3548
Langmuir 1993,9, 3548-3552
Computer Simulation of Adsorption from Binary Ar-Kr Solutions in Thin Slit Pores of Graphite E. M. Piotrovskaya' and E. N. Brodskaya Department of Chemistry, St. Petersburg, University, Petrodvoretz, St. Petersburg 198904, Russia Received March 5, 1993. In Final Form: July 21, 199P Grand canonical ensemble Monte Carlo calculations were held for Ar-Kr binary solution in slit graphite pores of different width at 88 K. The main aim of the present work was to investigate the dependence of the adsorption properties for binary fiis on the pore width. It was shown that the graphite pore sizes influence significantly the selective adsorption of krypton only in the range of pore widths less than six molecular diameters. For very thin pores the sizes of MC cells in x and y directions are quite important for adsorption characteristics. The study of adsorption in porous media is of great scientific and practical interest (namely, in catalysis, separation science, etc.). Over the past few years great success has been achieved in theoretical studies of the properties of fluids on solid surfaces, and the most serious attenton has been paid to the different modifications of the density functional theory192 as well as to computer simulation techniques-Monte Carlo (MC) and molecular dynami~s.3-l~Reviews of the main results on computer simulation of fluids on solid surfaces and in micropores of simple geometries (slits, cylinders, spherical cavities) up to 1985 are given in refs 3 and 4. These methods have been employed successfully in the studies of thermodynamic properties and structural characteristics of pure Lennard-Jones fluids in thin pores with adsorbing (wetted) walls,- the main subject of interest was the dependence of the film properties (density profile, radial distribution function, adsorption, disjoining pressure, diffusion coefficients) on the pore width. The later worksg-ll on molecular simulation of capillary condensation of pure and binary Lennard-Jones fluids in cylindrical pores are of particular interest. Quite recently, other more complex, but more realistic pore models have been studied, among them a pore of triangular cross section12and rough-walled cylindrical pores.13 Binary liquid mixtures on solid surfaces were not widely investigated by computer techniques but for the studies *Abstract published in Advance ACS Abstracts, September 1,
1993. (1)Tarasona, P.; Marini Bettolo Marconi, U.; Evans, R. Mol. Phys. 1987,60,573.Ball, P. C.; Evans, R. Mol. Phys. 1988,63,195. (2)Tan, Z.;Marini Bettolo Marconi, U.; van Swol, F.; Gubbins, K. E. J. Chem. Phys. 1989,90,3704. (3)Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics ofddsorption; AcademicPress: London and New York, 1982;p 398. (4)Brodskaya, E. N.; Piotrovskaya, E. M. In Chemistry and Thermodynamics of Solutions (in Russian); LGU Publishing House: Leningrad, 1986; Vol. 6,p 54. (5)Snook, J.; van Megen, W. J. Chem. Phys. 1980,72,2907;1981,74, 1409; 1981,75,4738. (6)Lane, J. E.;Spurling, T. H. A u t . J. Chem. 1980,33, 231. (7)Magda, J. J.; Tirrell, M.; Davis, H. T. J. Chem. Phys. 1986,83, 1888. (8)Schoen, M.; Cuehman, J. H.; Diestler, D. J.; Rhykerd, C. L., Jr. J. Chem. Phys. 1988,88,1394;Mol. Phys. 1989,66,1171. (9)Peterson, B. K.; Gubbins, K. E. Mol. Phys. 1987,62,215. (10)Peterson, B. K.; Gubbins, K. E.; Heffelfnger, G. S.; Marini Bettolo Marconi, U.; van Swol, F. J. Chem. Phys. 1988,88,6487. (11)Heffelfinger,G.S.;Tan,Z.;Gubbm,K. E.;MariniBettoloMarconi, U.; van Swol, F. Mol. Simul. 1989,2,393. (12)Bojan, M. J.; Steele, W. Langmuir 1993,9, 2569. (13)Bojan, M. J.; Vemov, A. V.; Steele, W. Langmuir 1992,8, 901. (14)Vinogradova,G. B.; Piotrovskaya, E. M.; Smirnova, N. A. Vestnik LOU 1987,No. 18, 53 (in Russian).
