Piotr Kowalczyk,*, Piotr A. Gauden, and Artur P. Terzyk

Transcription

1 J. Phys. Chem. B 2008, 112, Cryogenic Separation of Hydrogen Isotopes in Single-Walled Carbon and Boron-Nitride Nanotubes: Insight into the Mechanism of Equilibrium Quantum Sieving in Quasi-One-Dimensional Pores Piotr Kowalczyk,*, Piotr A. Gauden, and Artur P. Terzyk Applied Physics, RMIT UniVersity, GPO Box 2476V, Victoria 3001, Australia, and Department of Chemistry, Physicochemistry of Carbon Materials Research Group, N. Copernicus UniVersity, Gagarin St. 7, Torun, Poland ReceiVed: January 24, 2008; ReVised Manuscript ReceiVed: April 25, 2008 Quasi-one-dimensional cylindrical pores of single-walled boron nitride and carbon nanotubes efficiently differentiate adsorbed hydrogen isotopes at 33 K. Extensive path integral Monte Carlo simulations revealed that the mechanisms of quantum sieving for both types of nanotubes are quantitatively similar; however, the stronger and heterogeneous external solid-fluid potential generated from single-walled boron nitride nanotubes enhanced the selectivity of deuterium over hydrogen both at zero coverage and at finite pressures. We showed that this enhancement of the D 2 /H 2 equilibrium selectivity results from larger localization of hydrogen isotopes in the interior space of single-walled boron nitride nanotubes in comparison to that of equivalent singlewalled carbon nanotubes. The operating pressures for efficient quantum sieving of hydrogen isotopes are strongly depending on both the type as well as the size of the nanotube. For all investigated nanotubes, we predicted the occurrence of the minima of the D 2 /H 2 equilibrium selectivity at finite pressure. Moreover, we showed that those well-defined minima are gradually shifted upon increasing of the nanotube pore diameter. We related the nonmonotonic shape of the D 2 /H 2 equilibrium selectivity at finite pressures to the variation of the difference between the average kinetic energy computed from single-component adsorption isotherms of H 2 and D 2. In the interior space of both kinds of nanotubes hydrogen isotopes formed solid-like structures (plastic crystals) at 33 K and 10 Pa with densities above the compressed bulk para-hydrogen at 30 K and 30 MPa. I. Introduction Many potential applications have been proposed for carbon (CNs) as well as boron nitride (BNs) nanotubes, including conductive and high-strength composites, energy storage and energy conversion devices, sensors, biochips, field emission displays and radiation sources, hydrogen storage media, nanometer-sized semiconductor devices, encapsulation of drugs, and others These tubular materials are characterized by extraordinarily fine structure on a nanometer scale. Moreover, due to cylindrical pore geometry, the solid-fluid potential inside of these nanoscale objects exerts a huge internal stress on the adsorbed guest molecules. Even at room temperature, this strong external field is responsible for the ordering of guest molecules into a quasi-one-dimensional structure such as plastic crystals, peapods, nanorods, nanowires, and so forth The properties of highly compressed guest molecules in the quasi-onedimensional internal space of nanotubes can be different compared to their bulk counterparts. For example, strong confinement changes various physicochemical properties of guest molecules including freezing/meting temperature, chemical equilibrium, solvation of organic molecules and ions, quantum delocalization, molecular transport, and so forth Therefore, it is not surprising that both CNs as well as BNs have been under extensive experimental and theoretical investigations. * To whom correspondence should be addressed. Tel: +61 (03) Fax: +61 (03) E72231@ems.rmit.edu.au. RMIT University. N. Copernicus University. The separation of hydrogen isotopes is a difficult and energy intensive process. Since the chemistry depends on the interactions of protons with electrons, the chemical properties of these isotopes are almost the same. The difference in mass of isotopes gives rise to differences in thermophysical properties such as vapor pressures or molecular diffusion rates. These differences have been used for the separation of hydrogen isotopes using thermal diffusion, cryogenic distillation, diffusion through alloys, formation of hybrids, gas chromatography, electrolysis, and others. 26 However, most of these techniques have low selectivity for separating hydrogen isotopes. More recently, Matsuyama et al. 27,28 developed and applied a new separation technique for the separation of hydrogen isotopes, called self-developing gas chromatography. The advantages of this method are operation near room temperature and high efficiency of separation. However, the chromatographic column is filled with Pd-Pt/ Pd-Cu alloys, inevitably increasing the price of the separation if one considers an industrial scale. The separation of hydrogen isotopes at cryogenic temperatures via the quantum sieving mechanism is one of the promising applications of those tubular nanoscale vessels The question arises, why are these nanomaterials promising quantum sieves? The high density of surface atoms, as well as cylindrical pore geometry of both CNs and BNs, creates high internal stress on the order of GPa, inevitably impacting the ordering/packing of adsorbed quantum particles. Moreover, very large potential gradients existing within small nanotubes impact on the quantum delocalization of light particles In the radial direction, the movement of light particles at cryogenic temperatures is /jp800735k CCC: $ American Chemical Society Published on Web 06/21/2008

