Simulations of Binary Mixture Adsorption in Carbon Nanotubes

Received May 15, 1997. In Final Form: October 27, 1997. The adsorption of equimolar binary Lennard-Jones gas mixtures into single-walled carbon nanotu...
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Langmuir 1998, 14, 880-890

Simulations of Binary Mixture Adsorption in Carbon Nanotubes: Transitions in Adsorbed Fluid Composition K. G. Ayappa Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India Received May 15, 1997. In Final Form: October 27, 1997 The adsorption of equimolar binary Lennard-Jones gas mixtures into single-walled carbon nanotubes is investigated using grand canonical Monte Carlo simulations. For mixtures whose species have different molecular diameters, the larger energetically favored species is adsorbed at higher temperatures. However, at lower temperatures and intermediate nanotube diameters, a complete exclusion of the larger species in favor of the smaller species is observed. This transition in nanotube fluid composition is accompanied by a decrease in the total potential energy of the system. Although adsorption of the smaller species is favored, both species adsorb at lower temperatures in the larger nanotubes. In situations where the molecular diameters are similar, the energetically favored species is preferentially adsorbed at all temperatures. Axial pair correlations are used to relate the composition in the nanotube with the structure of the adsorbed fluid.

1. Introduction 1

Carbon nanotubes are cylindrical structures that were first discovered during the direct current arc evaporation of graphite. They consist of rolled up graphite sheets and are typically multi-walled with their ends capped.2,3 Single-walled nanotubes are found to grow in the gas phase and have been observed to cluster, forming a bundle of straws.4,5 Tersoff and Ruoff6 show that the nanotubes retain their cylindrical shape when their internal diameters are less than 10 Å and flatten to form a honeycomb structure when the internal diameters exceed 25 Å. Recent research efforts have been directed at synthesizing aligned nanotubes on substrates7 and single-walled nanobundles.8 With the ability to introduce material into the nanotube, researchers have confined molten lead,9,10 decomposed silver nitrate,11 and studied the influence of metals on its electronic properties.12 Local density functional calculations have been carried out by Pederson and Broughton13 on a pair of HF molecules constrained to move along the nanotube axis, and Breton et al.14 have studied the adsorption of rare gases and alkali-metal atoms in the nanotube. Although the nanotubes typically have their ends capped, attempts to open the ends has met with some success.15,16 (1) Iijima, S. Nature 1991, 354, 56. (2) Iijima, S.; Ichihashi, T.; Ando, Y. Nature 1992, 356, 776. (3) Ajayan, P. M.; Ichihashi, T.; Iijima, S. J. Chem. Phys. 1993, 202, 384. (4) Iijima, S.; Ichihashi, T. Nature 1993, 363, 603. (5) Bethune, D. S.; Klang, C. H.; de Vries, M. S.; Goman, G.; Savoy, R.; Vazquez, J.; Beyers, R. Nature 1993, 363, 605. (6) Tersoff, J.; Ruoff, R. S. Phys. Rev. Lett. 1994, 73, 676. (7) Li, W. Z.; Xie, S. S.; Qian, L. X.; Chang, B. H.; Zou, B. S.; Zhou, W. Y.; Zhao, R. A.; Wang, G. Science 1996, 274, 1701. (8) Thess, A. Science 1996, 273, 483. (9) Ajayan, P. M.; Iijima, S. Nature 1993, 361, 333. (10) Seshadri, R.; Govindaraj, A.; Hemantkumar, N. A.; Rahul Sen; Subbanna, G. N.; Raju, A. R.; Rao, C. N. R. Curr. Sci. 1994, 66, 839. (11) Ugarte, D.; Chatelain, A.; de Heer, W. A. Science 1996, 274, 1897. (12) Galpern, E. G.; Stankevich, A. L.; Chistykov, A. L.; Chernozatonskii, L. A. Chem. Phys. Lett. 1993, 214, 345. (13) Pederson, M. R.; Broughton, J. Q. Phys. Rev. Lett. 1992, 69, 2689. (14) Breton, J.; Gonzalez-Platas, J.; Girardet, C. J. Chem. Phys. 1994, 101, 3334. (15) Ebbesen, T. W.; Ajayan, P. M.; Hiuri, H.; Tanigaki, K. Nature 1994, 367, 519.

The inner diameters of the carbon nanotubes range from 5 to 50 Å with lengths up to 104 Å. This range of inner diameters is similar to those found in a number of zeolites and other nanoporous materials commonly used for adsorption, separations, and catalysis. The grand canonical ensemble (µVT) is ideally suited for studying adsorption in nanopores and nanoporous materials. In a grand canonical Monte Carlo (GCMC) simulation the nanopore of a fixed volume V is equilibrated with a bulk fluid whose chemical potential µ and temperature T are held fixed. In addition to obtaining adsorption isotherms, GCMC simulations have been used to determine the factors that control and influence the selectivity of a nanoporous host toward a specific component from a bulk fluid mixture. In this regard mixture simulations have been carried out in slit pores,17-23 in zeolites,24,25 and to a lesser extent in cylindrical pores.17,18 The carbon nanotube is an ideal candidate to investigate adsorption characteristics in a cylindrical geometry, and recently simulations have been used to study adsorption of pure fluids26 and mixtures25,27 in carbon nanotubes. The emerging picture from the literature on mixture GCMC studies is that adsorption of the energetically favored species is preferred at bulk gas conditions.17,25 At high bulk pressures or when the bulk fluid is a liquid, the situation is more complex. GCMC simulations on the adsorption from a bulk liquid mixture of cyclohexane and octamethylcyclotetrasiloxane (OMCTS) in smooth slit mica pores show that OMCTS, the larger and more energetically (16) Tsang, S. C.; Chen, Y. K.; Harris, P. J. F.; Green, M. L. H. Nature 1994, 372, 159. (17) Keffer, D.; Davis, H. T.; McCormick, A. V. J. Phys. Chem. 1996, 100, 638. (18) Nicholson, D.; Gubbins, K. E. J. Chem. Phys. 1996, 104, 8126. (19) Somers, S. A.; McCormick, A. V.; Davis, H. T. J. Chem. Phys. 1993, 99, 9890. (20) Curry, J. E.; Cushman, J. H. Mol. Phys. 1995, 85, 173. (21) Cracknell, R. F; Nicholson, D.; Quirke, N. Mol. Phys. 1993, 80, 885. (22) Somers, S. A.; Ayappa, K. G.; McCormick A. V.; Davis, H. T. Adsorption 1996, 2, 33. (23) Piotrovskaya, E. M.; Brodskaya, E. N. Langmuir 1993, 9, 3548. (24) Van Tassel, P. R.; Davis, H. T.; McCormick, A. V. Langmuir 1994, 10, 1257. (25) Maddox, M. W.; Sowers, S. L.; Gubbins, K. E. Adsorption 1996, 2, 23. (26) Maddox, M. W.; Gubbins, K. E. Langmuir 1995, 11, 3988. (27) Ayappa, K. G. Chem. Phys. Lett., in press.

