Defect-Mediated Reduction in Barrier for Helium Tunneling through

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Defect-Mediated Reduction in Barrier for Helium Tunneling through Functionalized Graphene Nanopores Murugan Lalitha, Lakshmipathi Senthilkumar, and Suresh Kumar Bhatia J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b05567 • Publication Date (Web): 24 Aug 2015 Downloaded from http://pubs.acs.org on August 31, 2015

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The Journal of Physical Chemistry

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Defect-Mediated Reduction in Barrier for Helium Tunneling through Functionalized Graphene Nanopores

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Murugan Lalitha1,2, Senthilkumar Lakshmipathi1,2, Suresh K. Bhatia1

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School of Chemical Engineering, The University of Queensland, Brisbane 4072, QLD, Australia. 2 Department of Physics, Bharathiar University, Coimbatore 641046, Tamil Nadu, India.

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ABSTRACT:

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We investigate the separation of helium isotopes by quantum tunneling through graphene

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nanopores, recently proposed as an alternative to conventional methods for 3He production. We

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propose here a novel defective nanopore created by removing two pentagon rings of a Stone-

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Thrower-Wales (STW) defect, which significantly decreases the helium tunneling barrier by

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50% to 75%. The barrier height is fine-tuned by adjusting the effective pore size, which is

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achieved by pore rim passivation using an appropriate functionalizing atom. This fine-tuning

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leads to positive deviation in the tunneling probability of 3He compared to 4He in the low energy

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region, and thereby to high selectivity and transmission of the former isotope. It is found that

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fluorine-passivated nanopores restrict helium atom penetration due to their highly reduced pore

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size. Defective nanopores in nitrogen and oxygen passivated structures exhibit relatively high

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transmission values of 10-3 for the oxygen variant and improved selectivity value of 669 for the

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nitrogen variant. It is demonstrated that defective nanopores passivated on both sides with

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oxygen are the most attractive for 3He/4He separation, based on their much higher flux values,

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while still providing good selectivity.

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Keywords: Helium isotope separation, Stone-Thrower-Wales defect, Nanoporous graphene, Defective nanopore, DFT

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*

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1. Suresh K. Bhatia: Email: [email protected] Tel.: +61 7 336 54263

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2. Senthilkumar Lakshmipathi: Email: [email protected]. Tel.:+91 9443702753

Corresponding Authors

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INTRODUCTION

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The discovery of graphene in 20041 holds much potential for advancements in science,

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technology and research2-5, due to its two dimensional structure and significant inherent

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electronic properties6-8. Graphene, a one-atom thick planar sheet is impermeable even to helium

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atoms, due to the repulsion exerted by the π-electron cloud of the aromatic rings, evident through

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both experiment9 and theory10. Nevertheless, there is much interest in its applications in

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separation, with graphene rendered permeable by the creation of nanosized pores. The nanopores

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can be created by top-down techniques such as electron beam irradiation11, or by the bottom-up

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self-assembly of porous graphene structures using suitable precursors12. When made permeable

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in this way, the nanoporous graphene structure retains the inertness and extraordinary

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mechanical strength of graphene, while offering permeance sufficiently attractive for gas

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separation. This has opened up a new thrust in the field of materials science for applications in

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gas and isotope separations13.

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The two key factors which determine the efficiency of gas separation through a membrane are its

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selectivity and permeability. Since the permeance of gas molecules is inversely proportional to

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the thickness of the membrane14, the one-atom thick graphene holds promise for achieving high

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permeance. As a result there is increasing recognition of the potential of nanoporous graphene in

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various applications such as water desalination15, traditional gas separation (e.g. H2/CH4,

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CO2/CH4, H2S/CH4)13,14,16,17, characterization of DNA18, hydrogen isotope separation19, noble

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gas separation20, and selective passage of ions21. The pores can also be asymmetrically decorated

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to provide energetically favorable pathways, as suggested for hydrogen isotope separation by

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Hankel et al19. Likewise, Hauser et al22 proposed a two-ring nanopore with rim terminated by

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nitrogen atoms; they estimated the permeation rates of CH4, N2, O2, CO2 and H2, and concluded 2 ACS Paragon Plus Environment

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that nitrogen doping gives better transmission rate for N2 than hydrogen termination. Lalitha et

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al.23concluded that edge functionalization of graphene sheets with hydrogen or fluorine atoms

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improves adsorption of carbon dioxide molecules and restricts that of water molecules.