0743-746319312409-3548$04.00/0
of the adsorption from Ar-Kr solution on single graphite surface,14 in slit pores,l5 and the works already cited above.lOJ1 The investigations of binary and multicomponent systems are stimulated, first of all, by the fact that the knowledge of the selective adsorption of the components is of particular interest for different technological processes such as rectification, phase separation, and so on. Here we report our computer results on the dependence of the adsorption of binary Lennard-Jones solution in slit pores with graphite walls on the pore width. For our calculations we used grand canonical MC simulations.
Intermolecular Potential Models and Method We consider a model pore with slitlike geometry, with parallel walls of infinite extent separated by a distance I,. The walls are assumed to be basal planes of graphite. The fluid-fluid interactions are described by the LennardJones potential aij(r)= 4e~[(uij/r)12 - (uij/#] (1) where eij and aij are the parameters of the potental and r is the distance between molecules,i j = Ar or Kr. LorentzBerthelot combination rules are used for the cross Ar-Kr interaction parameters. The cut-off radius of potential (1)was taken equal to 2.2ah-h. For the fluid-solid interactions our calculations were based on two types of potentials, 9-3 potentialls and 104-3 (see for example ref 3). 9-3 solid-fluid potential is = (33'2/2)eb[(~i$~)9 - (uJz)~] (2) where z is the distance between molecule i of the adsorbate and one of the graphite walls (i = Ar, Kr). Parameters cb and ub are determined from experimental data on the second virial coefficients of gases adsorbed on graphite.17 The 10-4-3 potential takes into account the layering structure of solid and is as follows &(z) = 2?rp,cbai~A[2/5(ui$z)''-
-
(a:/3A(0.61A + z ) ~ ) ](3) where pa is the solid density (for graphite pa = 114 nm-3) and A is the distance between crystal layers (for graphite (15)Piotrovskaya, E. M.; Brodakaya, E. N. Zh. Fiz. Khim. 1991,53, 673 (in Russian). (16)Steele, W.A. Interactions of Cases with Solid Surfaces;Pergamon Press: Oxford, 1974; p 323. (17)Sams,J. R.; Constabaris, G.; Halsey, G. D. J. Phys. Chem. 1960, 64,1689.
0 1993 American Chemical Society
Langmuir, Vol. 9, No. 12,1993 3549
Dependence of Adsorption Properties on Pore Width
e% 0 0 - 4 - 3
Table 11. Average Energies per Particle (a),Excess Adsorption of the Components (ria),Total Adsorption per Surface Unity (rit),Average Density (p), Average and Composition of the First Maximum Composition (h), (I#) in the Pore of the Width 1, at T = 88 K (0.733) and x h a = 0.754 1," -eAr -e& r p rKr- rArt rat p zh ~ ~ (
Y
I
Lg ( 9 - 3 )
Figure 1. Definition of the dead volume of the MC unit cell. Table I. Parameters of Model Potentials bAr Kr-Kr b K r AI-C elk (K) u
(A)
120.0 3.40
168.8 3.67
142.3 3.54
1107 1.91
Kr-C 1460 1.925