S0743-7463(97)00499-X CCC: $15.00 © 1998 American Chemical Society Published on Web 01/23/1998

Binary Mixture Adsorption in Carbon Nanotubes

favored species completely excludes cyclohexane at certain slit widths.19,22 On the other hand adsorption simulations of an argon-xenon binary mixture in zeolite NaA, whose host cavities are cuboctahedrons, show that at high bulk pressures the adsorption of argon is favored over the larger and more energetically favored species xenon.24 In the above studies superior packing abilities of the species was used to explain its greater selectivity at high loadings. If the Lennard-Jones (LJ) molecular diameters of the species are similar, both species have their potential energy minima located at similar distances from the pore wall. This situation occurs in LJ mixtures like methaneethane21,28 and argon-krypton.23 In these situations the adsorbates compete for similar regions of the pore space and packing effects are less important. The structure of the adsorbed fluid, which is related to its ability to pack or epitaxially layer, appears to play an important role in determining selectivity. With the exception of the molecular dynamics study by Somers et al.22 and GCMC results of Curry and Cushman20 the relation between adsorbed fluid structure, mobility, and selectivity has received little attention. In a recent work we reported GCMC simulations of adsorption from a LJ binary gas mixture into a singlewalled carbon nanotube.27 GCMC simulations were carried out over a range of temperatures for a fixed composition low-pressure bulk gas mixture. At high temperatures, the larger energetically favored OMCTS is adsorbed, and at lower temperatures and pressures, cyclohexane, the smaller species, completely eliminates the larger species from the nanopore. This complete transition toward the smaller species at low temperatures suggests a mechanism fundamentally different from previous mixture GCMC studies,17,24 where an increase in selectivity toward the smaller species is observed at high bulk pressures. The motivation for the present work is 2-fold. In addition to investigating the nanotubes potential as an effective separations material, we investigate in detail the reasons for the observed27 transitions in selectivity as a function of temperature and nanotube diameter. Since carbon nanotubes are experimentally observed over a range of internal diameters, simulations at different diameters not only are realistic but also can reveal a large spectrum of adsorption scenarios. By using four different binary gas mixtures and treating the molecules as LJ spheres we determine the conditions under which these temperature driven transitions in selectivity are likely to occur and show situations where the transitions will not occur. Using GCMC simulations with particle identity exchange, we study the adsorption of the following mixtures into carbon nanotubes of various diameters: cyclohexane (c)-OMCTS (o), where σoo/σcc ) 1.426 and oo/cc ) 1.059; methane (m)-ethane (e) where σee/σmm ) 1.036 and ee/mm ) 1.64; ethane (e)-propane (p) where σpp/σee ) 1.427 and pp/ee ) 0.996; and methane (m)propane (p) where σpp/σmm ) 1.479 and pp/mm ) 1.634. With the exception of the methane-ethane system, the adsorption sites for the other mixtures are located at different radial locations within the nanotube. The motivation for studying the above mixtures is discussed in the next section. In section 2 following the introduction we discuss the simulation procedure and potentials used for the fluidfluid and fluid-wall interactions. In results and discussion (section 3) we report radial density distributions, pore (28) Tan, Z.; Gubbins, K. E. J. Phys. Chem. 1992, 96, 845. (29) Tan, Z.; Van Swol, F.; Gubbins, K. E. Mol. Phys. 1987, 62, 1213.

Langmuir, Vol. 14, No. 4, 1998 881

a

b

Figure 1. Graphite sheet indicating modes of rolling and a segment of an armchair carbon nanotube of diameter 14.964 Å.

average densities, mole fractions, and total energies as a function of temperature for the various mixtures studied. In an attempt to correlate the adsorption selectivity with the adsorbed fluid structure, we also calculate the axial pair correlation functions for the confined fluid. 2. Simulation Procedure Fluid-Fluid and Fluid-Wall Interactions. Figure 1a illustrates a graphite sheet and the modes of rolling that give rise to the armchair and saw-tooth or zigzag configurations. Chiral configurations occur when the axis of rolling lies between the two axis indicated in the figure. We construct single-walled structured carbon nanotubes according to the armchair mode of rolling. Carbon nanotubes constructed in this manner have only certain allowed diameters. Note that the saw-tooth mode of rolling would produce nanotubes at different allowed diameters. Figure 1b illustrates a segment of an armchair carbon nanotube of diameter D ) 14.964 Å. We used a length of 76.405 Å for all simulations reported in this study. Using the C-C bond length of 1.423 Å corresponding to that of graphite, carbon nanotubes with diameters D ) 8.184, 9.538, 10.894, 12.25, 13.608, 14.964, 16.322, and 19.036 Å were constructed. D is the center to center distance of two diametrically opposite carbon atoms on the nanotube wall. The number of carbon wall atoms varied from 756 at D ) 8.184 Å to 1764 at D ) 19.036 Å. Fluid-fluid and fluid-wall interactions are assumed to follow a 12-6 LJ potential

[( ) ( ) ]

V(rij) ) 4ij

σij rij

12

-

σij rij

6

.

