4He Separation

Letter pubs.acs.org/JPCL

Nanoporous Graphene Membranes for Efficient 3He/4He Separation Andreas W. Hauser*,† and Peter Schwerdtfeger*,†,‡ †

Centre for Theoretical Chemistry and Physics (CTCP), The New Zealand Institute for Advanced Study (NZIAS), Massey University (Auckland Campus), Private Bag 102904, North Shore City, 0745 Auckland, New Zealand ‡ Fachbereich Chemie, Philipps-Universität Marburg, Hans-Meerwein-Straße, 35032 Marburg, Germany ABSTRACT: Nanoporous functionalized graphene membranes containing nitrogen atoms in a nanosized ring pore have the capability to separate 3He from 4He efficiently by quantum tunneling through the pore. Calculations on size-reduced model systems of single-graphene pores reveal that a balanced choice of tunneling barrier height and low gas temperature boosts the selectivity and keeps the helium gas flux at an industrially exploitable level.

SECTION: Molecular Structure, Quantum Chemistry, General Theory

A

dwindling supply of 3He paired with a growing demand in recent years led to a critical situation for low-temperature research institutes and cryogenic industries.1 A shortage of this highly precious gas, which is mainly used to fill neutron detectors (for detecting smuggled plutonium) or for large neutronscattering facilities, might also affect fundamental research in ultracold physics and chemistry, where 3He is used in dilution refrigerators. A more recent application of spin-parallel helium gas is the in vivo magnetic resonance imaging of human lungs, an upcoming new technique that allows for a detailed, timeresolved visualization of the pulmonary ventilation.2−4 As 3He is very rare in nature, the standard way of manufacture is to skim it from tritium reserves as a byproduct of the radioactive tritium decay. Because of the very limited production capability of this technique, even the possibility of lunar 3He mining has been considered.5,6 However, as traces of 3He helium can be found in the atmosphere and in some natural gas sources, there is the option of a direct recovery. Here we focus on the potential usage of nanoporous graphene derivatives for the purpose of 3He/4He separation, that is, the yet unsolved problem of helium-3 harvesting. The isolation of a free-standing sheet of graphene in 2004 opened a whole new branch in materials science,7 leading to new insights in physics, chemistry, and the applied sciences.8−13 It is the mechanical strength and chemical inertness of graphene combined with its extraordinary feature of being essentially 2-D, which makes this material a suitable candidate for porous membranes. However, a perfect sheet of graphene is impermeable even to helium atoms, which are repelled by the electron density of the aromatic rings.14 Leenaerts et al. report a barrier height of 11.67 eV for He escaping through a nondefect graphene sheet.15 Therefore, the artificial creation of nanosized holes by techniques such as electron beam treatment or the © 2011 American Chemical Society

design of suitable precursors for the self-assembling of porous graphene structures is currently being investigated.16,17 Although a cost-effective method of producing graphene with equally distributed pores of similar size is yet to come, its principal possibility already led to a few theoretical studies on the design of pores for potential applications such as the separation of hydrogen from methane,18 the separation of helium from other noble gases and methane,19 the selective passage of ions,20 the characterization of DNA,21 or the filtration of water.22 Here we suggest a novel type of nanoporous graphene that we predict to be capable of separating bosonic 4He from its highly valuable fermionic isotope 3He at an industrially exploitable level. The separation of 3He and 4He is achieved by taking advantage of slight deviations in the tunneling probabilities of the two isotopes when propagating through a suitably designed potential wall, in our case when passing through a functionalized pore in a graphene sheet, as shown in Figure 1. The reaction path describing the propagation of a single helium atom through such a pore corresponds to a simple 1-D potential problem and can therefore easily be treated by quantum mechanics. The main problem here is to find a functional form of the potential that emphasizes the different tunneling probabilities of the two isotopes and at the same time can be synthetically realized, for example, by designing a nanopore with a functionalized pore rim. In the simplest case, the pore rim may be passivated by the addition of hydrogen atoms. Whereas Schrier analyzes in some detail the 3He/4He isotope effect for the transmission through a pore of this type,19 Received: November 14, 2011 Accepted: December 29, 2011 Published: December 29, 2011 209