A = 0.335 nm). The cross parameters of solid-liquid interactions were calculated by using Lorentz-Berthelot rules. We considered the interactions of each molecule with both walls of the slit pore of the width 1,. All calculations were held in reduced units in which LennardJones parameters for Ar eh/k = 120 K and u h = 0.340 nm were taken equal to unity. Parameters of the potentials are given in Table I. It should be noted that the adsorbate adsorbent interactions for Kr are stronger than for Ar. There are different definitions of the pore width. For two adsorption potentials (2 and 3) these values are defined in Figure 1. However, close to the walls there is the dead space not accessible for the centers of the adsorbed molecules because of their finite size. The width of this dead space for potential 2 is -0.5uh and for potential 3 is -0.8uh. For the calculations of adsorption the volume of the dead space is not taken into account. The basic MC cell had the sizes 1, = 1, from 4.4 to 8.8 in x and y directions, the periodic boundary conditions were used for the system in these directions. The distances between solid walls (pore width) 1, varied from 2.0 to 13.0. The averaging of all calculated properties was held over (0.5-1.5) X 106 configurations, with about 0.5 X lo6 preequilibrium configurations being discharged. MC simulations of Ar-Kr liquid films in graphite pores were held in grand canonical ensemble at the temperature 88 K and chemical potentials of the components pi. The grand canonical ensemble describes an open system in material contact with a large reservoir for which the chemical potentials are known. The chemical potentials in our calculations correspond to the binary liquid solution in equilibrium with the vapor. Therefore, we consider the adsorption from the liquid solution with composition x k a = 0.754 and, correspondingly, the chemical potentials p h a = -9.18 and pfia = -17.44. Average number of molecules of each sort Ni, partial internal energies per molecule ei, adsorptions ri, as well as profiles of local characteristics were calculated. Excess adsorptions of the components ri=,total adsorptions per surface unity rit are calculated according to the equations
I?? = (Ni- xypaV)/2A = Ni/2A
(4)
(5)
where xi" and pKare the bulk mole fraction of component i and bulk density, respectively, corresponding to the same values of T and pi; V = lxZy(lz - 1) is the volume of the MC
2.0 2.01 2.16* 2.5 2.5* 3.0 3.0* 3.5 3.5* 4.0 4.0* 4.5 4.5* 5.0 5.5 6.0 6.5 7.0 7.0* 9.0 11.0 13.0
16.1 14.2 14.6 16.8 14.1 13.7 11.5 12.7 11.7 12.6 10.3 10.2 10.1 9.3 10.6 9.2 9.9 8.5 8.7 8.3 8.1 7.8
21.5 19.2 19.1 21.0 19.4 18.6 17.4 18.1 15.3 15.7 15.3 13.8 14.2 16.0 13.8 13.3 13.9 13.8 12.9 12.3 11.1 11.1
0.03 0.09 0.10 0.07 0.15 0.26 0.19 0.02 0.1 0.24 0.17 0.10 0.12 0.27 0.02 0.29 0.10 0.45 0.23 0.03 0.00 0.06
Potential 9-3 0.25 0.31 0.15 0.36 0.18 0.42 0.20 0.49 0.16 0.57 0.14 0.82 0.18 0.75 0.22 0.72 0.18 0.80 0.22 1.08 0.20 1.01 0.20 1.08 0.22 1.06 0.18 1.39 0.31 1.28 0.16 1.69 0.24 1.64 0.13 2.13 0.19 1.91 0.34 2.21 0.36 2.80 0.37 3.42
0.34 0.24 0.28 0.34 0.30 0.32 0.36 0.45 0.41 0.50 0.48 0.52 0.54
0.54 0.72 0.62 0.74 0.68 0.74 1.07 1.28 1.47
0.648 0.48 0.48
0.603 0.650 0.661 0.702 0.764 0.737 0.670 0.695 0.789 0.749 0.712 0.708 0.775 0.726 0.770 0.735 0.804 0.758 0.729 0.743 0.752
0.60 0.60 0.59 0.65 0.72 0.67 0.61 0.66 0.68 0.67 0.68 0.66 0.72 0.64 0.72 0.69 0.76 0.72 0.67 0.69 0.70
0.60 0.60 0.60 0.65 0.63 0.60 0.57 0.66 0.67 0.61 0.70 0.71 0.55 0.67 0.67 0.63 0.58 0.64 0.60 0.63 0.62
Potential 10-4-3 3.0* 14.7 18.7 0.07 0.14 0.49 0.28 0.619 0.63 0.64 4.0* 11.9 16.1 0.06 0.20 0.76 0.43 0.679 0.64 0.60 Asterisk indicatesthe pore of the width 1, for which 1, = ly = 8.8; in other cases 1, = 1, = 4.4.