(1)

For dissimilar species, interaction parameters are cal-

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Ayappa

Figure 2. Fluid-wall (Ufw) potential energy profiles for the four mixtures studied: (a) cyclohexane (c)-OMCTS (o); (b) methane (m)-ethane (e); (c) ethane (e)-propane (p); (d) methane (m)-propane (p). The ratios of the LJ parameters are indicated in the figures. The angular and axial positions of the molecule had little influence on the radial variation of Ufw. Table 1. Lennard-Jones Parameters for Various Molecules Used in the Simulations species

ii/k (K)

σii (Å)

cyclohexane19 (c) OMCTS19 (o) methane30 (m) ethane30 (e) propane25 (p) carbon30 (w)

324 343 148.1 243 242 28

5.4 7.7 3.81 3.95 5.637 3.4

Ξ)

culated using the Lorentz-Berthelot mixture rules, where σij ) (σii + σjj)/2.0 and ij ) (iijj)1/2. The total potential energy, N-1 N

U(r) )

a carbon nanotube of diameter 14.964 Å. The ratios of the LJ interaction energies and molecular diameters are also shown in the figure. In the cyclohexane-OMCTS mixture (Figure 2a), the fluid-wall interaction energies are comparable; however, the larger size of OMCTS results in it having a deeper potential energy minima due to an increased number of OMCTS-wall interactions. This situation is independent of the pore geometry30 and occurs in smooth pores19 as well. For methane-ethane (Figure 2b), due to the small difference in molecular diameters, the potential energy minima are located at almost identical distances from the nanotube wall, with ethane the energetically favored species having the deeper well. For the ethane-propane system (Figure 2c) the interaction strengths and size ratios are similar to the cyclohexaneOMCTS system; however, the fluid-wall interaction strengths are lower. For the methane-propane system (Figure 2d) the energetic advantage of propane, unlike the case of OMCTS in the cyclohexane-OMCTS mixture, results from its larger diameter and greater interaction energy than methane. In all cases the minimum for the fluid-wall potential energy interaction is located at the center of the nanotube for small tube diameters and shifts toward the wall as the nanotube diameter increases. Pore GCMC. The GCMC (µVT) simulation is carried out by equilibrating the carbon nanotube of a fixed volume V, with a bulk fluid mixture whose species chemical potentials, µi, and temperature, T, are held constant. The grand canonical partition function, ∞

VNz1N1z2N2

N1,N2)0

N1!N2!

∑

zi )

(2)

where the first term represents fluid-fluid interactions and the second term the fluid-wall (Ufw) interactions, both calculated using the LJ potential, eq 1. N is the total number of fluid particles in the nanotube and Nw is the total number of carbon atoms in the nanotube wall. We used a potential cutoff of 3.0σ of the larger particles for the cyclohexane-OMCTS and methane-ethane mixtures and a half box length cutoff for the other two mixtures. Exceptions to the above cutoff values are mentioned in the text. The LJ parameters used for the four fluid mixtures studied here are shown in Table 1. For the carbon wall (w) atoms in the nanotube, we used the LJ parameters for graphite.30 Cyclohexane and OMCTS parameters are similar to those used in previous slit pore GCMC simulations19 and in our mixture simulations in carbon nanotubes.27 The LJ parameters for methane, ethane, and propane are similar to those used for adsorption studies in slit pores21 and in carbon nanotubes.25 Although LJ parameters differ slightly depending on the source, using the above values enables us to compare our results with other nanopore adsorption studies where possible. Figure 2 illustrates the radial variation of the reduced fluid-wall potential energies for the various mixtures, in

(3)

where N1 is the number of molecules of species 1, N2 is the number of molecules of species 2, N ) N1 + N2, V is the pore volume based on the diameter D defined earlier, s is a scaled variable where dsi ) dx′i,dy′i,dz′i, and x′i ) x/V1/3, y′i ) y/V1/3, and z′i ) z/V1/3. The activity for the ith species

Nw N

V(riw), ∑ ∑ V(rij) + w)1 ∑∑ i)1 j)i+1 i)1

∫ ... ∫ exp[-βU(s)] ds1, ..., dsN,

exp(βµi) , Λ3i

(4)

where µi ) µiid + µiex is the sum of the ideal gas and excess chemical potential contributions. The probability distribution that is used to generate the Markov chain during the GCMC simulations is

F(N1,N2,s) )

VN1+N2z1N1z2N2 exp[-βU(s)] . N1!N2!Ξ

(5)

During the grand canonical Monte Carlo simulation a new state in the Markov chain is generated by carrying out a cycle consisting of a random displacement, insertion, deletion, and particle exchange moves. The probabilities for these moves are based on the probability distribution given in eq 5. The GCMC moves are carried out by accepting moves with probability

exp(-β∆ψ),

(6)

according to the standard Metropolis algorithm.31 For (30) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon: Oxford, 1974. (31) Allen, M. P; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987.

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Langmuir, Vol. 14, No. 4, 1998 883

displacing a randomly chosen molecule

∆ψ ) ∆U,

(7)

for inserting a molecule of species i

∆ψ ) ∆U -

(

)

z iV 1 ln , β Ni + 1

(8)

i ) 1, 2 for deleting a molecule of species i

∆ψ ) ∆U -

( )

Ni 1 ln , β ziV

(9)

i ) 1, 2 and for exchanging a molecule of species i with species j

∆ψ ) ∆U -

(

)

zjNi 1 ln . β zi(Nj + 1)

(10)

i, j ) 1, 2 In the above equations, ∆U ) Unew - Uold is the change in the potential energy of the system after executing a given step. Potential energy calculations are carried out in units reduced with the LJ diameter of the smaller species, and periodic boundary conditions were used along the axial direction of the nanotube. The particle exchange moves have been shown to speed up equilibration19,24 and reduce the fluctuations in the particle numbers once equilibration has been achieved.21 At lower temperatures or when a transition in selectivity occurs, the first 3 to 4 million cycles were discarded and averages were accumulated over the next 4-6 million cycles. At high temperatures where only one component is adsorbed, equilibration is much quicker, and the first 2 million cycles were discarded with averages taken over the next 2 million. We tested our simulation procedure with the GCMC results of Somers et al.19 for smooth slit mica pores and the condition for which number densities were reported by Cracknell et al.21 The ensemble average particle numbers and density distributions obtained by us were in good agreement with those reported in the above studies. We also carried out simulations with and without the particle identity exchange for the cyclohexane-OMCTS mixture at D ) 14.964 Å, at different temperatures. At low and high temperatures where predominantly one species is adsorbed, the number of moves to achieve equilibrium were similar to simulations carried out with the exchange steps. At the transition temperature where selectivity changes from one species to the other, fluctuations in the particle numbers about their mean values was the greatest. These fluctuations around the transition temperatures have been reported in our earlier work,27 where fluctuations in the cyclohexane mole fractions were as larger as 40% at D ) 14.964 Å. For the cyclohexaneOMCTS mixture at D ) 14.964 Å, 12 million cycles were insufficient to equilibrate the system around the transition temperature without particle exchange moves. We did not attempt to equilibrate the simulations without the particle exchange at these conditions. To investigate the effect of system size we carried out simulations at T ) 300 and 389 K at D ) 14.964 Å, with nanotube lengths of 76.405 and 118.4 Å. Particle densities were about 5% greater for the longer pore. Differences in mole fractions

