Computer Simulations of the Structural and Kinetic Characteristics of

Feb 1, 1996 - Thermodynamic, structural, and kinetic properties of bulk and adsorbed binary Ar-Kr ... pores of width lz, which was varied in the range...
0 downloads 0 Views 184KB Size
Download PDF
+

+

1578

Langmuir 1996, 12, 1578-1584

Computer Simulations of the Structural and Kinetic Characteristics of Binary Argon-Krypton Solutions in Graphite Pores A. V. Klochko, E. M. Piotrovskaya,* and E. N. Brodskaya Department of Chemistry, St. Petersburg State University, Petrodvoretz, St. Petersburg 198904, Russia Received June 23, 1995. In Final Form: November 20, 1995X Thermodynamic, structural, and kinetic properties of bulk and adsorbed binary Ar-Kr systems in the slitlike graphite pores were investigated by Monte Carlo and molecular dynamics methods at the temperature T ) 88 K. Special attention was paid to the influence of solution composition on the local density and the coefficients of diffusion parallel to the walls of the pore. It was found that the dependence of the diffusion coefficients on the width of the pore is defined mainly by the behavior of the local density in pores. It was shown that the increase of the concentration of krypton in the systems decreases the diffusion coefficients of both components.

Introduction The present work is the continuation of the paper1 which was devoted to the investigation of adsorption of a binary solution of simple liquids in thin graphite pores by computer simulations. The main goal of the present work is the investigation of the influence of the width of the pore and the composition of adsorbed solution on the diffusion coefficients of a binary mixture. This problem is of great interest for the explanation of the molecular mechanisms of the selective adsorption and in particular for the selectivity of membranes widely used in different fields of chemistry. In the early works devoted to the investigations of the properties of simple liquids by molecular dynamics methods, the coefficient of selfdiffusion of a Lennard-Jones fluid was calculated,2,3 and the results were in good agreement with experimental data for argon. The investigation of diffusion coefficients of a binary mixture Ar-Kr4-8 appeared much later, and the results of different authors correlate only qualitatively. The obtained data show a higher mobility of argon in comparison with krypton. The diffusion in pores was studied only for a pure substance.9-15 A nonmonotonous dependence of the diffusion coefficients along the pore on the width of the pore was observed in refs 12 and 13. The differences in the values of the diffusion coefficients in ref 12 and ref 13 are connected, first with different values of temperature and second with different ways of describing * E-mail: [email protected]. X Abstract published in Advance ACS Abstracts, February 1, 1996. (1) Piotrovskaya, E. M.; Brodskaya, E. N. Langmuir 1993, 9 (11), 1837. (2) Rahman, A. Phys. Rev. A 1964, 136, 405. (3) Levesque, D.; Verlet, L.; Ku`rkijarvi, J. Phys. Rev. A 1973, 7, 1690. (4) MacGowan, D.; Evans, D. J. Phys. Rev. A 1986, 34, 2133. (5) Vogelsang, R.; Hoheisel, C. Phys. Rev. A 1987, 35, 3487. (6) Gardner, P. J.; Heyes, D. M.; Preston, J. R. Mol. Phys. 1991, 73, 141. (7) Pas, M. F.; Zwolinski, B. J. Mol. Phys. 1991, 73, 483. (8) Heyes, D. M. J. Chem. Phys. 1992, 96, 2217. (9) Rowley, L. A.; Nicholson, D.; Parsonage, N. G. Mol. Phys. 1976, 365. (10) Lane, J. E.; Spurling, T. H. Chem. Phys. Lett. 1979, 67, 107. (11) Snook, I. K.; van Megen, W. J. Chem. Phys. 1980, 72, 2907; 1981, 74, 1409; 1981, 75, 4738. (12) Magda, J. J.; Tirrell, M.; Davis, H. T. J. Chem. Phys. 1985, 83, 1888. (13) Shoen, M.; Cushman, J. H.; Diestler, D. J.; et al. J. Chem. Phys. 1988, 88, 1394. (14) Demi, T.; Nicholson, D. Mol. Simul. 1991, 5, 363. (15) Somers, S. A.; Davis, H. T. J. Chem. Phys. 1992, 96, 5389.