2 8276 J. Phys. Chem. B, Vol. 112, No. 28, 2008 Kowalczyk et al. Figure 1. Solid-fluid potential for the selected single-walled carbon nanotubes: (7,7)-equivalent pore diameter 9.43 Å; (9,5)-equivalent pore diameter 9.55 Å; (10,10)-equivalent pore diameter Å; (11,11)- equivalent pore diameter Å. Solid lines represent the calculations for infinitely long structureless carbon nanotubes, and open circles correspond to infinitely long atomistic carbon nanotubes. For comparison, we attached the solid-fluid potential computed for the (6,6)-CNequivalent pore diameter 8.09 Å (see Figure 9 in ref 31). Figure 3. Left panel: Single-component isotherms in (7,7), circles, and (11,11), triangles, infinitely long single-walled carbon nanotubes at 33 K computed from PI-GCMC. The gray circles denote D 2, the open circles denote H 2, and the open squares denote the experimental equation of state for para-h 2 at 30 K. Right panel: Average gyration radius of Feynman s cyclic polymers quantizing D 2 (gray circles) and H 2 (open circles) adsorbed in the (7,7) single-walled carbon nanotubes at 33 K. Red and blue circles represent quantum delocalization of hydrogen and deuterium in the bulk fluid at 33 K, respectively. Figure 2. The average kinetic energy (computed for 33 K) of the H 2 (open circles) and D 2 (gray circles) molecules placed in the infinitely long atomistic (symbols with error bars) and structureless single-walled carbon nanotubes. The right axis (black circles) presents the difference between the the average kinetic energies of H 2 and D 2. All closed stars correspond to single-walled CNs displayed in Figure 1 (blue (7,7); red (9,5); green (10,10); yellow (11,11)). The classical kinetic energy, E kin ) (3/2)k BT, at 33 K is K. quantized, whereas in the longitudinal direction, the particles can move freely. The combined effect of quantum delocalization and strong confinement differentiates hydrogen isotopes, and this can be used for their efficient separation. In other words, we can adjust the quantized energy levels of confined quantum particles by manipulation of the nanotube type, the external operating conditions, as well as the pore size. Indeed, Tanaka et al. 34 showed that single-walled carbon nanohorns differentiate D 2 from H 2 under the adsorption equilibrium at 77 K. This phenomenon can be explained by different quantum fluctuations of both components in the adsorbed phase. In a series of papers, Wang et al. 29 and Challa et al. 30,31 investigated the application of an idealized model of carbon nanotube bundles for the quantum sieving of hydrogen isotopes. According to these studies, the interstitial channels formed by adjacent carbon nanotubes can be used for the efficient separation of hydrogen isotopes under thermodynamic equilibrium at 20 K. Moreover, in contrast to experimental reports showing hydrogen isotopes quantum sieving on zeolites, Figure 4. Left panel: Pore density variation of the average kinetic energy of H 2 (open circles and red crosses) and D 2 (gray circles and blue crosses) adsorbed in (7,7), infinitely long, atomistic single-walled carbon nanotube at 33 K. Circles denote simulations in the canonical ensemble, and crosses denote simulations in the grand canonical ensemble by the path integral method. The solid line presents the difference between the average kinetic energy of single-component isotherms, and the dashed line corresponds to the classical kinetic energy, E kin ) (3/2)k BT, which at 33 K is K. Right panel: Pressure variation of the difference between the average kinetic energy of H 2 and D 2 adsorbed in a (7,7), infinitely long, atomistic single-walled carbon nanotube at 33 K. charcoals, carbon molecular sieves, silicas, or single-walled carbon nanohorns, the authors predicted that the equilibrium selectivity of T 2 over H 2 or D 2 over H 2 increases during the filling of the pore space BNs have many of the superior properties of CNs, such as a high Young s modulus and thermal conductivity, but unlike CNs, they exhibit high resistance to oxidation and wide band gap regardless of chirality. 3 5 The advantage of BNs in relation to the adsorption/separation of quantum fluids is their strong nonunifrom solid-fluid potential field. At cryogenic temperatures, the heterogeneous landscape of the external potential field generated from boron and nitrogen surface atoms of BNs inevitably affects delocalization (i.e., high-temperature density matrix) of adsorbed light particles. The quantum particle fluctuating close to the heterogonous BNs surface is exposed to a stronger solid-fluid potential in comparison to that of the

3 Hydrogen Isotopes in Single-Walled CNs and BNs J. Phys. Chem. B, Vol. 112, No. 28, Figure 5. Equilibrium snapshots of D 2 (left panel) and H 2 (right panel) adsorbed in a (7,7), infinitely long, atomistic single-walled carbon nanotube at 0.87 MPa 33 K (i.e., nanotube saturation). Figure 6. Equilibrium selectivity (computed at zero coverage) of D 2 over H 2 as a function of the slit-shaped graphite (stars) and boron nitride (circles) pore width. uniform CNs surface. Note that, the lower the temperature, the higher the delocalization of quantum particles. Moreover, these particles are localized closer to the surface. Therefore, it is clear that at cryogenic temperatures, the atomistic details of the solid surface strongly impact the small hydrogen isotope molecules fluctuating in the solid-fluid potential minimum well depths. Accounting for the effect of the strength as well as heterogeneous nature of the solid-fluid potential generated from the nanotube pore walls on the equilibrium selectivity of D 2 over H 2 is one of the main aims of our work. Besides the size and type of the nanotube, the operating conditions are crucial for delocalization of light quantum particles. As we demonstrate, these parameters govern the efficiency of the quantum sieving. In the current study, we simulate equimolar mixture adsorption of H 2 and D 2 in selected single-walled CNs and BNs at 33 K using first-principle path integral grand canonical molecular simulations (PI-GCMC). Additionally, we compute the exact selectivity and high-temperature density matrix (i.e., wave function times Boltzmann factor) at zero coverage to explain an unexpected delocalization of hydrogen isotopes in selected cylindrical single-walled CN and BN pores. In contrast to the Figure 7. Equilibrium selectivity (computed at zero coverage) of D 2 over H 2 as a function of the single-walled carbon nanotube (stars) and boron nitride nanotube (circles) pore diameter. previous works of hydrogen isotope separation by single-walled CNs at finite temperatures and pressures, 29 31,44 we relate the mechanism of this equilibrium process to the variation of the difference between the average kinetic energy of H 2 and D 2 computed from single-component adsorption isotherms. Both calculations at zero as well as finite pressures reveal that singlewalled BNs localize hydrogen isotopes more strongly in comparison to equivalent single-walled CNs. Moreover, we show that in the interior space of both kinds of nanotubes, hydrogen isotopes form solid-like structures (plastic crystals) at 33 K and 10 Pa with densities above the compressed bulk para-hydrogen at 30 K and 30 MPa. 45 II. Simulation Details II.1. Potential Models. Following the Feynman s path integral formalism, we used the quantum classical isomorphism in which each particle becomes equivalent to a chain or necklace of P classical beads r (1) i, r (2) i,..., r (P) i that accounts for the quantum delocalization of the particle When a system contains more than one type of quantal particle, decisions