and potential energies were less than 1%, well within statistical deviations. Bulk Fluid. The bulk state for all the mixtures studied was a 50/50 gas mixture with particle number densities, F1 ) F2 ) 3.0962 × 10-6 Å-3. This density corresponds to a total pressure of 0.338 atm at T ) 400 K. We carried out pore GCMC simulations by keeping the bulk density fixed with temperatures typically ranging from 100 to 400 K. Simulations carried out in this manner result in exposing the pore to a bulk gas of fixed density whose pressure decreases as the temperature is lowered. In all cases we carried out a series of bulk GCMC simulations with large and small simulation cell volumes to ensure that the bulk state was a gas at the simulation temperature. The cyclohexane-OMCTS mixture was found to be a gas at 200 K, which was the lowest temperature used for that mixture. All other bulk mixtures were in their gas states at 100 K. Since the bulk gas pressure is low, we assumed the gas to be ideal and set the species excess chemical potentials to zero. With this assumption 3 µi ) µid i ) kT ln(FiΛi ),

(11)

and the species activity (eq 4)

zi ) Fi.

(12)

We use zi as inputs while carrying out our GCMC simulations. To check the influence of the excess chemical potential on the simulations, we calculated the excess chemical potential in the bulk gas mixture by using the test particle insertion technique in the canonical ensemble (NVT). GCMC simulations were carried out using these excess chemical potentials and compared with the zero excess chemical potential results for a few different temperatures. Differences between the two were found to be insignificant. 3. Results and Discussion Angle averaged radial density distributions, Fi, for the ith fluid species is

∆r ∆r N (r- ,r+ )〉 〈 2 2 , F (r) ) i

i

2πr∆rL

(13)

where L is the length of the nanotube, Ni is the number of particles of species i in a cylindrical shell of thickness ∆r, located at a radial distance r from the center of the tube. ∆r ) 0.02σ, where σ is the molecular diameter of the smaller molecule, was used to calculate density distributions. Fi(r) obtained using eq 13 represents averaging in the angular as well as the axial directions of the nanotube. The mole fraction xi of the ith species is

xi )〈Ni〉/〈N〉,

(14)

where 〈Ni〉 is the ensemble averaged number of species i in the nanotube and 〈N〉 is ensemble average of the total number of particles in the nanotube. The pore averaged density for the ith species

Fi ) 〈Ni〉/V.

(15)

The axial pair correlation (APC) between two fluid particles of the the ith species, gii(z), which yields information about the structure of the adsorbed fluid along the axial direction of the nanotube was calculated in the

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Ayappa

Figure 3. Mole fractions, xo, and averaged densities, Fjo*, of OMCTS (o) for the cyclohexane-OMCTS mixture as a function of the diameter of the carbon nanotube. Fjo* ) Foσcc3.

canonical (NVT) ensemble using

gii(z) )

1 〈Ni(z,z+∆z)〉 , 〈Ni〉 2∆z〈Ni〉/L

(16)

where 〈Ni〉 is the ensemble averaged number of particles of species i obtained from the GCMC simulation and ∆z is the thickness of the circular disk located at a distance z from the particle of interest. The factor of 2 in the denominator accounts for the presence of two such disks. In situations where the fluid particles lie toward the wall of the nanotube, we split the nanotube into four quarters while calculating gii(z). This was used to prevent contributions from particles with similar z coordinates but different angular locations while evaluating gii(z). When the number of fluid molecules of a particular species was too small to get reasonable statistics, we did not plot the APC. The APC between two different fluid species was not evaluated. Cyclohexane-OMCTS Mixture. In all the figures we have used the following reduced quantities: r* ) r/σ, z* ) z/σ, F* ) Fσ3, and U* ) U/. Unless stated otherwise all the reduced units are with respect to the smaller species in the fluid mixture. Figure 3a illustrates the mole fraction of OMCTS, xo, for the cyclohexane-OMCTS mixture as a function of the nanotube diameter, D, at different temperatures. At D ) 9.538 Å, xo is zero at all temperatures and OMCTS is excluded due to molecular sieving. At D ) 10.894 Å, OMCTS is favored at low temperatures and at 389 K, xo decreases to 0.2. As we will see, this is the only situation in all the mixtures investigated here where the larger energetically favored species is preferred at low temperatures. At D ) 12.25 Å, xo is close to unity for all the temperatures studied. This situation persists for temperatures even as low as 100 K and cyclohexane is unable to access the nanotube. At D ) 13.608 Å, OMCTS is preferred at higher temperatures; however at 200 K the nanotube favors cyclohexane, completely excluding OMCTS from the nanotube. As the nanotube diameter is further increased, both OMCTS and cyclohexane are adsorbed at 389 K, and in all cases a complete exclusion of OMCTS occurs at 200 K.

Figure 4. Radial density distributions in the carbon nanotube for cyclohexane (solid) and OMCTS (dashed) at 200 and 389 K: (a) and (b) D ) 10.894 Å; (c) and (d) D ) 12.25 Å; (e) and (f) D ) 13.608 Å. Axial pair correlation functions for the fluid in the nanotube, goo (OMCTS-OMCTS) and gcc (cyclohexane-cyclohexane) are shown in the insets. Fo* ) Foσcc3, r* ) r/σcc, and z* ) z/σcc.