the structure of the walls, which were smooth in ref 12 and structured in ref 13. The previous work1 was devoted to the investigation of the adsorption of the mixture of argon and krypton enriched by argon. The references on the investigations of adsorption in binary systems are mostly given in ref 1, except for the latest works.16-18 Local partial densities, energies, and local compositions were calculated for the pores of width lz, which was varied in the range from 2 up to 13 molecular diameters at the temperature T ) 88 K. One of the aims of the present work is the investigation of the influence of the composition of the solution on the adsorption characteristics. Computer simulations of adsorption were carried out for the almost equimolar ArKr mixture at the same temperature, 88 K. Taking into account that the density and composition in the adsorption pores are local characteristics, it was necessary to calculate also the local diffusion in separate adsorbed monolayers to obtain the correlations of diffusion coefficients with the local density and composition. As stated above, the literature data on diffusion coefficients of the bulk Ar-Kr solution are rather contradictory. This was the reason for investigating also the properties of the bulk Ar-Kr solution at the temperature 88 K for two compositions when the mole fraction of argon xRAr was approximately equal to 0.50 and 0.70. Computational Method and Models of the Systems Binary Ar-Kr systems in slitlike pores of graphite were investigated at the temperature 88 K computer simulation methods. The intermolecular adsorbate-adsorbate interactions were described by a truncated Lennard-Jones potential with a cutoff distance rc

[( ) ( ) ]

Φij(rc) ) 4ij

σij r

12

-

σij r

6

(1)

Here r is the intermolecular distance and ij and σij are the parameters of the potential. The parameters of ArKr interactions were obtained from the Lorentz-Berthelot mixing rules. We consider a graphite wall to be a (16) Somers, S. A.; McCormick, A. V.; Davis, H. T. J. Chem. Phys. 1993, 99, 9890. (17) Cracnell, R. F.; Nicholson, D.; Quirke, N. Mol. Phys. 1993, 80, 885. (18) Sokolowski, S. Mol. Phys. 1993, 75, 999.

+

+

Binary Ar-Kr Solutions

Langmuir, Vol. 12, No. 6, 1996 1579

Table 1. Parameters of the Model Potentials /k (K) σ (nm)

Ar-Ar

Kr-Kr

Ar-Kr

Ar-C

Kr-C

120.0 0.340

168.8 0.367

142.3 0.354

1107 0.191

1460 0.1925

continuum with a constant density; thus, the adsorbateadsorbent interactions are described by a (9-3) potential

Ψis(z) )

[( ) ( ) ]

σis 3x3  2 is z

9

-

σis z

3

(2)

where z is the distance between a molecule of the adsorbate and the graphite wall. The parameters is and σis of the potential 2 were defined from the experimental data on the second virial coefficients of the gases adsorbed on graphite. It is necessary to mention the stronger selectivity of graphite to Kr (Kr-C/Ar-C = 1.3). The values of the parameters of the potentials 1 and 2 are given in Table 1. The slit pores we deal with in this work are formed by two infinite solid surfaces parallel to the xy plane, and the distance between two adsorbing surfaces (width of the pore) changed from lz ) 2.0σAr up to lz ) 4.5σAr. The sizes of a basic cell of the computer simulations in x and y directions were equal to lx ) ly ) 13.2σAr for the pores with the widths lz ) 2.0σAr and lz ) 2.5σAr, and for the rest of the pores, lx ) ly ) 8.8σAr. Periodic boundary conditions were used for the systems in these directions. For computer simulations the influences of the sizes of the basic cell on the results of the calculations are of special importance. It is obvious that it is more important for the thin pores when their widths are comparable to the sizes of the adsorbate molecules, and we observed this situation earlier.1 That is why the sizes of the basic cells of computer experiments were enlarged up to lx ) ly ) 13.2σAr for the pore widths lz ) 2.0σAr and lz ) 2.5σAr. The probability of the fluctuations increases for the larger sizes of the basic cell, thus bringing more reliable results. The averaging of the results for Monte Carlo (MC) simulations was performed over the last 3 × 106 configurations. The investigations of thermodynamic and kinetic properties were carried out for bulk binary Ar-Kr mixtures of two compositions, when the mole fraction of Ar (xRAr) was approximately equal to 0.50 and 0.70. These results gave the possibility of comparing the calculated diffusion coefficients for adsorption systems with the data for the bulk solutions, as well as of checking the existing literature data on diffusion coefficients for the bulk solutions. The length of the edge of the cubic basic cell was l ) 6.6σAr. Thermodynamic properties of the binary Ar-Kr solution both in the pores and in the bulk were investigated at the temperature 88 K and values of the chemical potentials µi corresponding to the bulk liquid-vapor equilibrium, and the calculations were held by grand canonical MC methods. The transport properties were calculated by molecular dynamics (MD) methods which are based on numerical integration of the classical equations of motion for all molecules of the system at the given initial and boundary conditions. The equations of motion have been integrated according to the algorithm19 with the time step ∆t ) 10-14 s. At the beginning of the calculations, the system was kept in the “thermostat” for about 5 × 10-9 s. It means that the value of the kinetic energy of the system Ek ) 3/ NkT (k is the Boltzmann constant, T is the temperature, 2 and N is the number of particles) was under control at this stage, which was done by the standard procedure of (19) Schofield, P. Comput. Phys. Commun. 1973, 5, 17.

