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J. Phys. Chem. 1996, 100, 638-645
Effect of Loading and Nanopore Shape on Binary Adsorption Selectivity D. Keffer, H. Ted Davis, and Alon V. McCormick* Department of Chemical Engineering and Materials Science, UniVersity of Minnesota, 421 Washington AVenue SE, Minneapolis, Minnesota 55455 ReceiVed: July 5, 1995; In Final Form: October 4, 1995X
Grand canonical Monte Carlo (GCMC) computer simulations are employed to predict selective adsorption from binary mixtures into slit, cylindrical, and spherical nanopores. The mixtures are Lennard-Jones fluids representative of Ar, Xe, and tetramethylsilane (TMS), allowing us to gauge the effects of adsorbate size, energy well depth, and mass. The adsorption selectivity can oscillate from favoring one component to the other as nanopore size increases. Beyond simple molecular sieving, two distinct mechanisms can be responsible for size selectivity. At low pore densities, selectivity depends mainly on which species is more energetically favorable in the nanopore. At high pore densities, though, the ability to pack well within the pore is most important so the component size becomes the main criterion for selectivity. In high-pore density processes (many separations, catalytic processes, and templating during zeolite synthesis), the last mechanism can be active and requires consideration.
I. Introduction We have previously reported the adsorption and placement of a single-component Lennard-Jones fluid (LJ) in slit, cylindrical, and spherical nanopores.1 Pure fluids showed adsorption preference for smaller, more curved pores at low chemical potentials but showed preference for larger, less curved pores at high chemical potentials. We attributed these preferences to energetic and entropic contributions. In this work, we extend this investigation to the selective adsorption of binary mixtures and test whether the trends established for pure fluids to explain the features of selective adsorption with respect to pore shape, pore size, and pore density. We define the selectivity for component one of a binary mixture as
S1 )
xpore,1/xbulk,1 xpore,2/xbulk,2
(1)
where x is the mole fraction. Monson has studied selective adsorption of square-well mixtures in one dimension.2 In those systems, he finds that the qualitative form of the distribution of fluid within the pore is determined principally by molecular size effects and the strength of adsorbate-pore interactions. Attractive adsorbate interactions have a significant quantitative effect. In terms of selectivity, he finds that molecular size differences promote the adsorption of the smaller component. Given similar adsorbatepore interactions, the component with stronger adsorbateadsorbate interactions is favored. Somers et al.3,4 have used Monte Carlo and molecular dynamics simulations to study the selectivity of slit micropores from a binary mixture of cyclohexane and octamethylcyclotetrasiloxane (OMCTS) at 1 atm and 303 K. They found that the composition of the pore fluid oscillates strongly with slit pore width. The oscillations reflect the differing ability of each species to pack as layers. In some pores wide enough to admit both components, they observed a surprisingly absolute preference for the larger component (superselectivity) even though purely energetic interactions could not account for such selectivity. X
Abstract published in AdVance ACS Abstracts, December 15, 1995.