Figure 3b illustrates the pore averaged density for OMCTS as a function of temperature and nanotube diameter. A peak in the average density is observed at D ) 12.25 Å, where the particles are packed along the axis of the nanotube. At intermediate diameters, although the pore volume is increasing, the change in the number of particles is small due to packing constraint, and the average density decreases. At larger nanotube diameters, however, the number of adsorbed particles increases and the average density rises. In general the trends in the average density are similar to the mole fraction. Note however that the pore densities are about 4 orders of magnitude greater than the bulk values (Fo* ) 4.875 × 10-4 ). Figure 4 illustrates radial density distributions for the cyclohexane-OMCTS system for D ) 10.894 (Figure 4a,b), 12.25 (Figure 4c,d) and 13.608 Å (Figure 4e,f), at 200 and 389 K. The corresponding APC’s are shown as insets. At D ) 10.894 Å and 200 K, predominantly OMCTS is adsorbed along the axis of the nanotube (xo ) 0.74) and the APC indicates that the fluid is solid-like in the nanotube. At 389 K the nanotube has a cyclohexane mole fraction of 0.8, and the APC indicates that it is more like that of a solid. At 450 K (not shown) xo reduces to 0.183. At D ) 12.25 Å (Figure 4c,d) OMCTS is present in the nanotube at both 200 and 389 K and the APC’s clearly indicate that OMCTS is solid-like in the nanotube at both temperatures. At 389 K although OMCTS is still solid, it is comparatively loosely packed as seen by the broader

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Langmuir, Vol. 14, No. 4, 1998 885

Figure 6. Cyclohexane (solid) and OMCTS (dashed) radial density distributions for D ) 19.036 Å at different temperatures. Axial pair correlation functions are illustrated as insets. F* ) Fσcc3, z* ) z/σcc and r* ) r/σcc.

Figure 5. Cyclohexane (solid) and OMCTS (dashed) radial density distributions for D ) 16.322 Å at different temperatures. Axial pair correlation functions are illustrated as insets. F* ) Fσcc3, z* ) z/σcc and r* ) r/σcc.

peaks in both the APC and density distributions. At this nanotube diameter, the difference between the fluid-wall potential well depths, ∆Ufw* ) -27.3, is the largest, and OMCTS is energetically preferred at both low and high temperatures. A similar situation is observed at D ) 9.538 Å in the ethane-propane (Figure 10) and methanepropane (Figure 12) systems where propane the larger energetically favored species is present in the nanotube at all temperatures. The adsorption of the larger molecule has also been observed in slit mica pore simulations of the cyclohexane-OMCTS bulk liquid system,19,22 where the packing ability of OMCTS determined its adsorption from a bulk liquid. As seen in Figure 3a the situation changes for nanotubes with diameters greater than 12.25 Å, where cyclohexane is able to access the pore at lower temperatures. At D ) 13.608 Å, (Figure 4f) only OMCTS is present above 250 K; however, at 200 K pure cyclohexane is present and OMCTS is completely eliminated from the nanotube. The APC’s in Figure 4f clearly indicate that OMCTS is frozen at higher temperatures and cyclohexane is highly structured, though not quite frozen at 200 K. Figure 5 illustrates the density distributions in the nanotube of diameter 16.322 Å at various temperatures. At low temperatures only cyclohexane is adsorbed forming the outer fluid layer. As the temperature is increased, the amount of OMCTS adsorbed increases, with OMCTS occupying the inner layer. Since the nanotube diameter is large, both species are able to coexist within the pore and the transition occurs over a broad temperature range. APC’s illustrate that at low temperatures cyclohexane is

solid-like in the nanotube and turns more like a liquid as the temperature is increased. This transition seems to occur from 200 to 260 K. A definite liquid-like cyclohexane structure, which persists until about 350 K, is observed in the nanotube at 260 K. The structure of cyclohexane indicates that it condenses into the pore as the temperature is lowered. Similar fluid behavior for cyclohexane and OMCTS was observed in our simulations27 at D ) 14.964 Å where we first reported this temperature driven transition in selectivity. Figure 6 illustrates the density distributions for D ) 19.036 Å at different temperatures. Here again both species coexist in the nanotube at higher temperatures with the amount of OMCTS increasing with temperature. At 200 K we find for the first time, that in addition to the outer layer, a central layer of cyclohexane is able to form. Although the density peak for the outer layer is lower, there are about 69 cyclohexane molecules present with only 14 molecules at the nanotube axis. The APC at 200 K shown only for the outer fluid layer indicates that the fluid is frozen in the nanotube. A similar behavior is observed for the fluid located along the axis of the nanotube. APC’s indicate that cyclohexane is completely frozen at 200 K and both species appear to coexist in a liquid-like state at the other temperatures. Figure 7 illustrates the variation in xo and the total energy U* as a function of temperature for D ) 13.608 (Figure 7a,b) 14.964 (Figure 7c,d), and 16.322 Å (Figure 7e,f). In all cases the fluid-wall interactions contribute to over 90% of the total potential energy with the contribution of fluid-fluid interaction being the greatest at low temperatures when the nanotube is filled with a larger number of cyclohexane molecules. The change in xo as a function of temperature for D ) 13.608 Å is shown in Figure 7a, and the total potential energy is shown in Figure 7b. The transition in selectivity which occurs between 215 and 220 K (Figure 7a) is accompanied by a decreases in the total potential energy of the system as seen in Figure 7b. Although the difference between the fluid-wall potential energy minima, ∆Ufw* ) -22.1, the increased area for adsorption is able to offset this difference and a larger number of cyclohexane molecules adsorbs at lower temperatures, favoring a lower energy state as

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Figure 7. Mole fraction of OMCTS, xo, and reduced potential energies, U* ) U/cc, as a function of temperature for D ) 13.608, 14.964, and 16.322 Å. In all situations the potential energy of the system decreases as the temperature is lowered. A potential cutoff of half box length was used at D ) 14.964 and 16.322.

