consequence of
192
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gas /wall (adsorbate /surface) interactions [12] and apparently does not persist for n ]/3 from either wall. The density profile rn increases with the increase of the bulk density rb (see Fig. 6(a)). In the wall region the extent of layering (i.e. adsorption) diminishes with increasing T which is reflected by a decreasing peak height in r1. This is due to the higher (average) kinetic energy of the molecules of the gas A at higher temperature T which enables them to ‘escape’ the regime of attractive gas/surface interactions more easily. In the core region nothing spectacular is observed, the core region density is constant at rn $/rb , that is the core region is homogeneous and its properties are, therefore, identical to those of the bulk reservoir of the gas in thermodynamic equilibrium with it. This is expected because core gas region properties are dominated by molecule /molecule interactions (see Fig. 6(b)).
Acknowledgements The author A. Saber is grateful to ‘ICTP Programme for Training and Research in Italian Laboratories, Trieste, Italy’ for the financial support and wishes to extend his thanks to Professor G. Furlan head of TRIL program, and to Professor P. Mazzoldi for the fruitful discussion. This work has been partially supported by the P.F. MSTA II-CNR and by MIUR (COFIN ex 40%).
References [1] [2] [3] [4]
[5] [6] [7]
4. Conclusion [8]
In conclusion, the adsorption phenomena of gases in solids have been studied using the lattice model described by the Ising model in the frame of the mean field theory. The dependence of the layer concentration and Gibbs adsorption as a function bulk density and temperature have been discussed in this paper. We studied also the profile of the layer concentration.
[9]
[10] [11] [12]
E. Von Ising, Z. Physik 31 (1925) 253. L. Onsager, Phys. Rev. 65 (1944) 117. C. Domb, Adv. Phys. 9 (1960) 150. C. Domb, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, vol. 3, Academic Press, London, New York, 1974, pp. 1 /95. M.E. Fisher, Rep. Prog. Phys. 30 (1967) 731. T.L. Hill, Introduction to Statistical Thermodynamics, Dover Publications, New York, 1986, p. 246. T.L. Hill, Statistical Mechanics, Principles and Selected Applications, Dover Publications, New York, 1987, pp. 341 /345. G.L. Aranovich, M.D. Donohue, Phys. Rev. E 60 (1999) 5552. S. Ono, S. Kondo, Molecular theory of surface tension in liquids, in: S. Fugge (Ed.), Encyclopedia of Physics, vol. 10, Springer, Berlin, 1960. M.D. Donohue, G.L. Aranovich, Adv. Colloids Interf. Sci. 76 /77 (1998) 137. S.B. Kiselev, J.F. Eley, M.Yu. Belyakov, J. Chem. Phys. 112 (2000) 3370. M. Schoen, D.J. Diestler, J.H. Cushman, J. Chem. Phys. 101 (1994) 6865.
Multilayer adsorption of gases in solids Abderrahim Saber a,b,*, S. Dal Toe` a, S. Lo Russo a, G. Mattei a a
INFM, Dipartimento di Fisica, Universita` di Padova, Via Marzolo 8, 35131 Padua, Italy b The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy Received 26 August 2002; accepted 21 January 2003
Abstract Adsorption phenomena of gas in solids are investigated using the Ono /Kondo lattice model. Layer concentrations rn (n /1, 2, . . . L ) and Gibbs adsorption G are presented versus the bulk density rb and temperature. It is found that the density in core region (inner layers) and the density of the first layers next to the walls exhibit an opposite temperature dependence: while the density near the walls increases with decreasing temperature the density of the core region decreases to reach rb as T increases. The density profile of the layers is also presented. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Lattice model; Gibbs adsorption; Density profile
1. Introduction The study of the adsorption in porous materials has important application in fields such as separation and purification, chemical reactions and catalysis, the storage of gases, as well as many others. One popular approach for the modeling of adsorption phenomena is lattice /gas model, which provides the capability to predict adsorption isotherms and phase behavior in the pore including phase transitions (e.g. wetting, capillary condensation) and hysteresis effects.