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Nevertheless, the literature on helium isotope separation using nanoporous graphene is relatively

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scant, and very recent.

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Among the isotopes of helium only two, 3He and 4He are stable. While He is abundantly found in

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the cosmos, the proportion of 3Heis very small, varying from 1 ppm in He on earth to 100 ppm in

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interstellar space. 3He is used in neutron detectors, now of growing importance in security

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applications, and in dilution refrigerators for mK range cryogenics, besides applications in

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nuclear magnetic resonance instruments used in medical imaging and scientific research26. One

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of the major terrestrial sources of 3He is the radioactive decay of tritium (half-life about 12.5

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years), used in nuclear weapons. Industrially, 3He is separated from helium stockpiles and

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natural gas using methods such as cryogenic distillation or pressure-swing adsorption24, which

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are expensive due to the need to liquefy the gases. Nevertheless, due to the decline in production

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of 3He from tritium decay, there has been a drastic reduction in the availability of 3He, and this

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hassled to shortage and supply and demand imbalance of this precious isotope. As a result there

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is urgent need for more efficient technologies for the separation of3He from He existing in other

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terrestrial sources.

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While several chemical methods are available which can be used to separate gas molecules or

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isotopes and nuclear effluents27, they are inefficient in isotope applications. Quantum kinetic

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molecular sieving in nanoporous materials, relying on significant differences in de Broglie

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wavelength between isotopes at low temperature, has recently been proposed by Bhatia and co-

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workers based on theoretical calculations28-30; and subsequently experimentally verified by 3 ACS Paragon Plus Environment


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them31, using the example of H2/D2 separation. They showed that the heavier D2 diffuse

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significantly faster than H2 in nanoporous materials at low temperature when the pore size is

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comparable to the molecular size of H2 of about 0.27 nm28-31. In subsequent work19,32, it has been

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shown that efficient H2/D2 separation can be achieved in thin single layer nanoporous structures

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such as graphene membranes, and this has led to much recent interest in their use for isotope

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separation. This is particularly attractive because of the high efficiency and low energy cost of

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membrane-based gas separation33.

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While hitherto not examined for this application, a particularly attractive avenue would appear to

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be the use of defect-modified graphene membranes, because of reduction in the energy barrier

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due to the presence of defects. For instance, He atom escaping through a perfect graphene

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membrane having no defects experiences a barrier height of 11.7 eV, whereas in defective

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graphene, in which the sp2 hybridization is retained, the penetration barrier decreases

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exponentially with the size of the defect (Stone-Thrower-Wales, di-vacancy, tetra- hexa-, or

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deca-vacancy)10. Interestingly, as the number of carbon atoms included in the sp2 bonded defect

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ring increases, the barrier height decreases. Besides the reduction in penetration barrier height, it

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has been shown that the actual region of influence of the defect extends beyond the defect site34.

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Among the various kinds of defects, vacancies, substitutional impurities, adatoms and

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topological imperfections are formed during the growth of graphene35. The Stone-Thrower-

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Wales36,37 (STW) defect is a typical topological defect in carbons38,39, arising from bond rotation

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and rearrangement of four six-membered rings into two pentagons and two heptagons. Besides

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being formed during graphene synthesis, such defects can also be subsequently introduced by

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electron irradiation40-42, and have been imaged using transmission electron microscopy43. It is

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known that the Stone-Thrower-Wales (STW) defect distorts the graphene lattice, causing 4 ACS Paragon Plus Environment

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adjacent atoms to move out of the plane, thus producing buckling of the graphene sheet44. Hence,

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STW defects play a role in the intrinsic buckling of the graphene sheet45. Based on the results

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discussed above the STW defect would appear to be a prime candidate for incorporation in

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defect-modified graphene membranes for gas separation, but has yet to be investigated for such

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applications.