0022-3654/96/20100-0638$12.00/0
Tan and Gubbins5 studied methane-ethane mixtures in slit pores. They investigated selectivity as a function of LennardJones σ and , showing that the greater the difference between either of the parameters for the two components, the greater the selectivity. At temperatures well below the critical temperature and at high pressure, they observed the formation of distinct layers of the mixture. The compositions of the layers varied, becoming more methane rich with increased distance from the wall. The formation of additional layers eventually results in capillary condensation. Cracknell et al. has also studied LJ methane-ethane mixtures in slit pores.6,7 At pore sizes large enough to adsorb both components, they find that ethane (the energetically more favorable molecule) is selectively adsorbed. They also found that the selectivity versus pore size behavior changed when ethane was considered a LJ spherical molecule and a two-center LJ molecule. At pressures of 12.8 bar they see oscillations in the selectivity as a function of pore size. Curry and Cushman have studied binary mixtures in structured slit micropores.8 They demonstrate three mechanisms for selectivity: molecular sieving, liquidlike layering parallel to the pore walls, and solidlike epitaxial ordering due to the atomic structure of the pore wall. Dunne and Myers9 simulated a mixture of CCl2F2 and CO2 in a spherical pore, decorated to represent the zeolite 13X. At 323 K and 1000 kPa, they find that the selectivity for CCl2F2, the larger molecule, can be greater than 50. In a model zeolite A, Van Tassel et al.10 have simulated Xe/ Ar and Xe/CH4 mixtures. They find a preference for Xe at low loadings and a preference for the smaller molecule at high loadings. To our knowledge this was the first time in zeolite simulations that a reversal in selectivity was observed, indicating a possible change in the mechanism governing selectivity. The increased selectivity for the small molecule is partially attributed to its ability to better pack within the micropore. Also, CH4 was not able to displace Xe as well as Ar could. We conduct GCMC simulations of Xe/Ar and Xe/tetramethylsilane (TMS), (CH3)4Si, mixtures in slit, cylindrical, and spherical nanopores in order to generalize the principles of adsorption selectivity to nanopores of various shape. We also interpret these earlier results2-10 in terms of the generalized mechanisms for selectivity. We choose Xe/Ar and Xe/TMS in © 1996 American Chemical Society
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TABLE 1: Lennard-Jones Parameters ii/kb (K) σii (Å)
Xe
Ar
TMS
wall
221.0 4.10
119.8 3.405
349.5 5.75
496.67 4.10
particular because they respresent a mixture where Xe has a greater size, energy parameter, and mass than the other component (Ar) and a mixture where Xe has a smaller size and energy parameter but a larger mass than the other component (TMS). In this way we can distinguish size and energy effects from mass effects. II. Details of the GCMC Simulations The grand canonical Monte Carlo simulation method for mixtures is analogous to that used for a single component fluid, described elsewhere.1 The MC moves are divided equally between translations, insertions, deletions, and exchanges of both components. A total of 1.5 million MC moves were run, the first 500 000 of which were discarded. One million productions steps were sufficient to establish strong reproducibility. The standard deviations of the densities for each species were within 10%. The number of adsorbate molecules averaged about 150. The adsorbate-adsorbate (aa) potentials were taken as LennardJones potentials. The values of the LJ parameters are given in Table 1. The parameters for Xe and Ar are taken from Hirshfelder et al.11 The parameters for TMS were calculated using the corresponding states principle with the critical temperature and pressure of TMS taken from ref 12. The binary parameters were calculated using Lorentz-Barthelot mixing rules, i.e., a geometric average for ij and an arithmetic average for σij.13 The adsorbate-pore potential was taken as a Lennard-Jones potential integrated over the solid volume outside the slit, cylindrical, and spherical pores. To keep the conclusions general, we limit our investigation to pores with smooth walls, with no atomic structure. The effect of atomic structure of zeolites is to be presented in a later contribution. The adsorbate-pore potential parameters for each species were calculated using the values in Table 1 and the mixing rules. The density of the solid was Fs ) 1.0 molecules/σXeXe3. The forms of the external potentials are 9-3 potentials, representing pores in semiinfinite solids. (We have previously shown there are only minor differences in pore fluid structure in isolated and repeated pore networks as we have no long-range interactions.) The width of the pores are measured from the plane where the LJ external potential diverges at one side of the pore to the same plane on the opposite side. This represents the distance between centers of the first layer of atoms on opposite walls. These radii (or half-widths) of the pores studied ranged from 5 to 25 Å, dimensions representative of common zeolites. Figure 1 shows several external potentials as a function of adsorbate species, pore size, and pore shape. The total pore volume was chosen iteratively. We ran preliminary simulations to establish an approximate mean pore density for pore size and set of chemical potentials. Using that estimate, we defined the total pore volume by choosing the x and y dimensions (parallel to the walls) of the slit pore, the length of the cylindrical pore, and the number of spherical pores in such a way that the average total number of molecules would turn out to be about 150. For pores with larger widths and diameters, the transverse dimensions were reduced accordingly. For the slit pore, this translates into transverse dimensions from 25 to 400 Å. For the cylindrical pore, this led to cylinder lengths from 50 to 105 Å. For the spherical pores, this amounts to 1-104 distinct pores.