4 8278 J. Phys. Chem. B, Vol. 112, No. 28, 2008 Kowalczyk et al. concerning the appropriative value of P should be made separately for each particle. 49 For H 2 at 33 K and moderate pressures, we used P H2 ) 16 since this number of beads in a cyclic polymer reproduces experimental equation of state (see Figure 1S in the Supporting Information). Since the mass of D 2 is twice the mass of H 2, the number of beads in cyclic polymers quantizing the deuterium molecule is P D2 ) P H2 /2 ) 8. In the considered range of experimental equations of state, both the density as well as the mean kinetic energy of D 2 computed for 8 and 16 beads are within the error of simulation (see Figure 2S in the Supporting Information). In nanopores, ring polymers experience both an external potential, which is the sum of the fluid-fluid and solid-fluid interactions, and an internal potential, which comes from the intermolecular bonding interactions. The effective potential can be expressed according to the so-called primitive action W ) mp 2β 2 p 2 N P R)1 i)1 (ri - r i (R+1) ) P i<j P R)1 1 V ff (r ij ) + P P i<j R)1 V sf (r ij ) (1) where N is the number of atoms, β is the inverse of the temperature, and p is Planck s constant divided by 2π. Owing to the cyclic condition of the polymer chains, if R)P, then R + 1 ) For the case of the binary mixture simulation, the number of beads and the mass in eq 1 depends on the type of the component (here, H 2 or D 2 ) selected for perturbation. Similarly to Rabani et al. 50 and Wang et al., 29 we modeled the external interactions between the beads of ring polymers via the effective Silvera-Goldman (SG) potential 51 V ff (r ij ) ) exp(r-δr 2 ij - γr ij ) - ( C 6 r ij + C 8 + C ) f C(r ij ) + C 9 f 9 C(r ij ) (2) r ij r ij f C (r ij ) ) exp[-(r C /r ij - 1) 2 ]Θ(r C - r ij ) + Θ(r ij - r C ) Here, rij is the distance between the beads labeled R of two interacting polymer chains and Θ denotes the Heaviside function. The parameters of the potential are given elsewhere. 51 In nanospaces, the density of adsorbed/compressed hydrogen isotopes approaches the density of highly compressed liquid. The SG potential is a reliable representation of highly compressed hydrogen isotopes since this potential was adjusted to the solid properties of para-hydrogen (ortho-deuterium). 51 In the current work, we modeled CNs and BNs as infinitely long cylindrical tubes with detailed atomistic arrangement of the surface atoms. We adopted a Lennard-Jones potential for computation of the solid-fluid potential between the beads labeled R and surface atom 6] 52 (4) V sf (r i ) ) 4ε sf[( σ 12 sf - r i ) ( σ sf r i ) Here, r i is the distance between the selected bead of the polymer chain and the surface atom. The Lorentz-Berthelot mixing rule was used for the calculation of solid-fluid potential parameters. 53,54 For carbon atoms, we adopted the following values of parameters: σ C-C ) 0.34 nm and ε C-C /k b ) 28 K. 55 The Lennard-Jones parameters for the boron and nitrogen atoms were taken from ref 56 (σ B-B ) nm, ε B-B ) r ij (3) kj/mol, σ N-N ) nm, and ε N-N ) kj/mol). As previously, for molecular hydrogen and deuterium, we used σ ff ) nm and ε ff /k b ) 36.7 K (k b denotes the Boltzmann constant). 18,52 For selected carbon nanotubes, we performed additional calculations adopting a structureless model of the infinitely long carbon tube (for details see ref 8). II.2. Path Integral Monte Carlo Simulation of H 2 and D 2 in a Grand Canonical Ensemble. Single-component isotherms of H 2 and D 2 adsorbed in the single-walled CNs and BNs were simulated in a grand canonical ensemble (i.e., a fixed system volume V, temperature T, and the chemical potential µ p ). 53,54 We adopted three types of moves in our path integral GCMC simulations, particle displacement, creation, and deletion. The displacement step was realized by two independent algorithms. The first displacement perturbation is a classical bead-per-bead sampling with an additional movement of the whole selected cyclic path according to the displacement of the centroid. 57,58 The probability of accepting a trial displacement move is given by the standard Metropolis sampling scheme acc(s f s ) ) min{1, exp(-β U)} (5) where U ) U(r N ) - U(r N ) is the change in the total energy of the simulation system and β ) (k b T) -1. The displacement step size has been adjusted to give an acceptance ratio of 50%. It is well-known that this primitive sampling is very inefficient. 59 To overcome this sampling problem, we propose to sample the configuration of H 2 and D 2 adsorbed in cylindrical nanopores according to the bisection method. The details of the bisection sampling are given elsewhere The trial insertion/deletion of the particle was performed according to the Wang et al. 29 description. To maintain microscopic reversibility, we set up an equal probability for trial displacement, creation, and destruction of a selected particle. 53,54 Both the pressure and chemical potential of deuterium/hydrogen calculated in the canonical ensemble were used as an input in PI-GCMC simulations. The bead-bead intermolecular cutoff distance was 5σ ff. We did not add the long-range corrections because the fluid in pores is inhomogeneous and the potential energy rapidly approaches zero beyond the 5σ ff limit. Simulation runs were performed at gradually increasing chemical potentials, and the resulting configurations from one condition were used to initiate the subsequent calculations at higher chemical potentials. The grand canonical ensemble simulations utilized configurations, of which the first were discarded to guarantee equilibration. The stability of the results was confirmed by additional longer runs of configurations, with the equilibrium averages for hydrogen in pores fully reproducible. For each point on the simulated isotherm, we performed three independent PI-GCMC runs to average the final equilibrium averages. We used the same simulation set up for the case of H 2 /D 2 equimolar mixture adsorption in the single-walled BNs and CNs. The component of the binary mixture for displacement, insertion, and deletion perturbation was selected with equal probability. II.3. Equilibrium Selectivity of Deuterium over Hydrogen at Zero and Finite Pressures. In the zero-pressure limit, the equilibrium selectivity of D 2 /H 2 can be directly calculated from the path integral Monte Carlo simulation as a ratio between the partition function of D 2 over H R drrexp[-ud2 (r)/k b T] (6) dγ D2 dω D2 0 S 0 (p ) 0) ) R dγ H2 dω H2 drrexp[-uh2 0 (r)/k b T] where U denotes the configurational energy, r is the radial position of the ring center of mass, Γ and ω are the path s