scaling the velocities of the molecules. After 5 × 10-9 s the correction of the kinetic energy was stopped, and the system was moved to the standard conditions of the MD method in the microcanonical ensemble with the given values of the number of particles Ni, volume V, and total energy of the system E (NVE ensemble). The system relaxed for 5 × 10-9 s, and then the calculations of the average properties of the system were started. Diffusion in narrow pores is an anisotropic quantity depending on the direction of the motion. The Einstein equation can be used to define the R-component of the diffusion coefficient for the component i

DiR(t)

1 ) 2t

lim tf∞

Ni

1 Ni

〈[rjR(t) - rjR(0)]2〉 ∑ j)1

(3)

where Ni is the number of particles of the type i (i ) Ar, Kr), rjR is the R-component of the radius vector of the particle j (R ) x, y, z), 〈 〉 means the averaging over the number of initial configurations, and t is the time. In addition, there is the second definition of the diffusion coefficients with the help of the autocorrelation functions

DiR(t) )

1

Ni

1

〈vjR(t′)vjR(0)〉 dt′ ∫ ∑ 0 2t N j)1 t

(4)

i

where vjR is the R-component of the velocity vector of the particle j (R ) x, y, z). The autocorrelation function 〈vjR(t)vjR(0)〉 is vanishing with time. The coefficients of mutual diffusion, giving the total kinetic characteristics of the system, can be obtained according to the approximate equation.20 Kr ) xArDAr DAr-Kr R R + xKr DR

(5)

Due to the conditions of symmetry, two diffusion coefficients of the component i differ in thin pores: the coefficient of diffusion normal to the wall Di⊥ ∝ DiZ and the coefficient of diffusion parallel to the wall Di| ) 0.5(DiX + DiY). It is to be noted that eq 3 can be used strictly only to infinite system. The movement in the adsorbed layers parallel to the wall satisfies this condition quite well. For the movement perpendicular to the wall, eq 3 is approximate, although as it was shown in ref 13 this equation is a good approximation even for very thin pores. In the present work we deal with the pores of width lz e 4.5σAr. In such pores at the temperature of investigation (88 K), it makes sense to consider only the coefficients of diffusion parallel to the wall due to the behavior of the local density in the adsorption systems. The layering structure is typical for the density of the adsorbate in these pores, while the density between the monolayers is practically zero. It is evident that in such a case the interchange of the molecules between the layers almost does not exist. For the bulk systems the diffusion is an isotropic quantity, and hence Di ) (DiX + DiY + DiZ)/3. The calculations of the diffusion coefficients were carried out according to eqs 3 and 4. In order to make the calculations faster and to improve the statistics, the initial configurations were chosen each 40∆t. The diffusion coefficients according to the mean square displacements were calculated for the linear region from 200∆t up to 400∆t. The diffusion coefficients from the autocorrelation functions were obtained by integrating according to eq 4 in the region from 0 to 400∆t. The errors of these (20) Shoen, M.; Hoheisel, C. Mol. Phys. 1984, 52, 1029.

+

+

1580

Langmuir, Vol. 12, No. 6, 1996

Klochko et al.

Table 2. Average Energy per Particle ei, Density Gr, and R Composition xAr in the Bulk Solution at T ) 88 K (a) (b) a

-eAra

-eKra

FRb

R xAr

6.54 6.36

8.87 8.85

0.75 0.79

0.474 0.689

With factor Ar. b With factor σAr-3.