Figure 1. Integrated Lennard-Jones external potential. (a) as a function of adsorbate species for Xe, Ar, and TMS in the cylindrical pore of radius 6 Å; (b) as a function of pore size for Xe in cylindrical pores of radius 4, 6, and 8 Å; (c) as a function of pore shape for Xe in slit, cylindrical, and spherical pores of radii (or half-width) 6 Å.
III. Results A. Effect of Adsorbate Well Depth, Size, and Mass. We begin this section with an analysis of the effects of adsorbate well depth, size, and mass in a single-component fluid on adsorption. This is a useful preliminary step because it demonstrates which parameters favor adsorption at different pore densities. It is most convenient to uncouple the effects of these parameters by examining fictitious molecules. In Figure 2a we show the pore average densities in slit pores of width 2.0 and 3.2σXeXe of four LJ fluids at µa ) -45 kJ/mol: (1) LennardJones Xe, (2) pseudoweak Xe with aa ) 0.5XeXe, (3) pseudosmall Xe with σaa ) 0.5σXeXe, and (4) pseudolight Xe with ma ) 0.5mXe. The effects of these parameter changes are similar for both pore sizes. Even at this low, gaslike pore density, adsorbate size is the most important parameter (larger molecules favored), followed by energy and then mass. Note, however, all parameters are important. Larger adsorbate size enhances loading in these pores because it deepens the external potential well as shown in Figure 1a. A larger energy parameter enhances adsorption through both aa and ap interactions. The mass affects fluid density at a given chemical potential through the de Broglie thermal wavelength. This is because we keep the total chemical constant. For a monatomic ideal
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Figure 3. Mole fraction of Xe in a Xe/Ar mixture in slit, cylindrical, and spherical pores for µXe ) µAr ) -45 kJ/mol (xbulk,Xe ) 0.857).
Figure 2. Pore average density in slit pores of width 2.0 and 3.2σXeXe of four LJ fluids: (1) Xe, (2) Xe with aa ) 1/2XeXe, (3) Xe with σaa ) 1/ σ 1 2 XeXe, and (4) Xe with ma ) /2mX at (a) µa ) -45 kJ/mol and (b) µa ) 0 kJ/mol.
gas, the total chemical potential is14
µi(T,p,mi) ) µ0i(T,mi) + kT ln p
(2)
where the µ0i is given by
[(
µ0i(T,mi) ) -kT ln
) ]
2πmikT h2
3/2
kT
(3)
Thus for two gases of different masses, mi and mj, where µi ) µj and T is constant, the pressures and hence the densities will not be the same. In fact, for a monatomic ideal gas, the pressures (and densities) will differ by a factor of (mi/mj)3/2. Technically, the mass effect results in an increase in the probability of a Monte Carlo insertion or a decrease in the probability of a deletion of a heavier atom by a factor of (m2/ m1)3/2.15 Accounting for the de Broglie thermal wavelength of a heavier molecule can be thought of as accounting for the entropy of the adsorbed fluid by increasing the pore volume explored by the molecule. Figure 2b shows an analogous test of sorbate parameters at a high (liquidlike) pore density, where µa ) 0 kJ/mol. For both slit pores, the adsorbate size is now by far the most important parameter, favoring smaller molecules. We have shown previously that this is because the extensive entropic contribution to adsorption is greatest at high loadings. The ability to pack more of the smaller adsorbates into the pore is crucial to minimizing free energy.1 Secondarily, we see a slight favorability for the molecules with the lower aa. This is because molecules are packed tightly enough that an adsorbate with a steeper potential is less favored, as it becomes repulsive more readily. The mass has little effect at high chemical potentials, where the volume available is reduced to the molecular volume. In the following sections, we divide the discussion into adsorption at low and high pore densities, as guided by the extraordinary changes with pore density for pure fluids.1
Figure 4. Pore average density of (a) Xe and (b) Ar in slit, cylindrical, and spherical pores for µXe ) µAr ) -45 kJ/mol.