5 Hydrogen Isotopes in Single-Walled CNs and BNs J. Phys. Chem. B, Vol. 112, No. 28, Figure 8. High-temperature density matrix of H 2 (left panel), D 2 (middle panels), and T 2 (rights panels) adsorbed in the (8,0) single-walled BN nanotube (top panel) and the (8,0) single-walled CN nanotube (bottom panel) at 33 K. internal conformation and orientation in the pore, respectively, and R is the tube radius. In the current paper, we computed zero-pressure selectivity from eq 6 using the method proposed by Wang et al. 29 At finite pressure, equilibrium selectivity of D 2 over H 2 represents the ratio of the density of these two species in a porous material to the ratio of the density in the bulk phase 61,62 F p p D2 /F H2 S ) F b b D2 /F H2 where F i p and F i b are the average densities of species i in the tube and bulk phase, respectively. Values greater then unity imply that D 2 is preferentially adsorbed compared to H 2 ;in contrast, if the selectivity is smaller than unity, H 2 is preferentially adsorbed. We computed the equilibrium selectivity of D 2 over H 2 at finite pressures from PI-GCMC and eq 7. II.4. High-Temperature Density Matrix. Path integral simulation can be directly used for measurement of the hightemperature density matrix, that is, the wave function times the Boltzmann factor. 46,47 Let us consider one quantum particle immersed in a one-dimensional external potential, V(x). The effective potential can be expressed according to the so-called primitive action P W(x 1, x 2,..., x P ) ) mp (x - x (R+1) ) V(x ) 2β 2 p 2 P R)1 R)1 owing to the cyclic condition of the polymer chains, where x P+1 ) x 1. During the stochastic Monte Carlo process, we generate the sequence of equilibrium states ζ l, l ) 1, 2,... The efficient method used for evaluation of the high-temperature density matrix from {ζ l } is as follows. We divide the real space x into a large number of bins [m,(m+1) ], m ) 0, 1, 2,... After initial equilibration of the system, we monitored sequences of {ζ l } generated from the stochastic Monte Carlo process. The jth component of ζ l falls into some bin [m,(m+1) ]. A point x j ζ l, j ) 1, 2,..., P is stored for that bin. As the stochastic P (7) (8) Monte Carlo process continues, each bin accumulates points (i.e., path coordinates) with each successful entry or persistence in that bin. The high-temperature density matrix is then proportional to the total number of points in the bin. The extension of eq 8 for the two- and three-dimensional potential, V(r), is obvious. III. Results and Discussion III.1. Single-Component Isotherm of D 2 and H 2. Before discussing the various aspects related to the H 2 /D 2 equimolar mixture adsorption in single-walled CNs and BNs, let us focus on the single-component isotherms of these isotopes. We limited our discussion to the single-walled CNs due to the similar behavior of the single-walled BNs under the considered thermodynamics equilibrium. The solid-fluid potential inside of the investigated, infinitely long, structureless, and atomistic single-walled CNs is displayed in Figure 1. Note that for current study, we selected wider nanotubes that are now produced in large quantities. 1,2 The potential minimum in these carbon nanotubes is away from the center of the tube. Moreover, the height of the potential barrier at the center of the carbon nanotube depends on its size. The smallest energy barrier measured as the energy difference between the minimum and local maximum at the center of the (7,7) carbon nanotube is around 215 K (i.e., 6.5 k b T, T ) 33 K), whereas for the (11,11) carbon nanotube, this barrier reaches 515 K (i.e., 15.6 k b T, T ) 33 K). For comparison, the energy barrier in the (6,6) carbon nanotube is at around 30 K (i.e., 0.91 k b T, T ) 33 K), as presented in Figure 1 (see Figure 9 in ref 31). The average kinetic energy of a single H 2 and D 2 molecule placed in the interior space of the carbon nanotube strongly depends on its pore size, as shown in Figure 2. Interestingly, regardless of the size of the carbon nanotube, the average kinetic energy of both hydrogen isotopes exceeds its classical value at 33 K (i.e., K). The nonmonotonic variation of the difference between the average kinetic energy of H 2 and D 2 is characterized by the deep minimum near the carbon nanotube diameter at around 8 Å (the equivalent chiral vector of the nanotube is (6,6)). As we show later, this well-defined minimum is directly related to the

6 8280 J. Phys. Chem. B, Vol. 112, No. 28, 2008 Kowalczyk et al. Figure 9. High-temperature density matrix of H 2 (left panel), D 2 (middle panels), and T 2 (rights panels) adsorbed in a (5,5) single-walled BN nanotube (top panel) and a (5,5) single-walled CN nanotube (bottom panel) at 33 K. Figure 10. Left panel: The D 2/H 2 mixture isotherm in the (7,7), infinitely long, atomistic single-walled carbon nanotube at 33 K for an equimolar composition of a bulk D 2/H 2 mixture. Dark gray circles denote the total adsorption, white circles denote the densities of adsorbed H 2, and light gray circles correspond to densities of adsorbed D 2. Right panel: Pure-component isotherms of H 2 (white circles) and D 2 (light gray circles) in the same nanotube. Dashed lines present the comparison between the H 2/D 2 mixture density in the adsorbed phase (left panel) and the density of pure H 2 and D 2 (right panel) for selected pressures. reduction of the D 2 /H 2 equilibrium selectivity in this range of nanotube diameters. In agreement with experimental measurements, the pure-component isotherms of H 2 and D 2 indicated preferential adsorption of D 2, which can be explained by its lower average kinetic energy The comparison of both isotherms with bulk para-h 2 at 30 K showed that nanotubes drastically shift the pressure-density dependence to lower values of pressure. Moreover, the saturation density of H 2 as well as that of D 2 in the (7,7) single-walled CN is greater than the density of compressed bulk para-h 2 at 30 K and 30 MPa. 45 These properties of adsorbed phases result from the strong confinement and high compression of hydrogen isotopes