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The application of nanoporous graphene for the separation of 3He from 4He, in which various

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kinds of pore models and rim passivations are used, has found recent interest46-50. Schrier46

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examined the separation using a nanoporous graphene having one ring removed, and with the

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resulting edge atoms saturated with hydrogen, and calculated the quantum mechanical

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transmission of helium isotopes through the nanopore. Due to the small pore size, it exhibits a

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large barrier height of 50.6 kJ/mole, requiring relatively high temperature for acceptable

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transmission probability, but this makes the selectivity impractically low. In order to reduce the

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barrier height, Hauser et al.47 subsequently removed two hexagonal rings in graphene and also

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explored nitrogen functionalization of the rim atoms. The potential barrier height was adjusted

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by three different pore rim modifications, leading to3He/4He transmission ratio of 19, and a flux

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of 10-9 moles.cm-2.s-1, at a temperature of 10 K. Subsequently, Hauser48 et al. performed a

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comprehensive study of the two-ring removed nanopore, using both temperature and pressure

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difference driven approaches, finding that the tunneling probability of 3He is significantly higher

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at low temperature; however, no barrier was found for 3- and 4- ring holes, making these

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unattractive for separation. An isotopic enrichment ratio of 1:3 was obtained for thermally driven

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separation having TC=10 K and TH=20 K, with transmission rate exceeding 10-4moles.cm-2.s-1,

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allowing the system to reach steady state at an experimentally reasonable rate. Recently, Mandra

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et al.50 investigated bilayer graphene nanopore systems, and found that while the flux of 3He 5 ACS Paragon Plus Environment


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remains the same as in the monolayer membrane, the selectivity of 3He is doubled with respect to

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4

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nanoporous graphene is studied, with the pore rim functionalized with atoms such as hydrogen

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and nitrogen, while tuning pore size. Thus, pore rim functionalization has an important role in

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the flux and selectivity of helium isotopes.

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With the aim of improving flux and selectivity using single layer graphene, we propose here the

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incorporation of nanopores accompanied by defects on either side, as a potential alternative to

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the use of bi-layers. We show that the potential barrier height for helium is considerably reduced

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due to the reduced electron density in the defect region. Further, to improve the flux and

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selectivity of 3He over 4He, pore rim modification is investigated, considering functionalization

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with hydrogen, nitrogen, oxygen and fluorine atoms. In this way, the doped defective pore is

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characterized for efficient separation of 3He from 4He, and compared with the pristine nanopore.

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While extending the prior studies46-50 to defective nanopores, the results of this study

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demonstrate enhanced performance of single layer defective nanoporous graphene for the

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separation of 3He from 4He.

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COMPUTATIONAL DETAILS

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We considered a rectangular graphene sheet of length and width 14.73 Å and 11.34 Å

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respectively, incorporating three consecutive Stone-Thrower-Wales (STW) defects, with

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subsequent removal of two pentagon rings of the central STW defect to provide a defective

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nanopore, as depicted in Figure 1.Only a single layer of hexagonal rings surrounding the pore is

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considered for the present study. Further, in order to saturate the dangling bonds of the pore, the

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rim carbon atoms are either passivated with hydrogen or fluorine (which have only a single

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valence electron), or with atoms such as nitrogen, and oxygen (which have two valence

He, with the optimal interlayer spacing being 0.46 nm. In all the above studies, pristine


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electrons)by replacing unsaturated carbons. In all the considered structures the outer edges of the

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graphene sheet are terminated by hydrogen atoms to maintain its stability. The structures of the

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pristine and defective graphene sheets are optimized using the B97D functional of Grimme51 and

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a 6-311G** basis set using Gaussian 0952. Figure 2 displays the optimized geometry of pristine

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and defective nanopore structures with rim functionalizations. Scanning of the potential energy

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surface (PES) is subsequently performed along the central axis of the nanopore to obtain the

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potential energy profile, and the helium tunneling barrier, using the optimized structure of the

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graphene sheet. During the PES scan, the nanoporous graphene is kept frozen, as the time scale

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of relaxation of the graphene is larger than that for the interaction of helium atoms with the

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graphene layer (i.e. time scale of the motion of the helium); thus, the relaxation of graphene

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occurs after the helium atom has left the graphene layer, as discussed by Leenaerts et al10.

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THEORY:

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Permeance and selectivity are the two important parameters that determine the efficiency of gas

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and isotope separation through a membrane, with permeance being inversely proportional to

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membrane thickness. The total flux of molecules across a graphene pore may be decomposed

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into the direct flux and a surface flux, of which the surface flux is negligible for helium, since its

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two in-plane modes in the Schrodinger equation are essentially unoccupied48.This leads to

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reduced dimensionality of the system, and to a one-dimensional pathway for molecules crossing

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the pore. Here, the transmission factor of the helium isotope passing through the graphene

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nanopore is determined by numerically solving the one-dimensional Schrodinger equation, so as