B. Adsorption of Xe/Ar. From the principles outlined in section A, the ratio of sizes, σXeXe/σArAr ) 1.204, indicates that Xe should adsorb more favorably at low chemical potentials and Ar at high µ. The energetic parameter, wXe/wAr ) 2.259, also should favor Xe at low loading and Ar at high loading. Finally, the mass ratio, (mXe/mAr)3/2 ) 5.958, also should favor Xe. B.1. Low Pore Densities. To test these predictions, we simulated the adsorption of a binary mixture of Xe and Ar in slit, cylindrical, and spherical nanopores at the following conditions: T ) 298 K and µXe ) µAr ) -45 kJ/mol, where the bulk densities are gaslike (Fbulk,Xe ) 1.90 × 10-5 and Fbulk,Ar ) 3.17 × 10-5 molecules/Å3) and the mole fraction of xenon is xbulk,Xe ) 0.857 (nearly the composition of the ideal gas). Figure 3 shows the mole fraction of adsorbed Xe in slit, cylindrical, and spherical micropores ranging in width (or diameter) from 5 to 20 Å. Figure 4a,b shows the mean pore densities of Xe and Ar. The pore densities are higher than the
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Figure 5. Mole fraction of Xe in a Xe/Ar mixture in slit, cylindrical, and spherical pores for µXe ) µAr ) -5 kJ/mol (xbulk,Xe ) 0.309).
bulk but are not liquidlike; adsorbate separations are high enough that the adsorbates do not yet need to “pack”. The adsorption behavior in the slit, cylindrical, and spherical nanopores is qualitatively similar. As expected, once the pores are large enough to admit Xe, Xe is always selectively favored over Ar (i.e., the mole fraction of xenon in the pore is greater than the mole fraction of Xe in the bulk). The chemical potential is not high enough to force packing of adsorbates; thus, smaller size is not an advantage. In terms of molecular parameters, Xe is favored on all counts; it has greater size, energy, and mass than Ar. As the pore size is increased, the pore composition returns to the bulk composition. The slit pore returns most quickly because the adsorbate-pore attraction inside it is the least. The cylindrical and spherical pores will return to bulk compositions at larger pore sizes due to their greater ap attraction. B.2. High Pore Densities. At high pore densities, we expect the smaller size of Ar to become important, allowing it to displace Xe. We simulated the adsorption of a binary mixture of Xe and Ar in slit, cylindrical, and spherical micropores at the following conditions: T ) 298 K and µXe ) µAr ) -5 kJ/mol, where the bulk densities are liquidlike (Fbulk,Xe ) 6.48 × 10-3 and Fbulk,Ar ) 1.45 × 10-2 molecules/Å3) and the bulk composition is xbulk,Xe ) 0.309. Figure 5 shows the mole fraction of adsorbed Xe in slit, cylindrical, and spherical micropores ranging in width (or diameter) from 5 to 20 Å. Figure 6a,b shows the mean pore densities of Xe and Ar are now definitely liquidlike. Figure 7a-l shows selected density distributions. At slit pore sizes of 5 Å and smaller, only Ar is adsorbed due to molecular sieving. The cylindrical pore molecularly sieves Ar up to a pore size of 6 Å (the spherical pore up to 7 Å) because the potential well is farther from the wall due to the curvature of the wall (Figure 1c). Once the pore can admit Xe, Xe is favored. In the 8 Å spherical pore, the adsorbed fluid is over 99.4% Xe, representing a selectivity factor over the bulk of 35.7. This observation opposes our expectation that Ar would always be preferentially adsorbed at high pore densities. Xe is adsorbed preferentially because its size allows the most favorable packing. Although there are slight energetic and mass advantages for Xe, neither of them can explain the overwhelming selectivity for Xe. The ratio of the xenon size to the pore size is in fact the primary influence on the selectivity. The cylinder and slit pores show the same favorability for Xe as the spherical pores, but to a lesser extent since the ap attraction is not as strong. As we increase the pore sizes in Figure 2, the mole fraction of xenon drops back down to selectivities less than one.
Figure 6. Pore average density of (a) Xe and (b) Ar in slit, cylindrical, and spherical pores for µXe ) µAr ) -5 kJ/mol.