Table 3. Average Energy per Particle ei, Average Number of Molecules Ni, Average Density G, and Average Composition xAr in the Pores of Width lz at T ) 88 K and R xAr ) 0.474 lza

-eArb

-eKrb

NAr

NKr

Fc

xAr

2.0d

14.07 13.95 10.73 12.78 10.49 10.86

18.86 18.72 17.38 15.81 15.42 14.88

83.57 127.69 53.93 50.85 58.03 64.00

121.46 152.18 111.22 129.00 166.96 180.00

1.160 1.067 1.065 0.924 0.973 0.900

0.407 0.456 0.327 0.283 0.258 0.262

2.5d 3.0 3.5 4.0 4.5

a With factor σ . b With factor  . c With factor σ -3. d l is the Ar Ar Ar z pore of width lz for which lx ) ly ) 13.2σAr; in the rest of the cases lx ) ly ) 8.8σAr.

calculations were estimated as the mean square deviations over the series. Results and Discussion 1. Thermodynamic Properties and Structural Characteristics. At the given values of temperature and chemical potentials of the components, the average values of the density FR, composition xRAr, and energy per molecule eRi were calculated for bulk binary systems. These values are given in Table 2. In the paper1 for the systems enriched by argon (b), the values of the chemical potentials of the components were taken from the experiment for the given composition of the solution xRAr ) 0.75421 and the chemical potentials were estimated in the approximation of the ideal vapor phase. More precise MC calculations of the composition of this solution at the same values of the chemical potentials µi gave xRAr ) 0.689. This discrepancy may be explained both by the use of the supposition of the ideality of the vapor phase and by the fact that the parameters of the potential 1 for mixed argon-krypton interactions were obtained due to Lorentz-Berthelot rules, and it is obvious that this can differ significantly from the real mixed interactions. The experimental data on two-phase equilibrium for the equimolar mixture were not found in literature, and the existing experimental data were extrapolated into this region. The values of the chemical potentials of the components were also estimated in the approximation of the ideal gas phase. MC calculations showed that these values of the chemical potentials correspond to the solution with the mole fraction of argon xRAr ) 0.474. So, the adsorption was calculated for the binary solutions with xRAr ) 0.474 (a) and xRAr ) 0.689 (b). The average number of molecules Ni and the average energies per molecule ei, as well as the local characteristics, partial density profiles Fi(z) and composition profiles xi(z), were obtained for all systems under investigation. For the calculations of the profiles, the total space of the pore was divided into thin layers of width ∆z ) 0.1σAr parallel to the solid walls, and in each layer the system was considered to be uniform. The results are given in Table 3 and Figures 1-4. It is necessary to mention that the average density in the pore was calculated without the excluded volume near the walls which is unreachable for (21) Kogan, V. B.; Fridman, V. M.; Kafarov, V. V. Equilibrium between liquid and vapor; Nauka: Moscow, Leningrad, 1966; Vol. 1, p 225.

Figure 1. Dependence of the average number of molecules in the pore per surface area unit on the width of the pore lz: I, Ar; II, Kr; III, Ar + Kr.

the molecules. The width of the region was estimated to be equal to 0.5σAr near each wall. The dependence of the average number of molecules of the components in the pore per surface unit on the width of the pore for the adsorption from the solution with xRAr ) 0.474 is shown in Figure 1. The number of Kr atoms grows monotonously with the increase of the width of the pore, as observed in ref 1 for the case xRAr ) 0.689. The behavior of argon changes significantly with the changes in the composition of the solution. In comparison to ref 1 where the tendency of the quick growth of the number of argon atoms with the increase of the width of the pore was observed, in the case of the solution with the lower concentration of Ar xRAr ) 0.474, its content in the pore grows quite negligibly. The average composition in the pores (xAr) except for the most narrow pores (lz ) 2.0σAr and 2.5σAr) is much lower than the composition of the corresponding bulk solution xRAr ) 0.474 (Table 3). It can R be noted that the selectivity of the pores S ) (xArxKr )/ R (xKrxAr) toward krypton almost does not change with the composition of the solution. A small increase of the content of krypton in the two most narrow pores with the growth of its concentration in the solution can be explained by steric factors. The geometric limitations hinder the increase of the composition of krypton in the monolayers near the walls in spite of their preferential selectivity toward Kr. The local distribution of the molecules in the pores is shown in Figures 2 and 3. The layering structure of the adsorbate has been found for all the systems under investigation as it was already shown earlier in ref 1 (Figure 3). In the case of xRAr ) 0.474 (Figure 2), monolayers are enriched by krypton in comparison with the case xRAr ) 0.689 (Figure 3). It can be explained both by the increase of the mole fraction of Kr in a bulk solution (xRAr ) 0.474) and by the selectivity of adsorption potential toward Kr. The pore of width lz ) 3.0σAr (Figure 2c) for which the inner layer is enriched by argon for both