dx.doi.org/10.1021/jz201504k | J. Phys. Chem. Lett. 2012, 3, 209−213

The Journal of Physical Chemistry Letters

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Second, we started our quest for suitable molecular systems that could mimic the desired optimum tunneling potential. Benchmark calculations for small pores, where only one ring had been removed, did not yield suitable potentials. A full nitrogen passivation of such a one-ring hole considerably reduced its barrier from 4200 to 900 cm−1, but this was still too high for our purpose. Pores obtained by removing three or more rings made the barrier disappear. However, all but the largest pores, where the energy minimum corresponds to a helium atom sitting in the middle of the pore, showed the desired feature of a well-developed van der Waals minimum of >100 cm−1 at a helium distance between 1.9 and 2.5 Å above the pore. Finally, a closer look at the two-ring holes revealed the desired structure: a single-sided nitrogen-doped variant provided a suitable barrier height of 201 cm−1. Figure 1 shows this optimum pore type together with two closely related but still inappropriate structures. For all quantum theoretical treatments, we used molecular models of single-graphene pores. In the case of the two-ring holes, the pores consisted of 14 remaining rings in the direct neighborhood. Density functional theory (DFT) was applied, using the long-range corrected B97D functional of Grimme to account for the weak dispersive type of interactions.27,28 A correlation-consistent polarized triple-ζ basis set was used for all atoms, introducing a total of ∼2000 contracted Cartesian Gaussian basis functions.29,30 Each type of model pore was allowed to relax in a preliminary search and optimization.31 The obtained geometries for the graphene derivatives were then kept frozen in a follow-up scan for the helium tunneling path. Slight changes in the graphene geometry induced by the transit of a single helium atom turned out to be negligibly small (ΔE = 6 cm−1 for the single-sided N-doped pore). All energies are corrected for the basis set superposition error (BSSE) using the method of Boys and Bernardi.32 We note that the BSSE has its maximum value at the transition state and is of similar size for all three pore types shown in Figure 1. The correction increases the barrier height by 61 cm−1 for the single-sided N-doped pore. Figure 2 shows the calculated reaction paths for a single helium atom passing through three different pore types: the

Figure 1. Model systems for the analysis of the helium transmission through a functionalized graphene pore: (a) hydrogen-passivated pore, (b) single-sided nitrogen-doped pore, and (c) two-sided nitrogendoped pore. Carbon is printed in orange, hydrogen is printed in white, and nitrogen is printed in blue.

which is obtained by removing one ring of carbon atoms from the graphene sheet and saturating the carbon atoms with hydrogen atoms where required, he abandons this idea because of two contradicting tendencies: (i) Higher temperatures are necessary to keep the transmission probability at a reasonable value, which implies that the quantum isotope effect diminishes rather quickly, and (ii) at low temperatures (77 K) there is a suitably high 3He/4He transmission ratio of ∼3 but at the cost of an almost zero helium flux. This unwanted feature is directly related to the height of the potential wall, which is ∼4200 cm−1 for the hydrogen-passivated pore.19 In an attempt to overcome this problem, we decided to focus on the design of larger pores, which can be obtained by removing more than one ring or by replacing the C−H groups on the rim by nitrogen atoms. We found that a combination of both enlargement strategies provides a sufficiently smooth variation of the overall potential form and allows for a finetuning of the crucial tunnel barrier height. Furthermore, this choice of graphene modification reflects ongoing experimental research on the synthesis of nitrogen-doped graphene by chemical vapor deposition,23 arc-discharge,24 embedded C and N sources,25 and post-treatments.26 First, we tested several artificial model potentials that did not correspond to any known molecular pore structure and tried to design the optimum potential for an efficient 3He/4He separation. It turned out that among those potentials that may be realized later by suitable graphene modifications a single Gaussian-like maximum with a height of a few hundred wavenumbers, surrounded by two Gaussian-like minima of similar depth, is the most promising candidate. Whereas the minima do not noticeably affect the difference between 3He and 4He tunneling, their presence considerably improves the overall transmission probability by narrowing the central peak.