Increasing the pore size further, we find oscillations in the xenon mole fraction, reflecting the ability for Xe or Ar to pack more efficiently into the pore. In the density distribution of Figure 7a-l, we see that layers of adsorbate molecules form (planar layers in the slit pores, annular in the cylindrical pores, and spherical shells in the spherical pores). Additional layers of Xe and Ar appear at different pore sizes. The pore size which allows an integer number of layers of one component favors that component. As the pore size increases, the magnitude of the oscillation decreases, and the pore composition will reach the bulk composition. The oscillations in selectivity are offset for the slit, cylindrical, and spherical pores due to differences in accessible pore volume. Only at certain pore sizes does our expectation hold true that Ar would displace Xe. At certain other sizes, the pores display a change in selectivity because of competition in packing. For single component fluids at high loading, it has been shown that the smaller molecule will be favored, due to better packing (an entropic factor).1 Thus, Xe must overcome this barrier in order to displace the Ar. It is only able to do so when the pore size allows it to balance the entropic advantage for Ar (more Ar can enter the pore) against its own combined energetic and entropic contributions. Although we should not completely disregard the energetic influence to the selectivity, we do note that these selectivity patterns can be established even for nonattracting hard spheres. Figure 8 compares the selectivity in slit pores as a function of pore size for four different fluids: (1) Lennard-Jones binary mixture of Xe/Ar, (2) LJ mixture without attractive adsorbateadsorbate (aa) interactions, (3) LJ mixture without attractive adsorbate-pore (ap) interactions, and (4) LJ mixture without either aa or ap attractive interactions (hard sphere). In each case, the pore densities are comparable and the bulk mole fraction of Xe is 0.3. Here we see that all four fluids show oscillations in selectivity with respect to pore size. The aa and ap interactions only serve to shift and change the amplitudes of the oscillations. The ap interaction induces greater variations in selectivity than than the aa interaction because the energy
642 J. Phys. Chem., Vol. 100, No. 2, 1996
Figure 7. Density profiles of Xe and Ar in slit, cylindrical, and spherical pores at selected pore sizes.
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Figure 8. Mole fractions of Xe in a slit pore of (1) Lennard-Jones binary Xe/Ar, (2) LJ mixture without attractive aa interactions, (3) LJ mixture without attractive ap interactions, and (4) LJ mixture without either aa or ap attractive interactions (hard sphere). All fluids are nominally at same average pore density and in equilibrium with bulk fluids which are 30% Xe.
Figure 10. Pore average density of (a) Xe and (b) TMS in slit, cylindrical, and spherical pores for µXe ) µTMS ) -45 kJ/mol.
Figure 9. Mole fraction of Xe in a Xe/TMS mixture in slit, cylindrical, and spherical pores for µXe ) µTMS ) -45 kJ/mol (xbulk,Xe ) 0.638).
well next to the wall anchors the first layer of adsorbates. Including the aa interaction increases the density of the adsorbed fluid but does not induce any structure of its own accord. In fact, aa interactions inhibit oscillations in selectivity with pore size inasmuch as they always favor the energetically favorable component regardless of pore size. Finally, for the Xe and Ar mixture, although we do not explore different combinations of chemical potentials, we expect the extent of selectivity to depend on the bulk composition as shown by Somers et al.2 Thus, we expect to see the same qualitative selective behavior active for other combinations of chemical potentials where the extent of selectivity depends on bulk composition and density. C. Adsorption of Xe/TMS. For a Xe/TMS mixture, the ratio of sizes (σTT/σXeXe ) 1.402) indicates that TMS will be favored at low chemical potentials and Xe at high µ. The ratio of energetic parameters (wTMS/wXe ) 1.258) also favors TMS at low loading and Xe at high loading. The mass ratio ([mTMS/ mXe]3/2 ) 0.5885), though, favors Xe. C.1. Low Pore Densities. We simulated the adsorption of a binary mixture of Xe and TMS in slit, cylindrical, and spherical nanopores at the following conditions: T ) 298 K and µXe ) µTMS ) -45 kJ/mol, where the bulk densities are Fbulk,Xe ) 1.92 ×10-5 and Fbulk,TMS ) 1.09 × 10-5 molecules/Å3 and the bulk composition is xbulk,Xe ) 0.638. The bulk density of the mixture is gaslike. Figure 9 shows the mole fraction of xenon in slit, cylindrical, and spherical micropores ranging in width (or diameter) from 5 to 20 Å. Figure 10a,b shows the mean pore densities of Xe and TMS.