in cylindrical nanotubes, as shown in Figure 5. The variation of the average gyration radius of polymer chains during adsorption of both species in carbon nanotubes explains the enhancement of their average kinetic energies, as displayed in Figure 3. At low adsorbed densities, the quantum polymers shrink in comparison to equivalent polymer chains in the bulk fluid. Moreover, as adsorption proceeds further, increasing densities of adsorbed phases reduce the size of the polymer chains as well, as shown in Figure 3. That is why the difference between the average kinetic energy of both hydrogen isotopes is increasing with nanotube loading, as presented in Figure 4. Note that for the (7,7) single-walled CN at saturation, this difference is high, whereas the packing of the adsorbed H 2 and D 2 is very similar, as shown in Figures 4 and 5. Finally, for a selected (7,7) single-walled CN, we computed the difference between the average kinetic energy of adsorbed H 2 and D 2 as a function of the external pressure. Note that for the fixed external pressure, the density of adsorbed D 2 is always higher (or equal at saturation) than the density of adsorbed H 2 ; see the different curvatures of the single-component adsorption isotherms displayed in Figures 3 and 10. Interestingly, this dependence presented in Figure 4 is nonmonotonic with a small minimum at around Pa and is rapidly increasing above this value. As we show later, this nonmonotonic variation of the difference between the average kinetic energy of H 2 and D 2 at finite pressures is closely related to the variation of the D 2 /H 2 equilibrium selectivity upon the adsorption of their equimolar mixture. At the end of this paragraph, we want to underline that the variation of the average kinetic energy of both hydrogen isotopes upon the filling of the nanotubes (or other nanopores) is crucial for an understanding of their quantum sieving behavior under the thermodynamic equilibrium. During the adsorption, the differences between the quantized energy levels of adsorbed hydrogen isotopes are not constant. They can be lower or higher than these corresponding to zero coverage. Hence, one can expect the variation of the D 2 /H 2 equilibrium selectivity at finite pressures. III.2. Equilibrium Selectivity of D 2 over H 2 at Zero Coverage. We start by discussing the equilibrium selectivity of D 2 over H 2 in slit-shaped graphite and boron nitride (BN) pores computed at zero coverage. The key point is that regardless of the operating temperature, the BN surface enhanced

7 Hydrogen Isotopes in Single-Walled CNs and BNs J. Phys. Chem. B, Vol. 112, No. 28, Figure 11. Equilibrium selectivity of D 2 over H 2 at 33 K for selected single-walled carbon nanotubes (left panel) and boron nitride nanotubes computed from PI-GCMC. Solid lines denote calculations at zero coverage. Figure 12. Variation of the minimum D 2/H 2 equilibrium selectivity for selected single-walled carbon and boron nitride nanotubes computed from PI-GCMC at 33 K (see Figure 11). the equilibrium selectivity of D 2 over H 2 in comparison to the graphite surface. As one can expect, the lower the operating temperature, the higher the difference between the D 2 /H 2 equilibrium selectivity computed for the slit-shaped graphite and BN pores of the same size, as presented in Figure 6. The fluctuating quantum paths are exposed to a stronger solid-fluid potential as well as a heterogeneous surface landscape near the BN atomistic pore wall than near the graphitic one. Moreover, regardless of the type of the pore, we observe the deep minima of the D 2 /H 2 equilibrium selectivity near the pore width of 7.6 Å, as shown in Figure 6. In these slit-shaped pores, the solid-fluid potential minima generated from the opposed carbon as well as the BN pore walls merge inside of the pore (i.e., the double-well solid-fluid potential is transformed into the single one). This simply means that adsorbed quantum paths have a space for their delocalization, and they are not affected by confinement. Note that this quantum phenomenon does not depend on the pore wall composition, that is, the slit-shaped pores of pore width around 7.6 Å can be treated as virtual pores. Further extension of the pore size increases equilibrium selectivity due to creation of the potential barrier in the middle of the pore, as is shown in Figure 6. Clearly, the value of this potential barrier depends on the type of the porous material. Due to the higher potential barrier, wider BN slit-shaped pores enhanced the D 2 /H 2 equilibrium separation factor in comparison to the equivalent slit-shaped graphite pores, as displayed in Figure 6. For the case of single-walled BN as well as CN cylindrical nanotubes, the variation of the D 2 /H 2 equilibrium separation factor at zero coverage is very similar to that observed for the slit-shaped pore geometry, as shown in Figure 7. As previously, regardless of the type of the pore, we observed the deep minima of the D 2 /H 2 equilibrium selectivity near the pore diameter of 8 Å (note that this result is consistent with previous simulation studies; see Figure 1 in ref 29 and Figure 2 in ref 30). Moreover, this deep minimum of the D 2 /H 2 equilibrium selectivity is a consequence of the small difference between the average kinetic energy of H 2 and D 2 near the nanotube pore diameter of 8 Å (the equivalent chiral vector of the nanotube is (6,6)), as shown in Figure 2. Summing up, for both slit-like and cylindrical CN/ BN pore geometries, the steepness of the solid-fluid potential and not the size of the pore is a key parameter governing the efficiency of hydrogen isotope separation. This steepness of the solid-fluid potential is responsible for high localization and the enhancement of the average kinetic energy of hydrogen isotopes. Finally, the calculations at zero coverage do not show any advantage of nanotubes over ordinary slit-shaped carbon/bn materials in relation to quantum sieving of hydrogen isotopes at cryogenic temperatures. III.3. High-Temperature Density Matrix. The impact of the cylindrical pore type as well as its size on the localization of hydrogen isotopes at 33 K is displayed in Figures 8 and 9. At first, we found that for both single-walled BN and CN nanotubes the lighter hydrogen isotopes are more delocalized than the heavier ones. This effect is obvious and can be attributed to the higher quantum entropy of lighter hydrogen isotopes (i.e., higher zero-point motion of hydrogen molecules than that of deuterium or tritium molecules). However, the key point is that single-walled BNs localized hydrogen isotopes more strongly than equivalent single-walled CNs. Obviously, these differences strongly depend on the pore size, and the largest are observed for the smallest cylindrical pores. It is clear because in these cylindrical pores, the adsorbed quantum molecules are exposed to an enormous solid-fluid interaction potential characterized by the steep single minima. However, it is worth