cell occupied by the fluid (without the volume of the dead space); A = lxly is the surface area of a MC unit cell. In order to obtain the local characteristics, we divided the whole volume of the system into a set of sublayers with the width Az = 0.1 and the volume equal to &A, correspondingly. Then the local partial density pi(Z) is equal to the average number of molecules of the ith kind per a volume unit, the total local density p(z) is equal to Cpi(z), and the local composition Xi@) is equal to pi@)/ P@). Results and Discussion The main aim of the present work is to investigate the dependences of the adsorption properties for binary films on the pore width. The calculation results are given in Table 11, some partial density profiles are presented in Figures 2-6. The layering structure has been found in all investigated systems, and up to the pore width 1, < 7.0 these layers are well developed (Figures 2 and 3). Beginning with the pores l, > 9.0 minima appear with nonzero density values (Figure 4). The bulk phase in the middle of the pore is obtained only at 1, 3 9.0. Very low values of density between the first two, closest to the wall layers, mean that the probability of the molecule exchange between them is negligible. The displacement of the positions of the first maxima for Ar and Kr due to the differences in molecular diameters can be seen only for rather wide pores (1 2 5.0); for thinner pores it is not obtained because of the lack of free space. It is necessary to point out more strong selectivity of graphite pore to krypton. The composition of the inner layers in the slit approaches the composition of a bulk solution ( x h a = 0.754) for the pore widths more than 11.0 (Figure 4). It is well seen that the local composition near the walls differs significantly from the composition both of the inner layers and the bulk (Figures 2-6). Due to the adsorption potentials 2 and 3, the first monolayers are
1 )
Piotrovskaya and Brodskaya
3550 Langmuir, Vol. 9,No. 12, 1993
p(l)4.0 3.0
2.0
3.0
1 .o
2.0
0.0
li
f
1.o
0.0 3.0
Figure 2. Partialdensityprofiiesfor Ar (I)andKr (11)in graphite pores (potential 2) with 1, = ly = 4.4 of different width 1, = 2.0
2.0
(1);2.5 (2);3.0 (3);3.5 (4);4.0 (5). 1 .o
pb)
4.0
4
3.0
2.0
1.o
0.0
I
0.0
3
Figures. PartialdensityprofiiesforAr(1)andKr@)ingraphite pores (potential2)of differentwidth 1, = 2.0 (1);3.0 (2);4.5 (3); with 1, = ly = 4.4 (a) and 1, = ly = 8.8 (b).
3.0
1
3
1
!-
I
I-
f
Figure 3. Partialdensityprofiiesfor Ar (I)and Kr (11)in graphite pores (potential 2) with 1, = ly = 4.4 of different width: 1, = 4.5 (1);5.0 (2);5.5 (3);6.0 (4);6.5 (5).
3.0
2
2.0
1.o
0.0
Figure 4. Partialdensityprofiiesfor Ar (I)andKr (II)in graphite pores (potential 2) with 1, = 1 = 4.4 of differentwidth 1, = 7.0 (1);9.0(2);11.0(3);13.0(4). dereonlyhalfofthedensityprofiles
are displayed.