At pore sizes large enough to admit TMS, we see as expected from size and energy considerations that TMS is preferentially adsorbed. In slit pores larger than 8 Å (cylinders and spheres larger than 9 Å), the mole fraction of Xe inside the pore is less than 10%. In the 10 Å cylindrical pore, we find a selectivity for TMS of nearly 908 times over the bulk fluid. In all three pore shapes we see a small decrease in selectivity for TMS when two layers of Xe can fit in the pore. The fact that a broad range of pore sizes favors TMS is due to TMS having a greater energetic and size advantage over Xe. Only as the pores become very large will the composition return to a bulk composition. This return takes place first in the slit pore, then in the cylindrical pore, and last in the spherical pores because the energetic attraction with the pore wall increases with pore curvature (Figure 1c). We saw similar behavior in the Xe/Ar mixture at low chemical potential (Xe preferentially adsorbed at all pore sizes accessible to it) because Xe had the size, energy, and mass advantage. C.2. High Pore Densities. At high pore densities, we expect Xe to adsorb preferentially due to its smaller size. We simulated the adsorption of a binary mixture of xenon and TMS in slit, cylindrical, and spherical nanopores at the following conditions: T ) 298 K and µXe ) µTMS ) -5 kJ/ mol, where the bulk density is liquidlike (Fbulk,Xe ) 1.45 × 10-2 and Fbulk,TMS ) 2.18 × 10-6 molecules/Å3) and the bulk composition is nearly 100% Xe. Figure 11 shows the mole fraction of Xe in slit, cylindrical, and spherical micropores ranging in width (or diameter) from 6 to 20 Å. Figure 12a,b shows the mean pore densities of Xe and TMS. The same mechanism which caused the oscillations in the Xe/Ar high loading case are operative here but to a lesser extent. There is only a single oscillation as opposed to the multiple oscillations of Xe/Ar. The energetic ratio is smaller for TMS/ Xe than it is for Xe/Ar, and for TMS/Xe, the mass ratio favors Xe; both factors would inhibit the displacement of the smaller Xe by TMS.
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Figure 11. Mole fraction of Xe in a Xe/TMS mixture in slit, cylindrical, and spherical pores for µXe ) µTMS ) -5 kJ/mol (xbulk,Xe ) 1.000).
Figure 12. Pore average density of (a) Xe and (b) TMS in slit, cylindrical, and spherical pores for µXe ) µTMS ) -5 kJ/mol.
For the most part, our expectation that Xe is favored is valid, Xe is principally adsorbed except for a narrow range of pore sizes (9-10 Å in the cylindrical pores and 9-11 Å in the spherical pores). In the spherical pore, this range corresponds to one TMS (9 and 10 Å) and one TMS and one Xe (11 Å) in a pore. When the spherical pore is only large enough to accept one adsorbate, regardless of which species it is, TMS is adsorbed because it is energetically favored and because the loading advantage of Xe is nullified here. Similarly, when the spherical pore is only large enough to accept two adsorbates (a choice between one TMS and one Xe or two Xe), the combination with TMS is adsorbed because it is energetically favored and because the loading advantage of Xe is again nullified. We do not see subsequent oscillations as we increase cylinder or sphere pore size, as we did for the Xe/Ar case, and we see no oscillations in the slit pore. No TMS is adsorbed in the slit pore because (i) the energetic attraction there is less than the other pores and (ii) the Xe can pack tighter in the slit pore because it has less curvature than cylindrical or spherical pores.1
A. Effect of Pore Density. In the simulation results presented above, we have shown three mechanisms for selectivity. The first mechanism is molecular sieving, operative at both low and high chemical potentials. The second mechanism is due to an energetic advantage and dominates when the average pore density is low. The third mechanism for selectivity is due primarily to a size advantage in optimizing packing and dominates when the average pore density is high (liquidlike) since at high densities molecules are “in contact” with each other and with the wall and so must pack. Curry and Cushman proposed three mechanisms of adsorption which included molecular sieving, liquid layering, and solidlike epitaxial structure.8 We do not see epitaxial structuring because we do not have atomically structured