J. Phys. Chem. B 2008, 112, 8275–8284
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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, 87-100 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 D2/H2 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 D2/H2 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 D2/H2 equilibrium selectivity at finite pressures to the variation of the difference between the average kinetic energy computed from single-component adsorption isotherms of H2 and D2. 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.1–10 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.11–19 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.20–25 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) 99252571. Fax: +61 (03) 99255290. E-mail:
[email protected]. † 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.29–33 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.29–31 In the radial direction, the movement of light particles at cryogenic temperatures is
10.1021/jp800735k CCC: $40.75 2008 American Chemical Society Published on Web 06/21/2008
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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 13.46 Å; (11,11)equivalent pore diameter 14.81 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 2. The average kinetic energy (computed for 33 K) of the H2 (open circles) and D2 (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 H2 and D2. 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, Ekin ) (3/2)kBT, at 33 K is 49.73 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 D2 from H2 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,
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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 D2, the open circles denote H2, and the open squares denote the experimental equation of state for para-H2 at 30 K. Right panel: Average gyration radius of Feynman’s cyclic polymers quantizing D2 (gray circles) and H2 (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 4. Left panel: Pore density variation of the average kinetic energy of H2 (open circles and red crosses) and D2 (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, Ekin ) (3/2)kBT, which at 33 K is 49.73 K. Right panel: Pressure variation of the difference between the average kinetic energy of H2 and D2 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 T2 over H2 or D2 over H2 increases during the filling of the pore space.35–43 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
Hydrogen Isotopes in Single-Walled CNs and BNs
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Figure 5. Equilibrium snapshots of D2 (left panel) and H2 (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 D2 over H2 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 D2 over H2 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 H2 and D2 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 D2 over H2 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 H2 and D2 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 (2) (P) “necklace” of P classical “beads” r(1) i , ri ,..., ri that accounts for the quantum delocalization of the particle.46–49 When a system contains more than one type of quantal particle, decisions
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concerning the appropriative value of P should be made separately for each particle.49 For H2 at 33 K and moderate pressures, we used PH2 ) 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 D2 is twice the mass of H2, the number of beads in cyclic polymers quantizing the deuterium molecule is PD2 ) PH2/2 ) 8. In the considered range of experimental equations of state, both the density as well as the mean kinetic energy of D2 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”46–49 N
W)
P
∑∑
P
∑ ∑ Vff(rij(R)) +
mP (ri(R) - ri(R+1))2 + P1 2 2 2β p i)1 R)1 i