enriched by Kr in comparison with the bulk. As seen from Table 11, the composition of the first monolayer is x # ) H 0.6, but for some special cases 1, = 3.5 and 1, = 4.5. Certainly, the first monolayers are basically influenced by the adsorbent field;the interactions of these monolayers with the rest of the system have much less influence on the adsorbed monolayer properties. There exists one special case for the pore with I , = 3.0 for which the inner layer consists practically of pure Ar (Figure 2(3), Figure 5a,b(2)). It can be explained by the fact that the larger atoms of Kr are almost not able to form the third (inner) layer due to steric limitations. With the increase of the pore width from 3.0 to 3.5 the situation changes distinctly (Figure 2(4));i.e. the density of the inner
Figure 6. Partialdensityprofiiesfor Ar (I)andKr (11)ingraphite pores with I, = ly = 8.8 for differenttypes of adsorbateadsorbent potentials: (a) 10-4-3with 1, = 3.0 (1)and 4.0 (2);(b) 9-3with 1, = 2.5 (I)and 3.5 (2).
layer decreases and the content of Kr grows significantly. In the pores with 1, 3 7.0 the structure of the first monolayers stabilizes and does not change with the further increase of pore width. The composition of the first monolayer approaches 0.62 (Table 11). So, the selectivity of graphite toward Kr enlarges ita content in the first monolayers up to XK,(~) = 0.38 instead of X K , ~ = 0.246 in the bulk phase. The total adsorption of components Fit for thin pores has the stepped character of the dependence on the width (Figure 7); the same behavior was found earliel.sI6 for the adsorption of a one-component system. It means that the increase of the pore width from the integer value by half of the molecular diameter does not give any evident rise of the total adsorption. It is natural because of the steric
Dependence of Adsorption Properties on Pore Width
4.0
Langmuir, Vol. 9, No. 12,1993 3551
-
....
...'.
3.0 -
...
... :"/-
2.0 -
de.---. 1
I
0.0 0
4
2
12
10
8
6
l4
e i
Figure 7. Dependence of total adsorptions Fit on pore width 1,: I, Kr, 11, Ar;111, Ar + Kr. The results refer to potential 2.
-e, 25
i I
Ilj
-
\
y4
-
,
I_ - - - - - - - - _ _ .
,,,--
?.A:
i
6
0
2
4
6
8
10
12
14
e,
Figure 8. Dependence of partial energies ei on pore width :,Z I, Ar; 11, Kr. The results refer to potential 2.
limitations. Considering the behavior of the partial adsorption (Figure 7, curves I and 11) it is necessary to take into account not only the steric limitations but also the factor of the selectivity of the adsorbent field toward Kr. When these two factors contradict each other, the pore is characterized by an extremely low value of Kr adsorption. We have already observed the example of such behavior in the pore with 1, = 3.0 (see above). At the same time, when the two factors are summed up, the selective adsorption of Kr is growing significantly. So, the pore with 1, = 4.5 appeared to be that special case-this pore is characterized by anomalous adsorption of Kr. Such a value of Kr adsorption in the pore 1, = 4.5 (Figures 3 and 5, Table 11)is explained first of all by the proportionality of the pore width to Kr diameter, in this case four dense layers of Kr can be formed in the pore (4.5ak 4.la=). This geometry factor strengthens the effect of selective adsorption of Kr on graphite. We also observe the increase of Kr fraction in the inner layers, which decreases the absolute value of the partial energy of Kr (Figure 8). This effect is especially strong for the case when the sizes of the MC unit cell in x andy directions were also almost equal to four molecular diameters of Kr (e.g. 1, = 1, = 4.4). With the increase of 1, and 1, up to 8.8, this effect is smoothed but does not disappear (Table 11, Figure 5). We also found the same nonmonotonous dependence on the pore width for average particle density p and partial energies ei (Table 11,Figure 8). The lower density is typical for pores not with integer width values I , = 2.5, 3.5,4.5, and 5.5. This effect does not exist practically for 1, = 6.5. Again the main reason for this behavior is steric limitations. The average density in the most narrow pores (1, d 2.5) is lower than the bulk density of the liquid phase pa = 0.740. The pores of the width 1, B 3.0 are characterized by the higher density.