walls. We do, however, include the energetic factor for selectivity operative at all loadings but dominant at low loadings. While we state that the second mechanism (energy-dependent mechanism) is due to an energetic advantage, it is important to remember that the difference in size indirectly affects the energetic state of an adsorbate. Figure 2a,b shows that size alters the well depth because it dictates how close the molecule can approach the wall and thus where in the energy well it resides. Therefore, both the Lennard-Jones and σ parameters contribute to this energetic advantage at low chemical potentials. The third mechanism (the size-dependent mechanism) controls selectivity at high average pore densities. For Xe and Ar, the smaller adsorbate is entropically favored because it can pack a greater number of molecules into the same pore space.1 It is only when the size of the pore removes this advantage that the larger adsorbate can displace the smaller. For Xe/Ar, when the pore could accommodate only one layer of either molecule, Xe was preferred. The subsequent oscillations corresponded to choosing between two layers of Xe or two layers of Ar. In the case of Xe/TMS, TMS was able to displace Xe in small spherical pores. Those pores could have accommodated only one Xe or one TMS molecule; therefore, there was no packing advantage to Xe and TMS, the energetically favored molecule, adsorbed. High mean pore density, though, does not guarantee that the third mechanism will operate as shown by Xe/TMS in the slit pore, where no oscillations in mole fraction occur. This is because the slit pore does not have a strong enough external potential. If the smaller adsorbate preferentially occupies a pore in which the larger adsorbate would just fit, then the mechanism will not operate, as this is the optimum condition for it. Somers et al. demonstrated the size-dependent mechanism in slit pores for a mixture of cyclohexane and OMCTS at T ) 303 K, P ) 1 atm, and xbulk,C6H12 ) 0.42 where the bulk densities are Fbulk,C6H12 ) 9.72 × 10-4 and Fbulk,OMCTS ) 1.34 × 10-3 molecules/Å3.3 They found oscillations much like in the Xe/ Ar high density system. Tan and Gubbins demonstrated the energy-dependent mechanism with the methane/ethane simulated system in slit pores at zero pressure and T ) 148.1 K, showing a selectivity of up to 600 for ethane.5 At T ) 222.2 K, p ) 36.4 atm, and xbulk,C2H6 ) 0.5, they demonstrated the size-dependent mechanism, showing oscillations in selectivity as a function of pore size. Cracknell et al. demonstrated the energy-dependent mechanism at low loadings for methane and one-center ethane6 and the size-dependent mechanism at high loadings7 for methane and two-center ethane. We have demonstrated that sizedependent mechanism would have been operative regardless of whether ethane was treated like a one- or two-center LJ molecule.
Binary Adsorption Selectivity Dunne and Myers demonstrated the energy-dependent mechanism for a simulated mixture of CCl2F2 and CO2 in a spherical model zeolite 13X nanopore at 323 K and 1000 kPa.9 They showed selectivity for CCl2F2 the larger and energetically more favorable component. In a model zeolite A, Van Tassel et al. have demonstrated the energy-dependent mechanism at low pressures, preferring Xe over Ar, and the size-dependent mechanism at high pressures, preferring Ar over Xe.10 For the Xe/CH4 mixture, the energy-dependent mechanism operates at low pressure, favoring Xe, but the size-dependent mechanism operates to only a slight degree at high pressure due to the smaller ratio of adsorbate sizes (σXe/σAr ) 1.14 , σXe/σCH4 ) 1.09). The role of adsorbate size, aa and ap interactions, and loading in these three-dimensional systems is qualitatively the same as found in one-dimensional systems.2 The physics which drives the selectivity is the same in both cases. However, the onedimensional model will not be able to incorporate the differences in selectivity due to the three-dimensional pore shape without explicit knowledge of the way in which pore geometry affects ap potential well. B. Effect of Pore Shape. At low mean pore densities, the effect of greater nanopore curvature is to delay the return of the adsorbed fluid to a bulk state until larger pore sizes because greater curvature increases the ap interaction. This allows the energy-dependent mechanism for selectivity to