22092
2005, 109, 22092-22095 Published on Web 11/03/2005
Large Distinct Diffusivity in Binary Mixtures Confined to Zeolite NaY C. R. Kamala,† K. G. Ayappa,‡ and S. Yashonath*,† Solid State and Structural Chemistry Unit, and Department of Chemical Engineering, Indian Institute of Science, Bangalore - 560 012 ReceiVed: August 26, 2005; In Final Form: October 16, 2005
Mutual diffusion coefficients have been computed from molecular dynamics simulation of two different binary mixtures confined to zeolite NaY. In one of these mixtures, where one component is from the linear regime and the other from the anomalous regime of the levitation effect [S. Yashonath, P. Santikary, J. Phys. Chem., 1994, 98, 6368], the magnitude of distinct diffusivity, Dd, is unusually large and comparable to the mixture self-diffusivity Ds. Distinct van Hove correlations suggest that the large Dd seems to arise from the presence of distinct physisorption sites for the two components. The contribution from Dd might be important for achieving good separation of mixtures, for which zeolites are used extensively.
1. Introduction Properties of confined fluids differ from those of bulk or fluids adsorbed on a surface. Confined fluids have often shown surprising behavior such as single file diffusion1,2 or the levitation effect3 (LE). In the past decade, our understanding of the behavior of a single component fluid confined to porous solids has advanced considerably.4 However, confined fluids are often present as mixtures. Examples are fluids within living cells and lubricants. Separation of hydrocarbon mixtures using zeolites in petroleum refineries provides yet another instance. In the case of binary or multicomponent mixtures, selfdiffusivities alone are inadequate to describe the transport properties;5 instead, a knowledge of the mutual diffusion coefficient is essential. For transport in an isothermal, isotropic n-component mixture, the flux-force relatioship is given by5 n-1
∑DRβ∇cβ R,β ) 1, 2, ...n - 1 β)1
JR ) - Fm
(1)
where Fm is the total mass density of the DRβ are the mutual diffusivities relating the concentration gradients ∇cβ with the correpsonding fluxes JR. The microscopic description of the mass transport in multicomponent systems is nontrivial and has formed the subject of many theoretical investigations. The Enskog theory by Chapman7 describes transport in a highdensity single-component hard sphere fluid, which was extended to binary mixtures by Thorne7 and later to multicomponent mixtures by Tham and Gubbins.8 Bearman9 has developed a statistical mechanical theory of the transport processes in liquid solutions and has compared it with the theories of Hartley and Crank10 and Gordon.10 In a binary mixture, there is just one mutual diffusion coefficient D11 due to constraints on fluxes and forces. Mutual
diffusivity D11 in a binary mixture has contributions from the self-diffusivities of the two components as well as the distinct diffusivity:
D11 ) Ds + Dd
Ds is the mixture self-diffusivity, which is related to the selfdiffusivities D1, D2 of the two components 1 and 2 as
Ds ) Q(x2D1 + x1D2)
‡
Solid State and Structural Chemistry Unit. Department of Chemical Engineering.
10.1021/jp0548321 CCC: $30.25
(3)
where x1 and x2 are the mole fraction of species 1 and 2, respectively. The distinct diffusivity can be obtained from the microscopic properties through the relation
( [
Dd ) Q x1x2
f11 2
+
x1
f22
f12 -2 x2 x1x2 2
])
(4)
where f11, f12, and f22 are the integrals of the respective velocity cross correlation functions (vccfs).6 Hence
fRβ )
mixture,6
†
(2)
∫0∞CcRβ(t) dt R,β ) 1,2
(5)
where the vccfs, CcRβ(t), are
{
CcRβ(t) ) 1
NR N β
〈Wi(t + τ) ‚ Wj(t)〉τ if R, β ) 1, 2 and R * β ∑∑ j)1
3N i)1 N 1 R
NR
(6)
〈Wi(t + τ) ‚ Wj(t)〉τ if R ) 1, 2 ∑∑ j*i
3N i)1
where N ) N1 + N2 is the total number of particles in the system. In eqs 2 and 3 the thermodynamic factor Q can be computed from the radial distribution functions.6 We wish to point out that the recent literature on transport diffusivity in zeolites essentially discusses the above in a © 2005 American Chemical Society
Letters
J. Phys. Chem. B, Vol. 109, No. 47, 2005 22093
somewhat different language.12 The mutual diffusivity is similar to the transport diffusivity discussed by many of these authors. Recently, there have been measurements of transport diffusivity by Jobic and co-workers using neutron scattering and calculation by molecular dynamics simulation by Theodorou, Snurr and Sholl and their co-workers.13-15 The contributions from distinct diffusion Dd obtained from eq 3 can be either positive or negative, leading to an increase or decrease of D11 (eq 1). The ratio
R)
D11 Ds
(7)
indicates the contribution from cross correlations and provides a measure of the relative contribution of Dd to D11. R > 1 indicates that Dd is positive and enhances the mutual diffusivity while R < 1 implies a negative Dd and lowers the mutual diffusivity. R > 1 when unlike particles in the mixture dynamically dissociate while R < 1 when unlike species dynamically associate. The diffusivities in eq 5 and those reported in this work are
Ds ) Ds/Q and Dd ) Dd/Q
(8)
Figure 1. (a) Diffusion maximum is found for a guest diffusing within zeolite Y when the size of the guest molecule, σgg, is comparable to that of the 12-ring window diameter. (b) Illustrates the situations when σgg is comparable to the window diameter as well as when σgg is much smaller than the window diameter.