indicated by the total potential energies, and OMCTS is completely eliminated from the nanotube. The variation of xo and U* as a function of temperature at D ) 16.322 Å (Figure 7e,f), whose density distributions are shown in Figure 5, indicate that the transition from an OMCTS rich nanotube to pure cyclohexane occurs over a broader temperature range when compared with D ) 13.608 and 14.964 Å. In all cases however, the transition in selectivity is accompanied by a decrease in the total potential energy of the system which results from a switch in selectivity from OMCTS at high temperatures to pure cyclohexane at low temperatures. Although we did not study the temperature dependence in detail for D ) 19.036 Å, the total potential energy of the system U* at 200 K is -1196.3 and is significantly higher (U* ) -627.8) at 389 K indicating a considerable energetic benefit favoring the adsorption of cyclohexane at low temperatures. When a transition to pure cyclohexane occurs, our results indicate that the system is driven toward a lowenergy state due to a larger amount of cyclohexane molecules that can access the nanotube. To support this argument, we show that cyclohexane is unable to access the nanotube at D ) 12.25 Å, since the number of cyclohexane molecules needed to overcome the low potential energy state enjoyed by OMCTS is unfavorable at this nanotube diameter. Note that the difference between the fluid-wall potential well depths, ∆Ufw* ) -27.2, is the greatest at this diameter. Since the peak in the density distribution occurs at the location of the fluid-wall potential energy minima, a crude estimate of the fluidwall potential energy for a particular species can be made by simply multiplying the potential energy at the minima with the number of molecules adsorbed. On the basis of the depth of the fluid-wall potential, Ufw* ) -17.8 at D

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) 12.25 for cyclohexane, the number of cyclohexane molecules required to attain the same energy (U* ) -443) of the pure OMCTS system at 200 K is 22 molecules (allowing 10% for fluid-fluid interactions). Hence in the D ) 12.25 Å nanotube for cyclohexane to be energetically favored at 200 K it would have to adsorb in excess of 22 molecules. A canonical ensemble simulation at D ) 12.25 Å and 200 K with 25 cyclohexane molecules resulted in a U* of only -293.17 with repulsive fluid-fluid interactions accounting for about 15% of the total energy. Note that the number of cyclohexane molecules present at D ) 13.608 Å at 200 K was 25. As mentioned earlier, D ) 10.894 Å, is the only case where the larger energetically favored species is preferred at low temperatures and where both species are present at all temperatures studied. At D ) 10.894 Å, unlike other diameters, the fluid-wall potential energy profiles show that both species have their potential minima located in the center of the nanotube; however, the fluid-wall potential energy for cyclohexane is relatively flat across the nanotube with a weak minimum at the center. This results in a delocalization of cyclohexane in the nanotube as seen in its density distribution (Figure 4b), which is quite diffuse when compared to that of OMCTS. The increase in selectivity toward cyclohexane at higher temperatures appears to be driven in part by this entropic benefit. The difference between the fluid-wall potential energy minima, ∆Ufw* ) -7.7, for the two species at D ) 10.894 Å is the lowest among the nanotube diameters studied. The total potential energy, U*, is -278.5 at 200 K and -289 at 389 K indicating that energetic benefits, which play an important role in determining the selectivity at low temperatures in larger nanotubes, have a weaker influence at this diameter. In contrast, the difference between the potential energy minima, ∆Ufw*, is greatest at -27.2 for D ) 12.25 Å decreasing to -14.0 at D ) 16.322 Å. For the nanotubes which show a complete transition toward cyclohexane at lower temperatures, we estimated the size of the remaining space, ∆, in the nanotube to see if an OMCTS molecule could be accommodated purely on geometric considerations

∆ ) D - σw - (2 × 0.9)σcc

(17)

where σw is the diameter of the carbon wall atom and 0.9 accounts for a 10% compression of the molecules.19 At D ) 19.036 Å, ∆ ) 5.916 Å, indicating that only cyclohexane (σcc ) 5.4 Å) could be accommodated as is the case shown in Figure 6 at 200 K. At D ) 13.608, 14.964, and 16.322 Å, ∆ is smaller than 5.4 Å and only a single outer layer of cyclohexane molecules is adsorbed at low temperatures. The above calculation shows that OMCTS is unable to contribute to the low-energy state at low temperatures due to steric limitations. Hence one would expect that OMCTS would be present with cyclohexane at larger nanotubes at the low temperatures. Although we did not carry out simulations in larger nanotubes for the cyclohexane-OMCTS mixture, we show that this is indeed the case with the ethane-propane mixture at D ) 19.036 Å (Figure 11e,f). Packing effects if present are likely to occur at low temperatures. In order to evaluate the packing effectiveness, we calculated the layer packing fraction19 defined as the ratio of the effective pore diameter to the width occupied by the the fluid layers. In adsorption from a bulk liquid mixture of cyclohexane-OMCTS, Somers et al.19 found that the species with a layer packing fraction close to unity was favored. Allowing for a compression of

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Figure 8. Mole fractions, xe, and averaged densities, Fje*, for ethane for a bulk methane-ethane mixture vs nanotube diameter D. Ethane the energetically favored species is preferentially adsorbed at all nanotube diameters. Fje* ) Fjeσmm3.

10%19 for the fluid molecules, the layer packing fractions for D ) 13.608, 14.964, 16.322, and 19.036 (where transitions were observed) for cyclohexane were 1.05, 1.189, 1.329, and 1.072 and for OMCTS were 1.47, 0.834, 0.932, and 1.128, respectively. In all cases, except at D ) 16.322, cyclohexane is favored on packing grounds. However at D ) 12.25 where OMCTS is present at all temperatures, the packing fraction favors the adsorption of cyclohexane over OMCTS, and at D ) 10.894 where the packing fraction favors OMCTS, cyclohexane is adsorbed at low temperatures. Our high temperature results concur with previous GCMC mixture simulations of species with different LJ diameters from a bulk gas,17,24 where the species with greater fluid-wall interactions is preferentially adsorbed. However at low temperatures we observe that the adsorbate composition which makes up the lowest energy configuration dictates adsorption. Depending on the pore diameter this can result in the pore being enriched with either the small or large molecule. Calculations of the layer packing fractions show that the component adsorbed at low temperatures need not have a greater packing advantage. This differentiates our observations from previous mixture simulations with bulk liquids where packing is a dominant factor in determining pore selectivity.19,22 The observed shift in selectivity at lower temperatures suggests a mechanism quite different from the packing viewpoint attributed to increased selectivity toward the smaller species at high pore densities and higher bulk pressures.17,24 The qualitative trends that drives adsorption selectivity in our system appears to fit in with the features observed in lattice-gas models, where low-temperature states are dominated by the composition with the lowest internal energy.32 Methane-Ethane Mixture. Figure 8a illustrates the mole-fraction of ethane, xe, as a function of nanotube diameter for the methane-ethane mixture. Due to their similar molecular diameters, σee/σmm ) 1.036, both species (32) Chandler, D. Introduction to Modern Statistical Mechanics; Oxford University Press: Oxford, 1987.