operate up to larger pore sizes in pores of greater curvature. At low mean pore densities, the effect of increasing pore size is to strictly reduce the selectivity because there is less ap interaction per adsorbate in larger pores. At high mean pore densities, the effect of greater nanopore curvature is to enhance size-dependent selectivity, extending the oscillations in selectivity up to larger pore sizes for more highly curved pores and even causing oscillations in selectivity in the spherical pore where there are none in the slit pore. At high mean pore densities, the effect of increasing pore size is to alter the selectivity between one component and the other, based on their respective ability to pack well within the pore. Additional increases in pore size dampen the oscillations, allowing the adsorbed fluid to eventually return to bulk conditions. V. Conclusions We have shown the results from grand canonical Monte Carlo computer simulations of the adsorption of binary mixtures of Xe/Ar and Xe/TMS in slit, cylindrical, and spherical nanopores of sizes ranging from 5 to 25 Å from both bulk fluids at gas and liquid densities. In addition to molecular sieving, we have demonstrated two mechanisms for selective adsorption: an energy-dependent mechanism operative at low chemical potentials (gas bulk phase) and a size-dependent mechanism operative at high chemical potentials (liquid bulk phase). We have shown that selectivities can be explained in terms of the adsorption behavior observed for single component fluids. Also, we have shown that instances of adsorptive selectivity in nanopores observed in the literature can be interpreted via these two mechanisms. These findings have implications for separations, shape, and size selective catalysis and templating of solid phases. When
J. Phys. Chem., Vol. 100, No. 2, 1996 645 one wishes to predict the best “fit” of molecules into a pore space, there are three distinct features that must be considered: (a) what molecule may enter, (b) what molecules adsorb most exothermically at low loading, and (c) what molecules pack most efficiently at high loading. Acknowledgment. We acknowledge the Minnesota Supercomputer Institute and NSF (CTS-9058387). D. Keffer was partially supported by a Chevron Fellowship. Nomenclature h ) Planck’s constant kb ) Boltzmann constant mi ) atomic mass of species i n(x) ) local adsorbate density at position x, perpendicular to walls, in slit nanopores n(r) ) local adsorbate density at radial position r in cylindrical and spherical nanopores N ) number of adsorbate molecules p ) pressure Si ) selectivity for component i T ) temperature V ) volume U ) total internal energy w ) width of slit pore xbulk,i ) mole fraction of component i in pore xpore,i ) mole fraction of component i in pore aa ) Lennard-Jones adsorbate-adsorbate energy parameter ap) Lennard-Jones adsorbate-pore energy parameter Λi ) de Broglie thermal wavelength of species i µi ) total chemical potential of species i µ0i ) reference chemical potential of species i Fs) density of solid σaa ) Lennard-Jones adsorbate collision diameter σap ) Lennard-Jones adsorbate-pore collision diameter References and Notes (1) Keffer, D.; Davis, H. T.; McCormick, A. V. Adsorption 1996, 2, 11. (2) Monson, P. A. Mol. Phys. 1990, 70, 401. (3) Somers, S. A.; McCormick, A. V.; Davis, H. T. J. Chem. Phys. 1993, 99, 9890. (4) Somers, S. A.; Ayappa, K. G.; McCormick, A. V.; Davis, H. T., submitted for publication in Adsorption. (5) Tan, Z.; Gubbins, K. E. J. Phys. Chem. 1992, 96, 845. (6) Cracknell, R. F.; Nicholson, D.; Quirke, N. Mol. Phys. 1993, 80, 885. (7) Cracknell, R. F.; Nicholson, D.; Quirke, N. Mol. Simul. 1994, 13, 161. (8) Curry, J. E.; Cushman, J. H. Mol. Phys. 1995, 85, 173. (9) Dunne, J.; Myers, A. L. Chem. Eng. Sci. 1994, 49, 2941. (10) Van Tassel, P. R.; Davis, H. T.; McCormick, A. V. Langmuir 1995, 10, 1257. (11) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley & Sons: New York, 1954; pp 1110-3. (12) Silicon Compounds: Register and ReView, 5th ed.; Anderson, R., Larson, G. L., Smith, C., Eds.; Huls America Inc.: Piscataway, NJ, 1991; p 184. (13) Shukla, K. P.; Haile, J. M. Mol. Phys. 1987, 62, 617. (14) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976; p 87. (15) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, 1987; p 127.
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