8 8282 J. Phys. Chem. B, Vol. 112, No. 28, 2008 Kowalczyk et al. Figure 13. Equilibrium configuration of a hydrogen (purple spheres) and deuterium (light blue spheres) mixture in a (7,7) (top panel) as well as (12,12) (bottom panel) single-walled carbon nanotube (left panel) and single-walled boron nitride nanotube (right panel) computed at 33 K and a total pressure of equimolar mixture of 0.1 Pa. Figure 14. Equilibrium configuration of a hydrogen (purple spheres) and deuterium (light blue spheres) mixture in the (9,5) (top panel) as well as (10,10) (bottom panel) single-walled carbon nanotube (left panel) and single-walled boron nitride nanotube (right panel) computed at 33 K and a total pressure of equimolar mixture of 0.1 Pa. pointing out that wide cylindrical pores (i.e., open graphite and BN surfaces) also efficiently differentiate hydrogen isotopes due to the presence of a high potential barrier inside of the cylindrical pore, as shown in Figure 6. On the other hand, regardless of the type of studied nanotubes, the cylindrical pores of diameters at around 8 Å behave as virtual cylindrical tubes, that is, they have no impact on the quantum delocalization of hydrogen isotopes. These observations are consistent with variation of the D 2 /H 2 equilibrium selectivity computed at zero coverage as well as the variation of the difference between the average kinetic energy of H 2 and D 2 computed from the singlecomponent adsorption isotherms, as presented in Figures 2 and 7. Finally, our results displayed in Figures 2 4 and 8 and 9 explain recent experimental reports of the higher kinetic energy of confined quantum particles in comparison to the bulk ones. 63,64 Kinetic energy in quantum mechanics is a curvature of the wave function at zero temperature. Our simulations showed that the strong and steep solid-fluid potential reduced delocalization of hydrogen isotopes (i.e., increase the curvature of the high-temperature density matrix). The smallest the pore size, the more localized the state of confined quantum particle. Therefore, the quantum delocalization at the smallest cylindrical/ slit-shaped pores can increase its kinetic energy up to high values, as presented in Figures 2 and 4. III.4. Equilibrium Selectivity of Deuterium over Hydrogen at Finite Pressures. So far, we have considered the computations at zero coverage. These computations are obviously an approximation neglecting the important contribution of the fluid-fluid interactions at finite pressures. Let us now consider the adsorption of an equimolar mixture of D 2 and H 2 in atomistic single-walled CNs and BNs computed at 33 K and finite pressures up to 1 Pa, as presented in Figures In 2002, Challa et al. 30,31 gave an entire picture of the selective adsorption of hydrogen isotopes in single-walled CN and interstices between the individual nanotubes arranged into the simplified triangular lattice structure. The authors stated that under the thermodynamic equilibrium at 20 K, 31 selectivity in the nanotubes and interstices increases with pressure until the nanotube is saturated. Our PI-GCMC simulations indicated that the mechanism of hydrogen isotope separation in single-walled CNs and BNs under the thermodynamics equilibrium can be related to the difference between the average kinetic energy of H 2 and D 2 computed from the single-component adsorption isotherms. What is more important, this difference strongly depends on the nanotube pore size as a result of different packing of quantum particles (i.e., the high-temperature density matrix depends on the structure of the adsorbed phase). That is why the D 2 /H 2 equilibrium selectivity can be a monotonic (decreasing/increasing, as has been shown by the experiments and previous simulations 29 31,41 ) or nonmonotonic function upon the competitive adsorption of mixture components. For the cylindrical pore diameters considered in the current work, we observed a constant value of the D 2 /H 2 equilibrium selectivity at low pressures. Notice, that our PI-GCMC simulations reproduced the analytical calculations at zero coverage, as shown in Figure 11. Further rise in equimolar mixture pressure causes a decrease in the D 2 /H 2 equilibrium selectivity in both singlewalled CNs and BNs up to a well-defined minimum. As one can expect, this minimum is strongly correlated with both the type as well as the size of the single-walled nanotube, as displayed in Figure 6. Next, the position of the minimum is gradually shifted upon increasing the cylindrical pore diameter. Finally, after crossing of this minimum, the D 2 /H 2 equilibrium selectivity is increasing, as presented in Figure 11. The nonmonotonic shape of the D 2 /H 2 equilibrium selectivity versus the external pressure can be correlated with the variation of the difference between the average kinetic energy of H 2 and D 2 computed from single-component adsorption isotherms, as displayed in Figure 4. At low pressures, that is, in the Henry s region, the equilibrium selectivity of D 2 over H 2 is constant,

9 Hydrogen Isotopes in Single-Walled CNs and BNs J. Phys. Chem. B, Vol. 112, No. 28, and its value is governed by the difference between the enthalpy of hydrogen and deuterium adsorption. Here, we would like to stress that constant equilibrium selectivity of D 2 over H 2 at low mixture pressures was observed in numerous experimental studies The smaller the size of the cylindrical single-walled CNs or BNs, the higher the difference between the average kinetic energy of both components, as shown in Figure 2. On the basis of this argument, the highest equilibrium selectivity of D 2 over H 2 is expected in the smallest nanotubes considered here, that is, the (7,7) (the equivalent pore diameter is 9.4 Å). However, this pore size is located in the deep minima of the D 2 /H 2 equilibrium selectivity computed at zero coverage, as shown in Figures 2 and 7. This explained why the initial D 2 /H 2 equilibrium selectivity is higher for wider single-walled CNs or BNs. A further rise in the equimolar mixture pressure