Figure 9. Radial density distributions in the carbon nanotube for the methane (solid) and ethane (dashed) mixture at 100 and 400 K: (a) and (b) D ) 8.184 Å; (c) and (d) D ) 10.894 Å; (e) and (f) D ) 14.964 Å. Axial pair correlation functions for ethane in the nanotube, gee (ethane-ethane) are shown in the insets. F* ) Fσmm3 and r* ) r/σmm.

have their potential energy minima at nearly similar radial locations from the nanotube axis (Figure 2b) and hence compete for the same adsorption sites. Size effects are minimized and methane does not have the advantage of a greater area for adsorption. In this situation, quite unlike the other mixtures investigated here, we find that the energetically favored species ethane is preferentially adsorbed at both low and high temperatures for all nanotube diameters studied. The difference between the fluid-wall potential energy minima between methane and ethane is greatest at D ) 8.184 Å and decreases as the nanotube diameter is increased. Temperature has a greater effect at larger nanotube diameters, where although the densities are gaslike, a small increase in the mole fraction of methane is observed at higher temperatures. At 300 and 400 K, although methane is able to access the nanotube, desorption is high, and the overall pore densities are low as seen in Figure 8b. The pore averaged density (Figure 8b) shows that at smaller diameters, where the particles are toward the center of the nanotube, packing is maximum and temperature has little effect. At larger nanotubes the density changes significantly when the temperature is lowered from 300 to 200 K with little change in the density from 200 to 100 K. The density distributions for the methane-ethane system are shown in Figure 9 for D ) 8.184 Å (Figure 9a,b), D ) 10.894 Å (Figure 9c,d) and D ) 14.964 Å (Figure 9e,f). Ethane is the dominant species at all nanotube diameters. APC’s for all nanotubes show an increased

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Figure 10. Mole fraction, xp, and averaged densities, Fjp*, of propane vs nanotube diameter D for a bulk ethane-propane mixture, Fjp* ) Fpσee3.

structure in the pore fluid as the temperature is decreased. At D ) 10.894 Å (Figure 9c,d) ethane is highly structured in the nanotube at 100 K and more liquid-like at 400 K. APC’s evaluated at intermediate temperatures for this nanotube reveal that the pore fluid is more solid-like at 200 K, indicating that the transition occurs between 300 and 200 K. At this diameter the pore fluid density is quite insensitive to the temperature suggesting that a liquid to solid transition is occurring in this psuedo-onedimensional situation at fixed loading. The number of ethane molecules increases from 15 at 400 K to 18 at 100 K. At higher temperatures, although the density of ethane decreases, the probability of methane accessing the nanotube increases, as seen in the density distribution, Figure 9d. At D ) 14.964 Å (Figure 9e,f) ethane is able to adsorb both at the wall and along the axis of the nanotube at 100 K. This situation is similar to the 19.036 Å nanotube for the cyclohexane-OMCTS mixture. The inset in Figure 9e illustrates the APC for the outer fluid layer only. At this diameter desorption is dominant at 400 K, and the nanotube is almost empty; hence the density and APC are shown at 300 K (Figure 9f). Our observations for the methane-ethane mixture system are consistent with the observations in other mixture GCMC studies of species with similar molecular diameters, where the energetically favored species is preferred21,23,28 at both low and high bulk pressures. The preference of methane to form inner layers as seen in slit pores21,28 was not observed here. This is probably due to the low bulk pressures used in our study. Ethane-Propane and Methane-Propane Mixtures. Figure 10 illustrates the mole fraction and average density plots for the ethane-propane mixture. The ratios of the molecular diameters and energies are similar to the cyclohexane-OMCTS mixture; however, due to the lower energy parameters, the fluid-wall interactions are smaller (Table 1). At D ) 9.538 Å, propane is adsorbed at all temperatures, and the density distributions at 100 and 400 K (Figure 11a,b) show that propane adsorbs along the axis of the nanotube. At this nanotube diameter the

Figure 11. Radial density distributions in the carbon nanotubes for ethane (solid) and propane (dashed) at 100 and 400 K: (a) and (b) D ) 9.538 Å; (c) and (d) D ) 12.25 Å; (e) and (f) D ) 19.036 Å. Axial pair correlation functions for the fluid in the nanotube, gee (ethane-ethane) and gpp (propane-propane) are shown in the insets. F* ) Fσee3, z* ) z/σee, and r* ) r/σee.

difference between the potential energy minima of the two fluid species is greatest, and the situation is similar to the D ) 12.25 Å case for the cyclohexane-OMCTS mixture. Since the carbon nanotubes can be constructed at discrete diameters, we do not encounter potential energy profiles which lead to an increase in the mole fraction of ethane at higher temperatures, as was observed in the cyclohexane-OMCTS situation at D ) 10.894 Å. At larger nanotube diameters a complete transition to ethane is observed at 100 K, although at D ) 12.25 Å a nearly complete switch to ethane is seen to occur at a higher temperature of 200 K. The density distribution and APC for D ) 12.25 Å at 100 and 400 K are shown in parts c and d of Figure 11. The APC indicates that propane is liquid-like at high temperatures; however, at 300 K it is more solid-like (not shown). At D ) 14.964 Å (not shown) we find that ethane is able to form two layers at 100 K and the radial density of ethane is similar to that obtained for the ethane-methane mixture. At D ) 19.036 Å (Figure 11e,f) although the pore is being enriched with ethane, propane is present in the nanotube at 100 K and a transition in selectivity which occurs at smaller pores is not observed. This is a situation where steric effects do not inhibit the adsorption of propane when ethane is present in the nanotube at lower temperatures and both species contribute to the low energy state as the temperature is decreased. Density distributions for D ) 19.036 Å shown in parts e and f of Figure 11 illustrate that propane is able to form two inner layers at lower temperatures. The APC (not shown) for this pore