+

Binary Ar-Kr Solutions

+

Langmuir, Vol. 12, No. 6, 1996 1581

Figure 2. Local density profiles of Ar (I) and Kr (II) for xRAr ) 0.474 in slit graphite pores of the following widths: lZ ) 2.0σAr (a), lZ ) 2.5σAr (b), lZ ) 3.0σAr (c), lZ ) 3.5σAr (d), lZ ) 4.0σAr (e), and lZ ) 4.5σAr (f).

Figure 4. Dependence of the average energy per molecule on the width of the pore lZ: I, Ar; II, Kr.

Figure 3. Local density profiles of Ar (I) and Kr (II) for xRAr ) 0.689 in slit graphite pores of the following widths: lZ ) 2.0σAr (a), lZ ) 2.5σAr (b), lZ ) 3.0σAr (c), lZ ) 3.5σAr (d), lZ ) 4.0σAr (e), and lZ ) 4.5σAr (f).

compositions of the solution is considered to be a special case. It can be explained by the steric factors, because Kr atoms are not able to form an inner layer in this pore. It is necessary to mention that in general the changes in local density and composition in the monolayers near the walls are nonmonotonous. With the increase of the width of the pore from 2.5σAr up to 3.0σAr and from 3.5σAr up to 4.0σAr, the local density and the fraction of krypton atoms in the layers next to the walls increase significantly. The dependence of the average energy per molecule on the width of the pore is nonmonotonous, as was shown earlier in refs 1 and 11 (Figure 4). The pore with width lz ) 3.0σAr similar to the case in ref 1 is out of the general dependence. The substantial growth of the energy of Ar is explained by the increase of the mole fraction of Ar atoms in the middle of the pore, and their energy is much higher than the energy of Ar atoms in the monolayers next to the walls. It is necessary to mention that this effect is more pronounced for the solution with xRAr ) 0.474 for which the mole fraction of argon atoms is lower in the adsorption system than for the other solution. With the increase of the width of the pore from 3.0σAr up to 3.5σAr, the situation is changed (Figure 2d); in this case the density

Figure 5. Mean square displacements along the pore for Ar and Kr atoms in the pore of width lZ ) 3.5σAr: I, Ar; II, Kr.

of the inner layer grows significantly, and the number of Kr atoms in this layer increases. 2. Diffusion Coefficients. The time dependence of the mean square displacements of Ar and Kr atoms along the pore for the pore of width lz ) 3.5σAr (xRAr ) 0.474) is shown in Figure 5. It is obvious that the picture will be the same for the rest of the systems. The values of the diffusion coefficients Di| calculated according to the asymptotic dependence of the mean square displacements on time (eq 3) are given in Tables 4 and 5. Autocorrelation functions of velocity for Ar and Kr atoms parallel to the walls are shown in Figure 6. The values of diffusion coefficients Di| calculated according to eq 4 are also given in Tables 4 and 5. The comparison of the calculated results according to the mean square displacements and autocorrelation functions allow the conclusion that both methods give the same results for diffusion coefficients inside the error limits. Diffusion coefficients calculated for bulk systems are given in Table 4. Diffusion coefficients for argon are higher

+

+

1582

Langmuir, Vol. 12, No. 6, 1996

Klochko et al.

Table 4. Diffusion Coefficients Di and Coefficients of Mutual Diffusion DAr-Kr for the Bulk Solutions at T ) 88 K According to Mean Square Displacements (I) and Autocorrelation Functions (II)a

b

R xAr

DAr b (I)

DAr b (II)

DKr b (I)

DKr b (II)

DAr-Kr b (I)

DAr-Kr b (II)

0.474 0.689

7.82 9.71

7.78 9.68

6.41 7.94

6.33 7.94

7.08 9.16

7.03 9.14

a All values of the diffusion coefficients are multiplied by 102. With factor (ArσAr2/mAr)1/2.