Figure 2. Diffusion barrier for a single helium atom passing through a hydrogen- (red), a single-sided N- (blue), and a two-sided N- (green) passivated graphene pore. Distance refers to the height above the pore perpendicular to the graphene plane.

pure hydrogen-, the single-sided nitrogen-, and the two-sided nitrogen-passivated pore. The first and the last types of pore are expected to be good candidates for a separation of H2 from CH4,18 but they are not suitable for the purpose of helium isotope separation. Whereas the pure hydrogen-passivated pore 210



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is still too repulsive (625 cm−1 barrier height) to provide an acceptable helium flux at low temperature, the barrier almost disappears for the two-sided nitrogen variant: the transition state is found at 61 cm−1. With quantum tunneling being crucial for a noteworthy difference in the transmission probabilities for helium-3 and helium-4, the final adjustment in our approach was to restrict the nitrogen passivation to just one side of the pore. This last step yielded the desired potential form with a barrier height of 201 cm−1 and a van der Waals minimum of 134 cm−1. To derive the essential macroscopic quantities, namely, the 3 He and 4He flux, from the ab initio results, we solve the Schrödinger equation numerically for a single particle in the 1D potential33 calculated for the single-sided N-passivated pore. A Numerov recursion algorithm is applied to obtain the transmission probability t(E) as a function of the particle energy.34 To calculate pw(T), the thermally weighted transmission of particles, the function t(E) is then multiplied with p(E,T), the kinetic energy distribution in one dimension for a given temperature, followed by a numerical integration

pw (T ) =

∫ p (E , T )t (E ) d E

becomes more likely at higher particle velocities. For this reason, it makes sense to keep the gas temperature as low as possible, exploiting the advantageous conditions at the lowenergy sides of the transmission curves. The broad velocity distributions of higher temperatures cover both sides of the transmission curves, canceling any isotopic effects. This dictates an upper limit for the temperature and the potential barrier. The decrease in helium flux at lower temperatures may be compensated by a further reduction of the potential wall, but this adjustment has its limits as well: By lowering the barrier too much, the transmission function is shifted into the kinetic energy distribution and any isotopic effects vanish again. Our results are summarized in Figure 4, showing the weighted transmission of 3He and 4He as a function of the

(1) 3

4

The transmission functions t(E) for He and He going through the single-sided nitrogen-passivated pore are shown in Figure 3 together with the kinetic energy distributions of

Figure 3. Kinetic energy distribution (black) in one direction, plotted at different temperatures from 10 to 20 K. To obtain a thermally weighted transmission for a given temperature, we must multiply the corresponding energy distribution by the transmission probability (3He solid blue, 4He dashed blue), shown here as a function of the energy and integrated over the whole range of possible energies.

Figure 4. (a) Weighted transmission of 3He (solid lines) and 4He (dashed lines) through the single-sided N-passivated graphene pore as a function of temperature. (b) Ratio of quantum-mechanical to classical transmission probability. (c) 3He/4He transmission ratio drops down from 19 to its asymptotic value of 1 within a temperature range of 15 K. The classical transmission probability in panel a is shown as a dotted line.

helium for temperatures between 10 and 20 K. In our approach, we assume a Boltzmann distribution for the 1-D kinetic energies of both isotopes.35 This keeps us on the safe side for any predictions of the 3He/4He selectivity: distinguishing between translational energies of fermions and bosons in a mixed, nonideal (i.e., interacting) quantum gas is problematic, but if we could, we would find an even higher helium-3 selectivity because Fermi−Dirac statistics would lead to a broader helium-3 distribution than Bose−Einstein statistics would give for the bosonic helium-4. Hence, the thermally averaged transmission would be higher for helium-3 than for helium-4 and therefore the 3He/4He ratio even better. The Figure shows a significantly higher transmission probability for 3He at low kinetic energies. However, as the two transmission functions cross at the barrier height of 201 cm−1, the situation gets inverted and the tunneling of 4He