Here we enquire into the dependence of D11 and Dd on the diameters of the two components of the mixture in comparison to the diameter of the void in the host zeolite. We choose two mixtures. One mixture (termed mixture AL) consists of one component from LR and another from AR. For comparison, we also study another binary mixture where both the components are from LR (mixture LL). We carry out molecular dynamics simulations on these two mixtures and compute the mutual diffusivity, D11, and the contribution to D11 from Dd. 2. Methods
Equation 2 provides a description of the transport property for ideal and real mixtures; for ideal mixtures Dd is zero, while for real mixtures Dd is nonzero. The deviation of the ratio R from unity provides a measure of nonideality of the mixture. Previous investigations16 into binary mixtures of bulk fluids interacting via a Lennard-Jones(LJ) potential have shown that Dd has a negligible contribution toward D11 for most choices of the LJ interaction parameters, σ11, σ22, 11 and 22. For more asymmetric mixtures, employing Lorentz Berthelot (LB) mixing rules, with highly dissimilar LJ parameters for two components, the contribution from Dd to D11, although higher, is still only around 10%.17 Mixtures with non-LB mixing rules where 12 > x1122 or 12 < x1122 showed larger negative or positive contribution, respectively, from Dd to mutual diffusivity. The deviation from ideality is greater for ionic mixtures, which are generally found to have significantly larger contribution from Dd. However, in both these (ionic or highly asymmetric nonLB mixtures), the contribution from Dd does not exceed 30%. There have not been many investigations reporting mutual diffusivity of mixtures in confinement.15,18 Recent MD investigations by us of Ar-Kr mixtures confined to zeolite NaY showed a nonzero negative Dd(R ) 0.77) at around 200 K,18 although the same Ar-Kr mixture behaves ideally in bulk.6 A study of mixtures in slit pores19 also showed that the R value can significantly deviate from unity, depending on the guest and host parameters. These studies indicate that bulk mixtures showing ideal behavior can exhibit highly nonideal behavior upon confinement. This is consistent with many surprising results that confined fluids have exhibited such as single file diffusion,20 different types of dependence of self-diffusivity on sorbate concentration in zeolites and the levitation effect (LE).3 Levitation effect refers to the anomalous maximum in selfdiffusivity, D, observed for sorbates whose diameter is comparable to the void diameter (Figure 1). This regime where the anomalous maximum is seen is referred to as the anomalous regime (AR). If the diameter of the diffusant σgg relative to that of the void is small, then D ∝ 1/σgg2. This is referred to as the linear regime (LR). This maximum in self-diffusivity occurs in all types of porous solids irrespective of the chemical, geometrical, and topological features of the porous host.21
Zeolite NaY or faujasite crystallizes in the cubic space group Fd3hm. The structure consists of smaller sodalite and larger R-cages (or supercages) made up of corner shared AlO4 and SiO4 tetrahedra. The R-cages form an interconnected network of voids through which the molecules can diffuse. Each of the R-cages (diameter ≈12 Å) is connected in turn to four other cages in a tetrahedral fashion via a shared 12-ring window (diameter ≈7.5 Å). Each unit cell has eight supercages with a composition (Si/Al ) 3.0) and lattice parameter a ) 24.8536 Å.22 Simulations have been carried out at two different temperatures, 168 and 300 K, in the microcanonical (NVE) ensemble with cubic periodic boundary conditions and velocity Verlet algorithm. In the present study 2 × 2 × 2 unit cells of zeolite NaY were employed with a simulation cell length of 49.7 Å. The number of guest particles N is 256, corresponding to a concentration of 4 particles/R-cage. The composition studied here is equimolar composition: x1 ) x2. A potential cutoff of rc ) 14 Å has been employed for both guest-guest and guesthost interactions. A