results in competitive adsorption of both deuterium and hydrogen molecules, as shown in Figure 10. Note that for the fixed pressure of the D 2 /H 2 equimolar mixture, the equilibrium density of the hydrogen and deuterium mixture in the nanotube is always between the densities computed from pure-component isotherms, as shown in Figure 10. After the initial drop to the minimum, we observed further increasing in the D 2 /H 2 equilibrium selectivity. We related this behavior to the rapid increase in the difference between the average kinetic energy of H 2 and D 2 computed from single-component adsorption isotherms (see right panel in Figure 4). As we mentioned above, the experimental reports predicted decreasing selectivity of D 2 over H 2 upon filling of the zeolite, charcoals, silicas, and other porous materials Why are both single-walled CN and BN nanotubes so unusual? The key to understand this behavior is the analysis of the variation of the difference between the kinetic energy of H 2 and D 2 computed from single-component adsorption isotherms in carbonaceous pores of different shapes. Our results as well as previous ones 30,31 seem to be particularly important for practical applications of these nanomaterials as quantum filters since we expected high enrichment of the D 2 phase inside of both types of nanotubes upon increase in the mixture pressure (for comparison, see Figure 14 in ref 31). However, we bear in mind that PI-GCMC simulations assumed an adsorption process under the thermodynamics equilibrium. Since the quantum dynamics is an unsolved problem, we argue that the careful experiment is needed to estimate the equilibration time for this adsorption process. Finally, we compared the sieving properties of single-walled CNs and BNs at finite pressures and at 33 K. As for zerocoverage calculations, the entire mechanism of hydrogen isotope separation at finite pressures via the equilibrium quantum sieving mechanism for both investigated types of nanotubes are similar; see Figures 7 9 and However, due to a higher solid-fluid interaction potential as well as heterogonous composition of the pore walls, the operating pressure for single-walled BNs is shifted to smaller values. Next, regardless of the operating pressure, the D 2 /H 2 equilibrium selectivity is higher for singlewalled BNs. For the same equimolar D 2 /H 2 mixture pressure, the single-walled BNs contain more deuterium than the equivalent CNs that favors these nanomaterials for industrial applications, as displayed in Figures IV. Conclusions In this paper, we showed that quasi-one-dimensional pores of single-walled boron nitride and carbon nanotubes efficiently differentiate the adsorbed hydrogen isotopes at 33 K. We discussed the mechanism of the equilibrium quantum sieving of hydrogen isotopes at zero and finite pressures. Regardless of the type of nanotube considered in the current study, the variation of the D 2 /H 2 equilibrium selectivity upon the pore filling consists of three stages. At the first stage (i.e., in the Henry s region), the equilibrium selectivity of D 2 over H 2 is constant, and its value is governed by the difference between the enthalpy of hydrogen and deuterium adsorption. At the second stage, the D 2 /H 2 equilibrium selectivity drops to the minimum value. The position and the value of the minimal D 2 / H 2 equilibrium selectivity are strongly related to the nanotube pore diameter. Finally, after the initial drop to the minimum, we predicted further increasing of the D 2 /H 2 equilibrium selectivity. We related the nonmonotonic shape of the D 2 /H 2 equilibrium selectivity to the variation of the difference between the average kinetic energy of H 2 and D 2 computed from singlecomponent adsorption isotherms at finite pressures. Both calculations at zero coverage as well as at finite pressures revealed that BNs are more efficient for separation of hydrogen isotopes under the assumed thermodynamics equilibrium. This enhancement of the D 2 /H 2 equilibrium selectivity results from the higher localization of hydrogen isotopes in the interior space of single-walled BNs in comparison to that in equivalent singlewalled CNs. The operating pressures for efficient quantum sieving of hydrogen isotopes strongly depend on both the type as well as size of the nanotube. Our simulation showed that in the interior space of both kinds of nanotubes, hydrogen isotopes form solid-like structures (plastic crystals) at 33 K and 10 Pa with densities above the compressed bulk para-hydrogen at 30 K and 30 MPa. Acknowledgment. P.K. gratefully acknowledges Thanh X. Nguyen (Division of Chemical Engineering, The University of Queensland) and reviewers for fruitful discussions and suggestions on the content of the current paper. P.K. would like to thank the discussions and hospitality of E. Rabani (School of Chemistry, Tel Aviv University, Israel). P. A. Gauden and A. P. Terzyk acknowledge the use of the computer cluster at Poznan Supercomputing and Networking Centre as well as the Information and Communication Technology Centre of the Nicolaus Copernicus University (Torun, Poland). Supporting Information Available: The movies with the equilibrium composition of the D 2 /H 2 adsorbed phase computed from PI-GCMC in selected single-walled CN and BN nanotubes. The reproduction of the experimental equation of state for para- H 2 as well as variation of the mean kinetic energy versus density of bulk para-h 2. This material is available free of charge via the Internet at References and Notes (1) Saito, R.; Dresselhaus, G.; Dresselhaus, M. S. Physical Properties of Carbon Nanotubes; Imperial College Press: Singapore, (2) Dresselhaus, M. S.; Dresselhaus, G.; Eklund, P. C. Science of Fullerenes and Carbon Nanotubes; Academic Press: San Diego, CA, (3) Terrones, H.; Terrones, M. New J. Phys. 2003, 5, 126. (4) Terrones, M. Annu. ReV. of Mater. Res. 2003, 33, 419. (5) Khoo, K. H.; Mazzoni, M. S.; Louie, S. G. Phys. ReV. B2004, 69, (6) Golberg, D.; Bando, Y.; Tang, C. C.; Zhi, C. Y. AdV. Mater. 2007, 19, (7) Kowalczyk, P.; Holyst, R.; Terrones, M.; Terrones, H. Phys. Chem. Chem. Phys. 2007, 9, (8) Kowalczyk, P.; Solarz, L.; Do, D. D.; Samborski, A.; Macelroy, J. Langmuir 2006, 22, (9) Gauden, P. A.; Terzyk, A. P.; Rychlicki, G.; Kowalczyk, P.; Lota, K.; Raymundo- Pinero, E.; Frackowiak, E.; Begun, F. Chem. Phys. Lett. 2006, 421, 409. (10) Sholl, D. S.; Johnson, J. K. Science 2006, 312, (11) Mickelson, W.; Aloni, S.; Wei-Qiang, H.; Cumings, J.; Zettl, A. Science 2003, 18, 467.