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the transition in selectivity at low temperatures is accompanied by a decrease in the overall potential energy of the system. Simulations of methane-propane25 from a bulk mixture consisting of trace amounts of propane into a single-walled carbon nanotube (D ) 10.2Å) indicate that the selectivity toward propane increases dramatically as the temperature is lowered. This situation favoring propane is closest to the D ) 9.538 Å situation studied here where a preference for propane is observed at all temperatures (Figures 10a and 12a). 4. Summary and Conclusions

Figure 12. Mole fraction of propane, xp, and averaged densities of propane Fjp* vs nanotube diameter D from a bulk methanepropane mixture. Fjp* ) Fjpσmm3.

indicates that ethane is liquid-like at 100 K, and the APC for propane is more structured though not like that of a solid. At D ) 12.25 Å the transition to the smaller molecule ethane occurs at a higher temperature when compared with the transition at other diameters. A similar situation occurs at D ) 14.964 Å for the cyclohexane-OMCTS mixture. In both these situations we find that the potential minima for the larger molecule has just shifted from the center toward the wall. Figure 2a illustrates this case for the cyclohexane-OMCTS mixture. As a consequence the potential well is shallow and the fluid molecule is able to adsorb in a wider region of the nanotube. It is in these situations that the transition is seen to occur at a somewhat higher temperature. At larger pores where the fluid-wall potential well for the larger species is better defined, its elimination occurs at lower temperatures. We have not carried out simulations in order to determine the temperatures at which the elimination of propane takes place; however in all cases where a transition occurs, the total potential energy of the system is lowered due to the larger number of ethane molecules in the nanotube. This behaviour is similar to that observed in the cyclohexane-OMCTS system. The values of U* for D ) 9.538, 12.25, 14.964, and 19.036 Å at 100 K are -432.4, -811.23, -1478.8, and -1755.7 and at 400 K are -392.1, -282.56, -371.93, and -377.94, respectively. Unlike the cyclohexane-OMCTS and ethane-propane mixtures, in the methane-propane system, propane is energetically favored due both to its larger diameter and interaction strength. As a consequence the nanotubes are selective toward propane over temperatures ranging from 200 to 400 K (Figure 12). A transition in selectivity toward the smaller methane molecule occurs only at 100 K. Since ethane has a greater fluid-wall interaction energy than methane, ethane is able to access the nanotube at higher temperatures from the ethane-propane system than methane does from the methane-propane system. Hence we do not observe an increase in the methane mole fraction at 200 K as was observed for ethane in the ethanepropane system (Figure 10a). As in the other mixtures,

The adsorption from a binary gas mixture into a singlewalled carbon nanotube was investigated using GCMC simulations. We observe that a complete transition in selectivity can be acheived by simply lowering the temperature of a fixed composition bulk gas. In all cases the composition that makes up the low temperature adsorbed state is found to be the one with the lowest internal energy. For mixtures whose species have different molecular diameters we observe that at high temperatures the energetically favored species is adsorbed and at low temperatures and intermediate diameters the smaller species is able to eliminate the larger species from the nanotube. Since the smaller species always has its potential well located toward the wall, it has a larger surface area for adsorption. Hence when the smaller species accesses the nanotube at lower temperatures, it is able to do so in larger numbers and the transition in selectivity is always accompanied by a lowering of the total potential energy of the system. In nanotubes where the difference between the fluid-wall potential energies for the two species is large, the smaller species is unable to form the lower potential energy state and the larger species is adsorbed at both low and high temperatures. This situation occurs at D ) 12.25 Å for the cyclohexaneOMCTS mixture and at D ) 9.538 Å for the ethanepropane and methane-propane mixtures. At lower temperatures and larger nanotubes, when the bigger species accesses the pore in the presence of the smaller species, both species contribute to the low energy state, and although the larger species is not totally eliminated, an enrichment toward the smaller species is observed. An evaluation of layer packing fractions indicates that packing is subsidiary. For the methane-ethane system where the ratio of molecular diameters is nearly unity, both species adsorb at nearly identical locations from the nanotube wall. The adsorption area available to both species is similar, and ethane the energetically favored species is preferred at all temperatures. Methane is able to access the nanotube in small amounts at higher temperatures. Axial pair correlations which yield information about the structure of the adsorbed fluid in the nanotube indicate that the larger species can be either solid-like or liquidlike at higher temperatures. During the transition in selectivity toward the smaller species, the APC of the smaller species reveals a liquid-like state which gets more solid-like as the temperature is lowered. Although the APC is not sufficient to precisely determine the state of the adsorbed fluid, except in limiting situations, our studies suggest the presence of liquid-solid transitions in these pseudo-one-dimensional systems. Solid to liquid

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transitions of methane in slit pores have been the subject of a recent GCMC study.33 Since the energetic explanations for the selectivity patterns at lower temperatures have been based on spherical LJ approximations of the molecules, it is questionable whether such an effect will be observed with molecular fluid models. Nevertheless we feel that these transitions are most likely to be observed with smaller molecules where orientational effects are less important. The other point to note is the dependence of these transitions on the state point of the bulk binary fluid, particularly at low temperatures. Although we have not investigated this aspect in detail, the temperature at which the transitions in pore composition occurs is likely to be related to the proximity of the bulk state point to the two-phase coexistence region. Lastly, an interesting issue (33) Miyahara, M.; Gubbins, K. E. J. Chem. Phys. 1997, 106, 2865.

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is the influence of pore geometry on the transition. Cylindrical pore structures, like the carbon nanotubes and some zeolites which offer different adsorption areas for species with differing molecular sizes appear to be candidates in which these transitions are most likely to be observed. We are currently investigating the influence of more realistic molecular models and the importance of pore geometry on these transitions in selectivity. Acknowledgment. Computing resources were provided by the Supercomputer Education Research Center and the work was supported by grants from the Jawaharlal Nehru Center for Advanced Scientific Research and the Department of Science and Technology. The author is grateful to S. Yashonath and T. A. Abinandanan for several informative discussions on the observations reported here. LA970499J