time step of 10 fs has been used. Equilibration is for a period of 500 ps, and properties accumulated for a 1.5 ns run for obtaining various properties. Since Dd is a collective quantity, averages have been computed over 8 such independent runs to obtain a better estimate of the vccfs and therefore Dd. MD integration did not include atoms of the zeolite framework. Properties have been averaged from quantities stored every 20 fs. Mixture LL consists of two components whose LJ σ parameters viz., 3.07 Å and 3.48 Å, are both from the linear regime (see Figure 1). For mixture AL, σ11 ) 3.07 Å is from the linear regime and σ22 ) 6.0 Å is from the anomalous regime. For both the mixtures, 11 ) 22 ) 220 K and masses are m1 ) m2 ) 39.95 amu. LB combination rules have been used for the cross-interaction terms. Results and Discussion 3.1. Mutual, Distinct and Self-diffusivities from Velocity Correlations. Vccfs, Cc,11 Cc,22 and Cc12 (eq 6) for the two mixtures AL and LL are shown in Figure 2 at 168 K. Also
22094 J. Phys. Chem. B, Vol. 109, No. 47, 2005
Letters
Figure 2. The vccfs (eq 1), fijs (eq 6), Ds and Dd at 168 K for Mix A and Mix L. (a) vccfs averaged over 8 runs, (b) fijs in 10-8 m2/s and (c) Ds (from a single MD run) and Dd averaged over 8 MD runs in 10-8 m2/s for mixture AL at 168 K. (d),(e), and (f) are similar to (a), (b), and (c), respectively, but for mixture LL at 168 K. For mixture AL the contributions from both Ds and Dd are similar (c), resulting in a large value of R ) 1.85. In comparison, Ds is much larger for mixture LL (f).
TABLE 1: Self-diffusivities D1, D2, Ds, the fijs, and the Distinct (Dd) and Mutual (D11) Diffusion Coefficients in 10-8 m2/s and the Ratio R for Mixture AL and Mixture LL at 168 K AL D1 D2 Ds f11 f22 f12 D11 Dd R
LL
168 K
300 K
168 K
300 K
0.0931 0.0684 0.0807 -0.0564 -0.0093 -0.0674 0.1497 0.06896 1.85
0.4022 0.2048 0.3035 -0.2436 -0.0191 -0.1861 0.413 0.1095 1.36
0.5384 0.4699 0.5041 -0.3297 -0.2133 -0.2331 0.4273 -0.0768 0.85
1.554 1.321 1.437 -0.4505 -0.3655 -0.3877 1.396 -0.0406 0.97
shown are the integrals f11 f22 and f12 and the time-dependent self and distinct diffusivities Ds(t) and Dd(t). The values of fijs and the self-diffusivities D1 and D2 as well as the distinct diffusivity Dd and the mutual diffusivity D11 are listed in Table 1. In mixture LL (where both components are from the linear regime), f11 is found to be the largest in magnitude among the three integrals. Overall, it is found that Dd (see eq 4) is negative at both 168 and 300 K. In mixture AL at 168 K, the magnitude of contribution from unlike particle (f12) is larger than those from like particles (11 and 22) which leads to a positive contribution from Dd. In this case the magnitude of the mutual diffusivity, D11 is the smallest among the various mixtures investigated. More important, however, is the magnitude of the distinct diffusivity Dd relative to the self-diffusivity Ds. Note that since we have not evaluated the thermodynamic factor, fluxes cannot be evaluated directly. Hence we use R as a measure of deviation from ideality. For mixture LL the ratio R ) 0.85 at 168 K. The contribution of Dd is larger than that seen in the bulk mixture, which is expected to be close to zero.16 The results for mixture AL are, however, dramatic. The magnitude of Dd (0.06896 × 10-8 m2/ s) at 168 K is comparable to the mixture self-diffusivity Ds(0.0807 × 10-8 m2/s). The ratio R ) 1.85 at 168 K. To our knowledge, this appears to be the first time that such a large contribution from distinct diffusivity - almost equal to that from
Figure 3. Distinct van Hove correlation functions gdij(r,t) at 168 K at various times. (a) gd11(r), (b) gd22(r), and (c) gd12(r) for mixture AL. (d), (e) and (f) similar to (a), (b) and (c) respectively but for mixture LL. The key difference between the mixtures can be observed in (c) where the conspicuous absence of a peak at r ) 0 with time indicates that the two species have distinct noninterchangeable adsorption sites. In the case of mixture LL both species compete for the same adsorption sites.