* Corresponding author. Permanent address: De´partement de Physique, Faculte´ des Sciences, Universite´ Moulay Ismail, B.P. 4010, Mekne`s Morocco. Tel.: /39-049-827-7039; fax: / 39-049-827-7003. E-mail address: [email protected] (A. Saber).
Ising [1] gave an exact solution to the onedimensional lattice problem in 1925. Onsager [2] in 1944 was able to get an exact solution for behaviour of two-dimensional lattice at the critical density, but to date, no general solution has been obtained. The three-dimensional problem is even more complicated [3 /6]. No three-dimensional problem yet has a solution that is both complete and exact [6]. However, approximate methods, such as cluster expansion [3 /5], low-temperature expansions, and high-temperature expansions [7], have provided important insights. The article is organized as follows. In the next section the basic points of the model used to the present work, is briefly reviewed. This is followed by a presentation of numerical results, and their discussion, in Section 3. A brief conclusion is given in Section 4.
0927-7757/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0927-7757(03)00033-5
188
A. Saber et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 219 (2003) 187 /192
2. Model Consider a pure component (A) distributed on a fixed lattice with L layers, such situation is shown in Fig. 1. Each site may either be occupied by a molecule of A or may be empty. An expression of the free energy for the system, F /H/TS , can be derived by determining the Hamiltonian H and the entropy S for the gas confined in the lattice. Following Aranovich and Donohue [8], this is done using the mean field approximation [9] yielding: XX X Ho AA xnr xn?r? o AS xnr ; (1) n;n?
r; r?
nf1;Lg;r
where xnr is the occupation number on the lattice site (n, r ), (n , n ?) are plane indices and (r, r?) are different sites of the planes. xnr /1 if the lattice site is occupied by a molecule A and zero otherwise, oAS and oAA are, respectively, the energies for adsorbate /surface and adsorbate/adsorbate interactions. The gas density in the n -th layer, expressed as the fraction of lattice sites in layer n which are occupied by a molecule of A, is independent of r, so we can write rn x¯ nr which is the mean value of the occupation number. We now seek to find the entropy of the system. Consider taking an adsorbate molecule from the layer n and moving it to an empty site in the bulk (outside the pore). This is equivalent to the exchange of a molecule with a vacancy: An V Vn A
(2)
where A is the adsorbate molecule, and V is the vacancy that it fills (and vice versa). If this exchange occurs at equilibrium, then: DH TDS 0
(3)
where DH and DS are the enthalpy and entropy changes and T is the absolute temperature. The value of DS can be represented in the form: DS kB ln W1 kB ln W2
(4)
Fig. 1. Sketch of the lattice model used here for gas adsorption in solids.
where W1 is the number of configurations where the certain site in the layer n is occupied by an adsorbate molecule and the infinitely distant site is empty, W2 is the number of configurations where the infinitely distant site is occupied by an adsorbate molecule and the certain site in the layer n is empty, and k is the Boltzmann’s constant. If the overall number of configurations for the system is W0, then in the mean field lattice approximation: W1 =W0 rn (1rb )
(5)
and W2 =W0 (1rn )rb
(6)
where rn is the density that the certain site in the n-th layer is occupied by an adsorbate molecule A, and rb is the density of adsorbate in the bulk. Substituting Eqs. (5) and (6) into Eq. (4) we have: r (1 rb ) DS kB T ln n (7) rb (1 rn ) where kB is the Boltzmann constant. Using the Hamiltonian of the system Eq. (1) and in the frame of the mean-field approximation the enthalpy is:
DH
o AS o AA (z2 r1 z1 r2 z0 rb ) for n 1 o AA (z1 rn1 z2 rn z1 rn1 z0 rb ) for 2 5n5L1
(8)
A. Saber et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 219 (2003) 187 /192
where z1 and z2 are, respectively, the interlayer and intralayer coordination numbers (note that z0 / 2z1/z2 is the coordination number for the threedimensional lattice). From Eqs. (3), (7) and (8) it follows that: r1 (1 rb ) ln o AS =kB T rb (1 r1 ) (9) (z2 r1 z1 r2 z0 rb )o AA =kB T 0 for n 1 rn (1 rb ) ln rb (1 rn ) [z1 (rn1 rb )z2 (rn rb )
(10)
z1 (rn1 rb )]o AA =kB T 0 for
25 n5L1
The periodic conditions have to be satisfied, namely: (11)
r1 rL
Eqs. (9) /(11) constitute a nonlinear equations system of L equations in L unknowns. For a given values of the coordination numbers z1 and z2, (z1 /1, z2 /4 in the case of a simple cubic (sc) lattice; z1 /1, z2 /6 for simple hexagonal (sh) lattice; etc. . .), energy parameters EAS /oAS/kBT and EAA /oAA/kBT , and bulk density rb , this equation system can be solved for the density distribution vector r (r1, r2, . . . rL) describing the fractions of A in each of the L layers. The Gibbs adsorption for a lattice is given by: G
L X
(rn rb )
(12)
n1
By numerical treatment, iterative method of successive substitutions is used to solve the set of L nonlinear equations Eqs. (9) /(11) with respect to L unknowns, r1, r2, . . . rL , for a fixed parameters of oAA, oAS and the temperature T . Therefore, using Eq. (12), we can plot the adsorption isotherms. The Ono /Kondo lattice model is able to predict all known types of adsorption behavior, by changing two energetic parameters
189
(energies for adsorbate/adsorbate and adsorbate / adsorbent interactions).