temperature (a), the ratio of quantum-mechanical to classical transmission (b), and the 3He/4He transmission ratio (c) for the single-sided nitrogen-passivated graphene pore. In the temperature range from 10 to 20 K, the calculated transmission values deviate significantly from the classical prediction, which neglects any quantum tunneling, leading to the huge values for the ratio of quantum-mechanical to classical transmission plotted in panel b.36 The relevant 3He/4He ratio plotted in panel c raises almost exponentially with decreasing temperature from 2 at 20 K to 19 at 10 K. In ref 19, a hypothetical ratio of 3 is obtained at 77 K for the hydrogen-passivated pore but at the cost of an impractical transmission probability below 10−36. In our case, the transmission probabilities are in the range of 10−9, which corresponds to a total helium flux of more than 10−9 211



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moles cm−2 s−1 if we assume a 100% porous graphene sheet, ideal gas conditions at 10 K, and a pressure of 1 bar (collision rate approximately 1024 s−1 cm−2). In summary, we could show that a nitrogen-functionalized nanoporous sheet of graphene provides a suitable membrane for the separation of 3He from 4He. A balanced choice of temperature, barrier height, and potential form emphasizes the isotopic effect on the tunneling probability for helium atoms. Quantum-theoretical calculations on a size-reduced model system of a single graphene pore revealed that a nitrogen-doped variant of a two-ring hole yields a suitable potential form. Uncertainties of the barrier heights, introduced by the necessary simplifications within our computational approach, are compensated by the fact that the potential barrier is highly adjustable in reasonable small steps by pore rim modifications. To prove this, we presented three slightly different pore types with different barrier heights. For the most suitable type, we conservatively predict a respectable 3He/4He ratio of 19 at a temperature of 10 K and a still acceptable flux of 10−9 moles cm−2 s−1. Given these promising numbers, an industrial multistage purification of terrestrial helium sources at low temperature seems reachable: An effective membrane area of one square meter could yield more than 3 g of helium-3 per day.