self-diffusivity to D11 - has been observed in any LJ system obeying LB combination rules, confined or bulk.6,15,16,18 At 300 K, R is only 0.97 for mixture LL. This is to be expected, as at higher temperatures real mixtures approach ideal behavior. For mixture AL, however, R is 1.36, which is still far from that of an ideal mixture and quite significant for 300 K. We also carried out MD simulations at 200 K. We found that the ratio R for mixture AL is 1.5 (not listed in table). Thus, the nonideal behavior of mixture AL seen at low temperatures persists to a significant degree up to room temperature. 3.2 van Hove Correlations and Large Dd for Mixture AL. Mutual diffusivity of a mixture is a dynamic property, and the space-time correlations can be understood by studying the behavior of the van Hove correlation functions. The distinct part of the van Hove correlation function, gd(r,t) are the relevant quantities of interest here. Hence, we investigate the behavior of the cross-correlation functions gd11(r), gd22(r), and gd12(r). Figure 3 shows a plot of these three functions for mixtures AL and LL at 168 K at t ) 1, 3 and 10 ps. Plots at 300 K are not shown as they are quite similar. In mixture LL (Figure 3d,e,f), a peak near 5 Å is seen at t ) 1 ps for all three van Hove correlation functions, suggesting that a neighbor exists at 5 Å. With the passage of time, within about 10 ps, a new peak appears close to r ) 0 Å in all three, gd11(r), gd22(r) and gd12(r), suggesting that the first neighbor located at a distance of 5 Å occupies the position of that particle whose neighbor it was at t ) 0 ps. In mixture AL, (Figure 3a,b,c), a peak near 5 Å (approximately) is seen at t ) 1 ps for all three van Hove correlation functions. With the passage of time, however, a new peak appears close to r ) 0 Å in only gd11(r) and gd22(r). In gd22(r), which is for the larger anomalous (σ22 ) 6.0 Å) particle, the peak near r ) 0 Å appears slowly. By 10 ps, a noticeable peak is clearly visible. The third van Hove correlation function, which is between distinct particles of unlike or different types, gd12(r) reveals little or no decay of the 5 Å peak. Further, no peak near r ) 0 Å develops. Thus, the gd12(r) appears to evolve quite differently. Thus, in mixture LL, (Figure 3d,e,f) the van Hove correlation functions between both the like (gd11(r), gd22(r)) and unlike (gd12(r)) species behave in a similar fashion. In contrast, in
Letters mixture AL, (Figure 3d,e,f) the gd12(r) shows a very unusual behavior: a neighbor of the other type does not occupy the position of the original particle whose neighbor it is to start with. The reasons for this difference in the time evolution of gd12(r) for mixtures LL and AL may be understood by a study of the location of physisorption sites for components of the two mixtures. Previous studies3 reveal that the physisorption site for particles in the linear regime are found to lie well within the R-cage of zeolite Y. For two particles of different sizes but belonging to the linear regime, the same physical location within the R-cage acts as the physisorption site. In contrast, particles of any diameter from the anomalous regime, have physisorption sites at the 12-ring window interconnecting two neighboring cages, in addition to those within the R-cage. These findings have also found experimental confirmation from neutron diffraction studies of Fitch et al.22 who found that in the case of molecules such as benzene, which are nearly same size as the 12-ring window, the window is an additional adsorption site. For smaller molecules such as methane, which lie in LR, it is well-known that the physisorption sites are located only within the R-cage. Further, a careful investigation of the potential energy landscape for the linear and anomalous regimes suggests that the position where minima exist for LR are associated with maxima for particles from AR.3 Thus, for mixture LL, as the two components compete for the same physical region within the R-cage, one of the components can replace the other component while diffusing within the zeolite. This is reflected in the van Hove correlation function gd12(r). These arguments are applicable to the other distinct van Hove functions gd11(r) and gd22(r). In contrast, for mixture AL, while two particles belonging to the same component can replace each other (since their physisorption sites are located at the same physical region) this is not true for two particles hailing from different components (and also from different regimes: anomalous and linear regimes). The noninterchangeability of particles at the physisorption sites in case of particles from two different regimes can be seen in the fact that gd12(r) behaves very differently from gd11(r) and gd22(r). The distinct sites for two particles from two different regimes within zeolite NaY and the contrasting potential energy landscape leads to a dissociative tendency (Dd > 0) of the mixture AL during transport. For mixture LL, the value of R at both the temperatures studied is less than unity, suggesting that it is associative in nature. Further, they are close to unity and deviation from ideality is small. For mixture AL, the value of R is significanly larger than unity suggesting strong dissociation and greater deviation from ideality. In fact, the value of Dd we obtain is close to the value of Ds in magnitude for mixture AL. The present results suggests that the mixture AL exhibits large Dd as compared to the Ds. This is not the case for mixture LL.