10 8284 J. Phys. Chem. B, Vol. 112, No. 28, 2008 Kowalczyk et al. (12) Hirahara, K.; Suenaga, K.; Bandow, S.; Kato, H.; Okazaki, T.; Shinohara, H.; Iijima, S. Phys. ReV. Lett. 2000, 85, (13) Hummer, G.; Rasaiah, J. C.; Noworyta, J. P. Nature 2001, 414, 188. (14) Meyer, R. R.; Sloan, J.; Dunin-Borkowski, R. E.; Kirkland, A. I.; Novotny, M. C.; Bailey, S. R.; Hutchison, J. L.; Green, M. L. H. Science 2000, 25, (15) Sato, Y.; Suenaga, K.; Okubo, S.; Okazaki, T.; Iijima, S. Nano Lett. 2007, 7, (16) Guan, L.; Suenaga, K.; Shi, Z.; Gu, Z.; Iijima, S. Nano Lett. 2007, 7, (17) Murata, K.; Kaneko, K.; Kanoh, H.; Kasuya, D.; Takahashi, K.; Kokai, F.; Yudasaka, M.; Iijima, S. J. Phys. Chem. B 2002, 106, (18) Kowalczyk, P.; Brualla, L.; Zywocinski, A.; Bhatia, S. K. J. Phys. Chem. C 2007, 111, (19) Khlobystov, A. N.; Britz, D. A.; Briggs, G. A. D. Acc. Chem. Res. 2005, 38, 901. (20) Ohkubo, T.; Iiyama, T.; Nishikawa, K.; Suzuki, T.; Kaneko, K. J. Phys. Chem. B 1999, 103, (21) Suzuki, T.; Iiyama, T.; Gubbins, K. E.; Kaneko, K. Langmuir 1999, 15, (22) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; Sliwinska- Bartkowiak, M. Rep. Prog. Phys. 1999, 62, (23) Neimark, A. V.; Ravikovitch, P. I.; Vishnyakov, A. Phys. ReV. E 2000, 62, R1493. (24) Yang, Ch-M.; Kim, Y-I.; Endo, M.; Kanoh, H.; Yudasaka, M.; Iijima, S.; Kanko, K. J. Am. Chem. Soc. 2007, 129, 20. (25) Ravikovitch, P. I.; Neimark, A. V. Langmuir 2006, 22, (26) Vajsary, G. Tritium Isotope Separation; CRC Press: Boca Raton, FL, (27) Matsuyama, M.; Sugiyama, H.; Hara, M.; Watanabe, K. J. Nucl. Mater. 2007, , (28) Ueda, S.; Nanjou, Y.; Ito, T.; Tatenuma, K.; Matsuyama, M.; Watanabe, K. Fusion Sci. Technol. 2002, 41, (29) Wang, Q.; Challa, S. R.; Sholl, D.; Johnson, J. K. Phys. ReV. Lett. 1999, 82, 956. (30) Challa, S. R.; Sholl, D.; Johnson, J. K. Phys. ReV. B2001, 63, (31) Challa, S. R.; Sholl, D.; Johnson, J. K. J. Chem. Phys. 2002, 116, 814. (32) Hattori, Y.; Tanaka, H.; Okino, F.; Touhara, H.; Nakahigashi, Y.; Utsumi, S.; Kanoh, H.; Kaneko, K. J. Phys. Chem. B 2006, 110, (33) Tanaka, H.; Fan, J.; Kanoh, H.; Yudasaka, M.; Iijima, S.; Kaneko, K. Mol. Simul. 2005, 31, 465. (34) Tanaka, H.; Kanoh, H.; Yudasaka, M.; Iijima, S.; Kaneko, K. J. Am. Chem. Soc. 2005, 127, (35) Van Itterbeck, A.; Van Dingenen, W. Physics 1939, 4, 389. (36) Hansen, R. S. J. Phys. Chem. 1959, 63, 743. (37) Basmadjian, D. Can. J. Chem. 1960, 38, 141. (38) Basmadjian, D. Can. J. Chem. 1960, 38, 149. (39) Larin, A. V.; Parbuzin, V. S. Mol. Phys. 1992, 77, 869. (40) Polevoi, A. S.; Khoroshilov, A. V.; Yudin, I. P. Russ. J. Phys. Chem. 1983, 57, (41) Polevoi, A. S.; Yudin, I. P. Russ. J. Phys. Chem. 1982, 56, (42) Polevoi, A. S.; Yudin, I. P. Russ. J. Phys. Chem. 1982, 56, 745. (43) Polevoi, A. S.; Durneva, A. I.; Azizov, A. S. Russ. J. Phys. Chem. 1987, 61, (44) Garberoglio, G. J. Chem. Phys. 2008, 128, (45) McCarty, R. D.; Hord, J.; Roder, H. M. Selected Properties of Hydrogen; National Bureau of Standards Monograph 168; National Bureau of Standards: Washington, DC, (46) Feynman, R. P.; Hibbs, A. Quantum Mechanics and Path Integrals; McGraw- Hill: New York, (47) Feynman, R. P. Statistical Mechanics; Benjamin: New York, (48) Binder, K.; Heermann, D. W. Monte Carlo Simulation in Statistical Physics; Springer-Verlag: Berlin, Germany, (49) Chandler, D. In Hansen, J. P.; Levesque, D.; Zinn-Justin, J. Eds.; Theory of Quantum Processes in Liquids; Elsevier: Amsterdam, The Netherlands, (50) Rabani, E.; Jortner, J. J. Phys. Chem. B 2006, 110, (51) Silvera, I. F.; Goldman, V. J. Chem. Phys. 1978, 69, (52) Kowalczyk, P.; Gauden, P. A.; Terzyk, A. P.; Bhatia, S. K. Langmuir 2007, 23, (53) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon: Oxford, U.K., (54) Frenkel, D.; Smit, B. Understanding Molecular Simulation Form Algorithms To Applications; Academic Press: London, (55) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, U.K., (56) Won, C. Y.; Aluru, N.R. J. Am. Chem. Soc. 2007, 129, (57) Ceperley, D. M.; Pollock, E. L. Phys. ReV. Lett. 1986, 56, 351. (58) Pollock, E. L.; Ceperley, D. M. Phys. ReV. B1987, 36, (59) Ceperley, D. M. Phys. ReV. Lett. 1992, 69, 331. (60) Brualla, L.; Boronat, J.; Casulleras, J. J. Low Temp. Phys. 2002, 126, (61) Nicholson, D. Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption, Academic Press: London, (62) Tan, Z.; Gubbins, K. E. J. Phys. Chem. B 1992, 96, 845. (63) Nemirovsky, D.; Moreh, R.; Kaneko, K.; Ohba, T.; Mayers, J. Surf. Sci. 2003, 282, 526. (64) Nemirovsky, D.; Moreh, R.; Anderson, K. H.; Mayers, J. J. Phys. Condens. Matter 1999, 11, JP800735K