3. Results and discussion In this paper, we fix the energies for adsorbate / adsorbate and adsorbate/adsorbent at oAA/kB / /430 K, oAS/kB /40 K, respectively, which are arbitrarily chosen and in the most of the study the number of layers is L /4. Let us begin with the evaluation of the layer concentration as a function of the bulk one for the case of four-layer problem and for a simple cubic and simple hexagonal lattices at two different temperatures 300 and 77 K. Fig. 2(a /b) show the plot of the layer density rn (with r1 (/r4) and r2 (/r3) for the reasons of symmetry) of lattice sites occupied by molecules of A as a function of the bulk mole fraction. We see that, the layer concentration is always increasing with the increase of the bulk concentration for a fixed temperature and it decreases with increasing the coordination number and the temperature for a fixed rb (layers (1 and 4) (see Fig. 2(a))), however, in the core region (layers (2 and 3)) Fig. 2(b), we have just the opposite dependence of coordination number and temperature. We can see clearly this behavior in Fig. 3, where the layer concentrations r1(4)(T ) and r2(3)(T) are depicted as function of temperature for (sc) and (sh) lattices and bulk mole fraction of rb /0.2. As T is lowered, the density increases in the surface layers. The increase is caused by stronger adsorption at the walls due to lower (average) kinetic energy of gas molecules while the core region the density remains constant and never exceeds the bulk density rb /0.2. However, as T is lowered further, r2(T ) eventually decreases because of low kinetic energy which prevents the depletion of the gas. Notice that the layer concentration decreases with the coordination number in the surfaces and we have the opposite tendency in the inner layers. Fig. 4 shows typical adsorption isotherms behaviour at 77 and 300 K and for sc and sh structures. These isotherms show standard type I behaviour of micropores materials [10]. At 300 K the gasadsorption at pores of both structures is mono-
190
A. Saber et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 219 (2003) 187 /192
Fig. 3. Layer concentrations as a function of temperature is shown for a simple cubic (sc) and simple hexagonal (sh) lattices for rb /0.2, oAA/kB //430 K, oAS/kB /40 K.
Fig. 2. (a /b) Layer concentrations as a function of the bulk concentration for L /4 and oAA/kB //430 K, oAS/kB /40 K at two different temperatures T/77 and 300 K and for two crystalline lattices simple cubic (sc) and simple hexagonal (sh).
tonically increasing, while lowering the temperature to 77 K, the adsorption behaviour is characterized by a steep increase in the adsorption-
Fig. 4. Adsorption isotherms as a function of bulk density rb for L/4, oAA/kB //430 K, oAS/kB /40 K at two different temperatures T /300 and 77 K and for the simple cubic (sc) and simple hexagonal (sh).