J. Phys. Chem. B, Vol. 109, No. 47, 2005 22095 Previously nonequilibrium simulation of two multicomponent mixtures have been reported in NaY.23 In the mixture containing components corresponding to mixture LL, the separation of the two components was found to be poor. In contrast, for another mixture containing components similar to mixture AL, the separation was markedly significant. This suggests that mixture AL separates well on passage through a zeolite column but not mixture LL, indicating that Dd (in comparisosn to Ds) and the large dissociative tendency (R ) 1.85) might play an important role in kinetic-based separation of multicomponent or binary mixtures. A greater understanding of the role of Dd in separation of mixtures is therefore required. In summary, we have shown that under certain circumstances mixtures can exhibit large distinct diffusivities (as large as the self-diffusivity!). The situation when large Dd is seen is determined not only by the interaction parameters of the mixtures as hitherto believed but also on the nature of the confining medium. Finally, separation achieved appears to be related to the relative contribution of Dd. Acknowledgment. The authors wish to acknowledge support from Department of Science & Technology, New Delhi and Inter-University Consortium, DAE, Mumbai. References and Notes (1) Hahn, K.; Karger, J.; Kukla, V. Phys. ReV. Lett. 1996, 76, 2762. (2) Keffer, D.; McCormick, A. V.; Davis, H. T. Mol. Phys. 1996, 87, 367. (3) Yashonath, S.; Santikary, P. J. Phys. Chem. 1994, 98, 6368. (4) Bates, S. P.; van Santen, R. A. AdV. Catal. 1998, 42, 1-114. (5) de Groot, S. R.; Mazur, P. Non-Equilibrium Thermodynamics; Dover: New York, 1984. (6) Zhou, Y.; Miller, G. H. Phys. ReV. E, 1996, 53, 1587-1601. (7) Chapman, S.; Cowling, T. G. The Mathematical Theory of Non Uniform Gases; Cambridge University Press: Cambridge, 1952. (8) Tham, M. K.; Gubbins, K. E. J. Chem. Phys. 1971, 55, 268. (9) Bearman, R. J. J. Phys. Chem. 1961, 65, 1961-1968. (10) Hartley, G. S.; Crank, J. Trans. Faraday Soc. 1949, 45, 801. (11) Gordon, A. R. J. Chem. Phys. 1937, 5, 522. (12) Theodorou, D.; Snurr, R.; Bell, A. In ComprehensiVe Supramolecular Chemistry; Alberti, G., Bein, T.; Eds.; Pergamon: New York, 1999; Vol. 7. (13) Papadopoulos, G. K.; Jobic, H.; Theodorou, D. N. J. Phys. Chem. B 2004, 108, 12748. (14) Jobic H.; Skoulidas, A.; Sholl, D. J. Phys. Chem. B 2004, 108, 10613. (15) Sanborn, M. J.; Snurr, R. Q. Sep. Purif. Technol. 2000, 20, 1-13. (16) Schoen, M.; Hoheisel, C. Mol. Phys. 1984, 52, 1029-1042. (17) van den Berg, H. P.; Hoheisel, C. Phys. ReV. A 1990, 42, 20902095. (18) Kamala, C. R.; Ayappa, K. G.; Yashonath, S. Phys. ReV. E 2002, 65. (19) Kamala, C. R.; Ayappa, K. G.; Yashonath, S. J. Phys. Chem. B 2004, 108, 4411. (20) Hahn, K.; Karger, J. J. Phys. Chem. 1996, 100, 316. (21) Bandyopadhyay, S.; Yashonath, S. J. Phys. Chem. 1995, 99, 4286. (22) Fitch, A.; Jobic, H.; Renouprez, A. J. Phys. Chem. 1986, 90, 1311. (23) Rajappa, C.; Yashonath, S. Mol. Phys. 2000, 98, 657-665.