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to include the quantum-mechanical tunneling effect for quantum objects. Thus, following

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Schrier45 and Hauser et al.47,48, we solve the time-independent one-dimensional Schrodinger

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equation 7 ACS Paragon Plus Environment


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d 2ψ 2m + [ E − V ( x)]ψ = 0 dx2 h2

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Assuming that helium atoms approach the nanoporous graphene from the left side ( x < X ,

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where X is the position of the graphene membrane), eqn (1) provides the asymptotic solutions48

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(1)

ψ ( x) = C1 expik ′x + C2 exp −ik ′x , x > X

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2 1/2 where k ' = k = (2mE / h ) , due to symmetry (since V ( x ) = 0 for x > X ), and

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C0, C1, C2 are constants. The transmission and reflection coefficients are given by

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C t= 0 C1

2

C , and r = 2 C1

2

(3)

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For a He atom of given energy E, the value of the transmission coefficient t(E) is obtained from

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the values of actual numerical solution for the known potential energy profile, V(x), following48

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t=

2 2

1 + C1 + C2

2

=

2 1 + Pav

(4)

2

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where Pav is the average of the oscillatory part of the far-field probability density, ψ ( x) on the

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incident side, and C0 has arbitrarily been taken as unity.

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Using the energy profile determined above, eqs. (1)-(4) were solved by the finite difference

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numerical method outlined by Cedillo53, by constructing a uniform grid of points, with grid size

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0.001 Å, to determine the transmission coefficient t(E) for both 3He and 4He for any particle

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kinetic energy E. The helium atom approaches the nanoporous graphene from a point 9Å away,


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crosses the barrier and terminates at a distance 9 Å on the other side. Out of the 18Å end-to-end

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distance, a vacuum region of 4 Å prevails on the either sides of the double well symmetric

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barrier. The flux of any isotope is readily estimated from J (T ) = t ( T ) Z coll

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(5)

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where t (T ) is the mean transmission coefficient, evaluated based on a canonical distribution of

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energies, following ∞

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1 e− E / kT t (T ) = t ( E )dE π kT ∫0 E

(6)

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and zcoll = P〈u〉 / 4kBT is the collision frequency assuming an ideal gas. Further, P is the pressure

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and 〈u〉 = 8kBT / π m is the mean molecular velocity54. This leads to the flux expression

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J (T ) =

P t (T ) 2π mkBT

(7)

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RESULTS AND DISCUSSION

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Structure and Pore Size

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Figure 2a shows the pristine graphene nanopore (also used in reference 47), in which two

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hexagon rings are removed. As an alternative not previously investigated, we propose here a

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defective passivated pore, displayed in Figure 2b (hydrogen passivated defective pore), given the

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potential for decreased barrier height and stronger adsorption affinity, discussed earlier. The

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defective nanopore is created by removing the two pentagon rings of a Stone-Thrower-Wales

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defect, from among three consecutive STW defects. Thus, the nanopore is accompanied by two

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heptagons on either side, in total consists of four heptagons. Further, the two heptagons on either 9 ACS Paragon Plus Environment


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side of the nanopore are associated with a pentagon. Hence, the proposed defective nanopore

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consists of four heptagons and two pentagons in total. The dangling bonds along the edge carbon

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atoms in the nanopore are passivated by hydrogen atoms. Likewise, the rim carbon atoms of the

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pore are also functionalized using hydrogen atoms. Passivation of some rim carbon atoms is also

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carried out using other functionalizing atoms, which effectively fine-tunes the pore size, and

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modifies the tunnel barrier height. This functionalization performed in two ways: (i) single

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sided, and (ii) both side of the pore rim, using atoms such as nitrogen (Fig. 2c and 2d), oxygen

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(Fig. 2e and 2f) and fluorine (Fig. 2g and 2h respectively), which are equivalent to the size to the

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carbon atom. In the case of oxygen and nitrogen, they replace unsaturated carbon atoms; while

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the fluorine atom is used for functionalization by saturating the dangling bond on the rim, as

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discussed earlier.