A. Saber et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 219 (2003) 187 /192
191
isotherms at low bulk densities, a maximum appeared at rb /0.1051 and 0.1391 for sc and sh, respectively, when the saturation of gas-molecules in the first adsorbed layers is reached and a monotonic decrease for higher bulk densities. rb is fixed, the adsorption isotherm increases with the decrease of the coordination number (a dense adsorbent is likely to have less sites of adsorption, since the process of densification reduces the number of sites available by blocking the pores) and temperature, respectively, this behaviour is clearly seen in Fig. 5 for the adsorption along isochores as function of temperature for rb /0.2. The temperature dependence of G (T ) is identical to that of the layer density r1(T ). Qualitative results has been found using the crossover theory [11]. Finally, in Fig. 6(a /b) we show the density profiles of the gas A in different layers for L /40. These is drawn for a two selected values of the bulk density rb /0.05 and 0.2 and at T /77 and
Fig. 6. (a /b) Layer concentration profiles for L /40 layers for simple cubic lattice, (a) T /77 K and for two values of the bulk density rb /0.2 and 0.05 (b) rb /0.2, at T /77 and 300 K. oAA/kB //430 K, oAS/kB /40 K. Fig. 5. Adsorption isochores as a function of temperature for L /4, rb /0.2 , oAA/kB //430 K, oAS/kB /40 K and for the simple cubic (sc) and simple hexagonal (sh).
300 K. rn exhibits an oscillatory structure in the vicinity of the walls that reflects layering of gas molecules. Layering is a direct consequence of
192
A. Saber et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 219 (2003) 187 /192
gas /wall (adsorbate /surface) interactions [12] and apparently does not persist for n ]/3 from either wall. The density profile rn increases with the increase of the bulk density rb (see Fig. 6(a)). In the wall region the extent of layering (i.e. adsorption) diminishes with increasing T which is reflected by a decreasing peak height in r1. This is due to the higher (average) kinetic energy of the molecules of the gas A at higher temperature T which enables them to ‘escape’ the regime of attractive gas/surface interactions more easily. In the core region nothing spectacular is observed, the core region density is constant at rn $/rb , that is the core region is homogeneous and its properties are, therefore, identical to those of the bulk reservoir of the gas in thermodynamic equilibrium with it. This is expected because core gas region properties are dominated by molecule /molecule interactions (see Fig. 6(b)).
Acknowledgements The author A. Saber is grateful to ‘ICTP Programme for Training and Research in Italian Laboratories, Trieste, Italy’ for the financial support and wishes to extend his thanks to Professor G. Furlan head of TRIL program, and to Professor P. Mazzoldi for the fruitful discussion. This work has been partially supported by the P.F. MSTA II-CNR and by MIUR (COFIN ex 40%).
References [1] [2] [3] [4]
[5] [6] [7]
4. Conclusion [8]
In conclusion, the adsorption phenomena of gases in solids have been studied using the lattice model described by the Ising model in the frame of the mean field theory. The dependence of the layer concentration and Gibbs adsorption as a function bulk density and temperature have been discussed in this paper. We studied also the profile of the layer concentration.
[9]
[10] [11] [12]
E. Von Ising, Z. Physik 31 (1925) 253. L. Onsager, Phys. Rev. 65 (1944) 117. C. Domb, Adv. Phys. 9 (1960) 150. C. Domb, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, vol. 3, Academic Press, London, New York, 1974, pp. 1 /95. M.E. Fisher, Rep. Prog. Phys. 30 (1967) 731. T.L. Hill, Introduction to Statistical Thermodynamics, Dover Publications, New York, 1986, p. 246. T.L. Hill, Statistical Mechanics, Principles and Selected Applications, Dover Publications, New York, 1987, pp. 341 /345. G.L. Aranovich, M.D. Donohue, Phys. Rev. E 60 (1999) 5552. S. Ono, S. Kondo, Molecular theory of surface tension in liquids, in: S. Fugge (Ed.), Encyclopedia of Physics, vol. 10, Springer, Berlin, 1960. M.D. Donohue, G.L. Aranovich, Adv. Colloids Interf. Sci. 76 /77 (1998) 137. S.B. Kiselev, J.F. Eley, M.Yu. Belyakov, J. Chem. Phys. 112 (2000) 3370. M. Schoen, D.J. Diestler, J.H. Cushman, J. Chem. Phys. 101 (1994) 6865.