Table 5. Diffusion Coefficients Di| and Coefficients of Mutual Diffusion DAr-Kr for the Systems in Pores at T ) | R R 88 K and xAr ) 0.474 (a), xAr ) 0.689 (b) According to Mean Square Displacements (I) and Autocorrelation Functions (II)a lzb 2.0 2.5 3.0 3.5 4.0 4.5

b

(a) (b) (a) (b) (a) (b) (a) (b) (a) (b) (a) (b)

c DAr | (I)

c DAr | (II)

c DKr | (I)

c DKr | (II)

c DAr-Kr | (I)

c DAr-Kr | (II)

0.98 1.73 1.43 2.31 1.37 1.96 2.42 3.23 3.02 3.63 3.98 5.08

1.04 1.86 1.49 2.31 1.34 2.01 2.41 3.54 2.97 3.89 3.98 5.11

0.84 1.68 1.42 2.67 1.18 1.54 2.45 3.27 2.78 3.00 3.32 4.51

0.88 1.77 1.50 2.72 1.12 1.62 2.42 3.44 2.79 2.95 3.38 4.67

0.90 1.72 1.43 2.45 1.24 1.82 2.44 3.24 2.84 3.44 3.49 4.89

0.94 1.82 1.50 2.47 1.19 1.88 2.42 3.51 2.84 3.58 3.52 4.97

a All values of the diffusion coefficients are multiplied by 102. With factor σAr. c With factor (ArσAr2/mAr)1/2

Figure 6. Autocorrelation functions of the velocity for Ar and Kr atoms parallel to the walls in the pore of width lZ ) 3.5σAr: I, Ar; II, Kr.

than for krypton, which is a natural result of the ratio of the sizes and masses of the atoms. The reason for the decrease of diffusion coefficients of both components with the increase of the mole fraction of heavier and larger atoms of Kr in the solution is the same. In this case diffusion coefficients of Ar and Kr become closer in their values. The same tendency was observed in refs 5 and 7. With the changes in composition from xRAr ) 0.689 to 0.474, diffusion coefficients for both argon and krypton change up to 20%. The dependence of the diffusion coefficients Di| on the width of the pore is shown in Figure 7. As a rule, diffusion coefficients Di| grow with the increase of the width of the pore. The only exception is the pore with lz ) 3.0σAr, for which diffusion coefficients of both components are lower than for the pore with lz ) 2.5σAr. This effect is more pronounced for the adsorption from the solution with the

Figure 7. Dependence of the diffusion coefficients Di| on the R width of the pore lZ: I, Ar; II, Kr. (a) xRAr ) 0.474; (b) xAr ) 0.689.

mole fraction xRAr ) 0.689 (Figure 7b). It can be explained by the changes in the local structure of the adsorbate in these two pores (see Figures 2 and 3). Though the average densities in these pores have practically the same values (Table 3), the local density in the monolayer next to the wall is significantly higher for the pore with lz ) 3.0σAr than for the pore with lz ) 2.5σAr. The reverse contribution of the inner layer with much less density in the pore with