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The two parameters that determine the separation performance are: (i) the pore size of the

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structure through which the isotope travels, and (ii) the kinetic diameter of the isotope under

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study. At one extreme, very small pore sizes do not allow molecules to penetrate while, on the

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other extreme, very large pore sizes offer no barrier. Hence, the pore size of the structure must be

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comparable to the size of the diffusing molecule. The kinetic diameter of the helium atom is

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about 2.60 Å, corresponding to a radius of 1.30 Å. Here, the pore sizes of the graphene

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nanostructures considered are measured using the sphere of maximum diameter that can fit into

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the pore in the electron density iso-surface plot, illustrated in Figure S1 (Supporting

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information). The measured pore sizes for the pristine and defective graphene nanopores are

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listed in Table 1. The pristine nanopore possess a pore radius of 1.46 Å, however on

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incorporating the defect, the pore size increases marginally to 1.48 Å. Likewise, in the case of

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rim functionalization with nitrogen and oxygen atoms, the pore radius increases for double side 10 ACS Paragon Plus Environment

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functionalization compared to single side. On comparing oxygen and nitrogen passivation, the

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pore size decreases marginally for oxygen functionalization, as oxygen is more electronegative

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than nitrogen. The single side fluorine variant offers pore radius of 1.30 Å, equal to the kinetic

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diameter of the helium atom; however the double side fluorine variant provides a radius of 1.16

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Å, which restricts helium atom penetration due to its small pore size. From Table 1, the double

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side nitrogen variant offers highest pore radius of 1.72 Å, among the nanostructures considered.

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Thus, all the considered nanopores except the fluorine variant possess radius greater than that of

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helium (1.30 Å), and are suitable for helium separation. Hence, the helium tunneling barrier is

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computed for each of the nanopore structures considered.

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Potential Barrier Calculations:

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To determine the potential energy profile for helium tunneling, scanning of the potential energy

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surface (PES) has been conducted for the optimized geometry of each of the proposed doped

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defective nanopores, with the helium atom initially placed at a distance 9Å away from the pore.

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While scanning the PES, the graphene nanopore is kept frozen and subsequently the potential

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energy of the helium atom sampled at various positions, using a small step size of 0.5Å. The

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potential energy profiles for the doped defective graphene nanopore along with its pristine

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counterpart, and for the fluorine variant, are depicted in Figure 3a and 3b respectively. From the

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figures, the helium tunneling barrier is in the form of a symmetric double well potential, (i.e. a

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Gaussian peak accompanied by two Gaussian minima, due to the symmetric structure of the

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nanopore. In other words, the barrier is surrounded by a deep van der Waals minimum on either

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side, which reduces the width of the tunneling barrier without altering the barrier height, and

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enhances the tunneling probability48. This is important, because increase in width of the barrier

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profoundly decreases the transmission probability, as the wave function decays exponentially 11 ACS Paragon Plus Environment


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inside the barrier. Moreover, in the case of oxygen and fluorine passivated structures, an

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asymmetrical potential barrier is observed due to the steric repulsion between the oxygen-

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oxygen/fluorine-fluorine atom pairs. The tunneling barrier height and the corresponding van der

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Waals minimum for the various doped nanopores are listed in Table2.

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As seen in Table 2, the pristine nanopore (defect-free), has a maximum barrier height of 9.0x10-

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21

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STW defect and the subsequent removal of two pentagon rings in the nanopore considerably

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reduces the barrier height, without reducing the van der Waals minimum value, as evident from

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Table 2. The helium tunneling barriers for the doped defective nanopore lie closer to each other,

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and range from 2x10-21 to 5x10-21 J/molecule for the hydrogen, nitrogen and oxygen variants.

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Thus, the defective nanopore reduces the helium tunneling barrier from 50% to 75% compared to

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the pristine nanopore. On the contrary, the fluorine passivated defective pore (Figure 3b) exhibits

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very high tunneling barrier of 20x10-21 J/molecule (for single side passivation) and 25x10-21

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J/molecule (for double side passivation). Hence, the fluorine-functionalized counterpart is

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unsuitable for helium isotope separation, in agreement with its smaller pore size. Potential

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energy curves for a helium atom passing through different nanopores are displayed in Figure 3,

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showing a Gaussian peak for all defective nanopores as well as for pristine structures except for

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the hydrogen passivated structure, where the central Gaussian peak is broadened. Further, the

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defective nanopore with rim passivation by the hydrogen atom exhibits barrier height of 3.36x

268

10-21 J/molecule, with its corresponding van der Waals minimum of 3.49x10-21 J/molecule. The

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protrusion of the rim passivated hydrogen atoms on either side of the pore (on the longer side of

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the nanopore) is out of the plane by upto 1Å, and influences the helium atom up to 2 Å. The