The decrease of partial energies ei per molecule with the pore widening is also nonmonotonous (Figure 8).Here it is possible to point out for Kr the pore of the width 1, = 4.5, as well as for adsorption. The noticeable energy increase for Kr atoms is explained by the significant increase of the Kr atom fraction in the second adsorption layers with higher energy than in the first monolayers (Figure 5). The same explanation can be given to the increase of Ar atom energy for 1, = 5.0 and 6.0. In the limit of infinitely wide pores the values of ei(1,) have to approach the values of eiu in the bulk liquid phase. According to the results for energies the pores with 1, 3 11.0 may be considered as such wide pores (instead of 9.0 for densities). It means that for the adsorption systems the energy heterogeneity of the pore inner part appears to be in the wider interval of values of pore width than the density heterogeneity. Special attention in computer simulations was paid to the estimation of the influences of the sizes of the basic MC cell in x and y directions on calculation results. It is obvious that this question is of most importance for the pores of the width comparable to the molecular diameters of the adsorbate molecules. That is why we performed additional computer experiments for the most thin films under investigation with 1, from 2.0 to 4.5 (Figure 5) when I, = 1, = 8.8 (therefore, the surface area was enlarged 4 times) and for 1, = 7.0 (Figure 4). Substantial differences are obtained only for the most thin pore I, = 2.0 and for 1, = 4.5. It can be explained by the fact that the used values of 1, = 1, = 4.4 promote the adsorption of Kr (a similar situation is mentioned above for 1, = 4.5). The fluctuations which can destroy this state are very large and therefore are improbable and seldom occur. For the larger cells (1, = 1, = 8.8) the probability of fluctuations increases and the calculation results must be more reliable. To check these results we have carried out the calculations for these pores increasing again the values 1, and 1,: I, = 1, = 17.6 (a further increase of the surface area by a factor of 4). The calculation did not show any significant changes in the obtained results. For wide pores the limitations connected with the sizes 1, and 1, become much less (Table 11). We can hope that in these cases the results of computer simulation give correct values of adsorption from binary solution, though the changes in local characteristics take place, but they do not influence the total adsorption properties. As mentioned above, we carried out the calculations for the case when solid-fluid interactions were described by the 10-4-3 potential (3). Considering the pores with different adsorption potentials (2 and 3), it is necessary to compare the pores with the same width without dead space. It means that we compare the results for the following pores: 1, = 3.0 (10-4-3) with 1, = 2.5 (9-3), and also 1, = 4.0 (10-4-3) with 1, = 3.5 (9-3) (Figure 6). As seen from Table I1 and Figure 6, the use of potential 3 instead of potential 2 brings some decrease of general adsorbate density. But the character of adsorption and relative adsorptions of components are very similar for both types of adsorbate-adsorbent potentials.
Conclusions It is possible to summarize the main results of the investigation. First, the layering structure of the adsorbate was observed for all investigated systems and the layers nearest to the walls are isolated relative to the mass exchange with the rest of the system. These layers are enriched by Kr compared to the bulk solution.
3552 Langmuir, Vol. 9,No. 12, 1993
Second, with the increase of the pore width the region with the bulk liquid density appears only in the pore with the width more than nine molecular diameters and the composition inside this region becomes equal to the composition of the bulk solution only for the pore width more than 11 molecular diameters. Consequently, the value of 1, equal to 11.0 may be accepted as an estimation of the width of the pore for which the properties of the adsorbed system do not depend on the pore size. Third, there are two factors influencing the selectivity of the Kr adsorption: the adsorbate field and the steric limitations related to the pore size. When these factors
F’iotrouskaya and Brodskaya are cooperative, one can obtan an abnormal value of Kr adsorption (the case of 1, = 4.5). In the opposite situation the abnormal adsorption of the second component Ar is observed ( I , = 3.0). Fourth, both adsorption potentials 9-3and 10-4-3used in the work give a similar adsorption picture with only slight differences in density profiles. Finally, a pure methodological remark can be made: The sizes of the MC unit cell in the directions parallel to the hard walls do not influence the results of the calculations when they are greater than nine molecular diameters.