+

+

Binary Ar-Kr Solutions

lz ) 3.0σAr to the diffusion only weakens the decrease of diffusion coefficients. It is obvious that the similar decrease of diffusion coefficients has to take place in the layers next to the walls for the change of the width of the pore from lz ) 3.5σAr to 4.0σAr. However, in this case the contribution of the inner layers of the adsorbate defines the behavior of the diffusion, and diffusion coefficients grow a little. It is necessary to mention that the nonmonotonous behavior of Di| for the fluid in the pore was also observed in refs 12 and 13, and it was explained by the nonmonotonous behavior of the local density in the pores. As it is seen from Tables 4 and 5, even for the widest pore under investigation, lz ) 4.5σAr, diffusion coefficients are almost twice as low as those for the bulk solution at the same values of the chemical potentials. For almost all systems under investigation, when xRAr ) 0.689 the ratio of the diffusion coefficients Di| for argon and krypton is the same as for the binary solution,4-8 Kr that is, DAr | g D| , although this ratio is partly not valid for the most narrow pores. So, for lz ) 2.0σAr and 3.5σAr, Kr Ar Kr DAr | ) D| , and for lz ) 2.5σAr, D| < D| , although this difference coefficients of Ar and Kr for the latter case is very small. The noticeable increase of the mobility of Kr atoms in comparison with that of Ar atoms for the pore lz ) 2.5σAr is connected with the fact that the partial local density of Kr decreases with the change of the width of the pore from lz ) 2.0σAr up to lz ) 2.5σAr, while the partial density of Ar increases. The reverse effect is observed for the behavior of the partial densities with the change of the width of the pore from lz ) 2.5σAr up to lz ) 3.0σAr (Figure 3). The same ratio of the diffusion coefficients Di| for argon and krypton was observed for the systems with xRAr ) Kr 0.474, which means that DAr | gD| , although the difference between them is much less. Diffusion coefficients of both components for all pores under investigation decrease with the enriching of the solution by krypton. It can be seen from Figure 8, where the dependence of the coefficients of mutual diffusion on the width of the pore for both compositions of the solution is shown. In general the decrease of diffusion coefficients in the pores is up to 30%, and it is more than for the bulk solutions, as mentioned above. It can be explained by the fact that the solutions in the pores are substantially enriched by krypton in comparison with the initial bulk solution. The calculations of the diffusion coefficients in layers were carried out to clarify the character of the motion of atoms in the adsorbed systems. These calculations were done for the pore of width lz ) 3.5σAr at 88 K and xRAr ) 0.689. For this purpose the pore was divided into three monolayers defined according to the minima of the density profiles. The data of the calculations are given in Table 6. As seen from the table, diffusion coefficients in the inner layer are twice as large as the coefficients Di| in the layer next to the walls. This means that the movement of the atoms in the second adsorbed layer is preferential, but still these coefficients are lower than those of the bulk solution. It is connected with the value of the local density in this layer, which is higher than the density of a bulk solution (Table 6). The obtained results correlate quite well with the data from ref 13, where diffusion coefficients in layers for argon in pores were calculated at 120 K. Diffusion coefficients in layers estimated in this work are a little higher than the results from ref 13, although the temperature in our calculations was lower. The differences can be explained by different models of the adsorbent; in ref 13 it was supposed to be as a crystal layer of immobile atoms of the adsorbent.

Langmuir, Vol. 12, No. 6, 1996 1583

Figure 8. Dependence of the coefficients of mutual diffusion on the width of the pore lZ for the solutions of the following R compositions: (a) xAr ) 0.474; (b) xRAr ) 0.689. Table 6. Average Density G, Average Composition xAr, Diffusion Coefficients Di| in the Layers Calculated According to Autocorrelation Functions for the Pore lz ) R 3.5σAr at T ) 88 K and xAr ) 0.689 (the Layers are Numerated from the Solid Wall)

1st layer 2nd layer total b

Fb

xAr

c DAr |

c DKr |

2.195 0.904

0.683 0.571 0.651

2.54 5.85 3.39

2.50 4.88 3.33

a All values of the diffusion coefficients are multiplied by 102. With factor σAr-3. c With factor (ArσAr2/mAr)1/2.

Table 7. Temperature Dependence of Diffusion Coefficients Di| Calculated According to Mean Square R Displacements for the Pore of Width lz ) 3.5σAr (xAr ) 0.689)a

b

Tb

c DAr |

c DKr |

0.733 0.850 1.0

3.23 3.87 7.02

3.27 4.23 7.23

a All values of the diffusion coefficients are multiplied by 102. With factor Ar/K. c With factor (ArσAr2/mAr)1/2.

The calculations for three different temperatures at the fixed density in the pore of width lz ) 3.5σAr (xRAr ) 0.689) were carried out to test the temperature dependence of diffusion coefficients. The data of the calculations are given in Table 7. They correlate quite well with the data from ref 12; the closest values for T ) 120 K were obtained in ref 8 for a bulk solution. It is necessary to mention that with the increase of the temperature the increase of the diffusion coefficients for this pore is observed together with the higher mobility of krypton in comparison with argon. This fact has been mentioned above for the pore of width lz ) 2.5σAr at 88 K. It seems that with the increase of the temperature the inner monolayer is enriched by Kr atoms at the expense of the layers next to the walls. It

+

1584

+

Langmuir, Vol. 12, No. 6, 1996

brings a decrease in the partial local density of Kr in the layers next to the walls and thus an increase in the mobility of Kr atoms in comparison with Ar atoms. Conclusions To summarize the results of the present work, it is possible to conclude that the local structure of the adsorbate in thin pores influences significantly the character of the diffusion and the values of diffusion coefficients. The dependence of the diffusion coefficients

Klochko et al.

Di| on the width of the pore is mainly determined by the local density. The changes in the composition of the bulk solution bring the quantitative changes in diffusion coefficients, although the main tendencies in dependencies of diffusion coefficients on the width of the pore are similar. Acknowledgment. The present work was partly supported by the International Science Foundation and the Government of Russia, Grant R22300. LA9505069