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(13) Bonaccorso, F.; Sun, Z.; Hasan, T.; Ferrari, A. C. Graphene Photonics and Optoelectronics. Nat. Photonics 2010, 4, 611−622. (14) Bunch, J. S.; Verbridge, S. S.; Alden, J. S.; van der Zande, A. M.; Parpia, J. M.; Craighead, H. G.; McEuen, P. L. Impermeable Atomic Membranes from Graphene Sheets. Nano Lett. 2008, 8, 2458−2462. (15) Leenaerts, O.; Partoens, B.; Peeters, F. M. Graphene: A Perfect Nanoballoon. Appl. Phys. Lett. 2008, 93, 193107. (16) Fischbein, M. D.; Drndic, M. Electron Beam Nanosculpting of Suspended Graphene Sheets. Appl. Phys. Lett. 2008, 93, 113107-1− 113107-3. (17) Kuhn, P.; Forget, A.; Su, D.; Thomas, A.; Antonietti, M. From Microporous Regular Frameworks to Mesoporous Materials with Ultrahigh Surface Area: Dynamic Reorganization of Porous Polymer Networks. J. Am. Chem. Soc. 2008, 130, 13333−13337. (18) en Jiang, D.; Cooper, V. R.; Dai, S. Porous Graphene as the Ultimate Membrane for Gas Separation. Nano Lett. 2009, 9, 4019− 4024. (19) Schrier, J. Helium Separation Using Porous Graphene Membranes. J. Phys. Chem. Lett. 2010, 1, 2284−2287. (20) Sint, K.; Wang, B.; Král, P. Selective Ion Passage through Functionalized Graphene Nanopores. J. Am. Chem. Soc. 2008, 130, 16448−16449. (21) Garaj, S.; Hubbard, W.; Reina, A.; Kong, J.; Branton, D.; Golovchenko, J. A. Graphene As a Subnanometre Trans-Electrode Membrane. Nature 2010, 467, 190−193. (22) Suk, M. E.; Aluru, N. R. Water Transport through Ultrathin Graphene. J. Phys. Chem. Lett. 2010, 1, 1590−1594. (23) Wei, D.; Liu, Y.; Wang, Y.; Zhang, H.; Huang, L.; Yu, G. Synthesis of N-Doped Graphene by Chemical Vapor Deposition and Its Electrical Properties. Nano Lett. 2009, 9, 1752−1758. (24) Panchakarla, L. S.; Subrahmanyam, K. S.; Saha, S. K.; Govindaraj, A.; Krishnamurthy, H. R.; Waghmare, U. V.; Rao, C. N. R. Synthesis, Structure, and Properties of Boron- and Nitrogen- Doped Graphene. Adv. Mater. 2009, 21, 4726−4730. (25) Zhang, C.; Fu, L.; Liu, N.; Liu, M.; Wang, Y.; Liu, Z. Synthesis of Nitrogen-Doped Graphene Using Embedded Carbon and Nitrogen Sources. Adv. Mater. 2011, 23, 1020−1024. (26) Wang, X.; Li, X.; Zhang, L.; Yoon, Y.; Weber, P. K.; Wang, H.; Guo, J.; Dai, H. N-Doping of Graphene Through Electrothermal Reactions with Ammonia. Science 2009, 324, 768−771. (27) Grimme, S. Semiempirical GGA-Type Density Functional Constructed with a Long-Range Dispersion Correction. J. Comput. Phys. 2006, 27, 1787−1799. (28) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A. et al. Gaussian 09, revision A.02; Gaussian, Inc.: Wallingford, CT, 2009. (29) Woon, D. E.; Dunning, T. H. J. Gaussian Basis Sets for Use in Correlated Molecular Calculations. IV. Calculation of Static Electrical Response Properties. J. Chem. Phys. 1994, 100, 2975−2988. (30) Dunning, T. H. J. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007−1023. (31) Optimization criteria were set to “Opt=Tight” in Gaussian 09, corresponding to an rms force threshold of 10−5 au. (32) Boys, S. F.; Bernardi, F. The Calculation of Small Molecular Interactions by the Differences of Separate Total Energies. Some Procedures with Reduced Errors. Mol. Phys. 1970, 19, 553−566. (33) The two in-pore vibrational modes were calculated in a frequency job using the B97D functional and the cc-pVTZ basis set. We obtained 72 and 164 cm−1 for the two in-plane modes of the He atom due to the elliptic shape of the pore. Both vibrational levels are practically unoccupied at the relevant temperature range of 10−20 K. Hence we concluded that multidimensional effects are of minor importance for the description of the helium propagation at these very low temperatures. (34) A Matlab script is used to solve the 1-D time-independent Schrödinger equation for particles moving in the z direction, that is, towards the pore, using the DFT barrier as input potential. For the

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; peter.schwerdtfeger@ gmail.com.

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ACKNOWLEDGMENTS We thank Ulrich Zülicke and Gabriele Jaritz for helpful discussions. P.S. is grateful to the Alexander von Humboldt Foundation for financial support.

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REFERENCES

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Numerov recursion, a finite difference method of fourth order, we choose a step size of Δz = 0.0125 Å. (35) High accuracy in the numerical calculation of the transmission function close to zero energy is essential for obtaining correct results. To improve the accuracy of our calculations, we chose the more suitable domain of velocities for the numerical integration, avoiding the singularity of the energy distribution at zero energy. In this domain, the transition barrier occurs at different velocities for 3He and 4 He, but the distribution is of the Gaussian type (centered around zero) and therefore less problematic. (36) The overall transmission strongly depends on the accuracy of the calculated transmission probability curve at low energies. To be on the safe side for our predictions, tunneling probabilities lower than 1 × 10−7 are set to zero.

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4He Separation

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