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attractive nature between the passivated hydrogen and the helium atom is responsible for the

Joules, with a corresponding van der Waals minimum of 4.4x10-21 Joules. Inclusion of the


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broadening of the potential barrier. Hence, for the hydrogen passivated defective nanopore, even

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though the barrier height is reduced considerably, enhanced transmission probability will not be

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obtained due to the broadened potential barrier. Besides this, among the doped defective

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nanopores, the potential barrier height is reduced for double side passivation compared to the

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single side passivation due to the large pore size of the double side variant. The He tunneling

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barrier is reduced for the oxygen variant compared to the nitrogen variant, and hence higher

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transmission factors of helium are expected for the oxygen passivated structure. The decreasing

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order of the potential barrier values from Table 2 for the considered doped and pristine structures

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is as follows:Pristine > S-Nitrogen > D-Nitrogen > Hydrogen > S-Oxygen > D-Oxygen

281

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The barrier height of the hydrogen passivated and the single side oxygen (S-oxygen) variant is

283

the same, but due to the broadened peak, the H-passivated structure precedes S-oxygen.

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Transmission Probability:

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The flux and selectivity in helium tunneling can be obtained by calculating the transmission

286

probability, as discussed above, through eqs (1)-(7). Using these equations, the transmission

287

probability t(E) for 3He and 4He has been calculated using the finite difference method of

288

Cedillo53 for both the pristine and defective graphene nanopore structures, as displayed in

289

Figures4a-f.

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From these figures it is seen that the incorporation of defects leads to increase in transmission of

291

low energy He particles. Incorporation of the defect changes the hybridization in the defect

292

region (carbon atoms), which in turn weakens the electron density of the hexagon and heptagon

293

rings surrounding the pore (c.f. Supporting Information Figure S1). This subsequently reduces 13 ACS Paragon Plus Environment


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Page 14 of 32

294

the repulsion between the electron cloud in the pore region and the transmitting helium atom,

295

leading to higher tunneling probability. Further, the transmission curves show that 3He particle is

296

more favored at low kinetic energies, whereas the 4He particle is slightly more favored at high

297

energies. We find that the transmission of 3He and 4He coincides very slightly above the barrier

298

height energy, though not precisely at the barrier height. Thus, for example, for the pristine

299

nanopore in the present work, the helium tunneling barrier value is 9.025 x 10-21 J/molecule,

300

whereas the intersection of 3He and 4He occurs at about 9.48 x 10-21 J/molecule. We believe this

301

small difference is due to the minor asymmetry of the fitted potential energy profile which, while

302

closely matching the theoretical profile (with correlation coefficient of 0.9996), does show very

303

small deviation a little away from the barrier, as depicted in Figure S2 of the Supporting

304

Information, for the pristine graphene sheet nanopore. Indeed, Hauser et al.48 reported tunneling

305

probabilities of the two helium isotope to coincide when the kinetic energy of the particle is

306

exactly the same as the barrier height, with preference for 3He at lower energies and 4He at

307

higher energies. Convergence of our results over grid sizes of 0.001-0.00001 Å was obtained,

308

and the minor deviation such as that in Figure S2 does not have any impact on our results.

309

Among all the considered nanopore structures, the double side oxygen variant requires lowest

310

kinetic energy particles to penetrate the barrier, because of relatively low barrier height.

311

The effect of temperature on the transmission probability can be studied from the thermally

312

weighted transmission probability, through which the flux and 3He/4He selectivity is calculated.

313

314

315 14 ACS Paragon Plus Environment

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316

Permeability and Selectivity

317

The permeability is assessed from the temperature dependent transmission values calculated

318

using eqs. (6) and (7), based on the transmission values t(E). To perform the integration, the

319

kinetic energy distribution values f(E,T) were calculated following

f ( E, T ) =

320

1 e − E / kT π kT E

(8)

321

The f(E,T) curves showed that He particles are predominantly in the low kinetic energy region

322

for the entire range of temperatures considered, and the f(E,T) curve vanishes at high kinetic

323

energy, evident from Figure S3 in Supporting Information (SI). Inspection of the product

324

function f(E,T)t(E)also showed that the3He transmission exceeds that of4He at low energies, but

325

the two approach each other at high energies. This is illustrated for the double side oxygen

326

variant at 100 K in Figure S4 of SI, in which the 3He transmission is higher up to energy of about

327

1500 J/mol, beyond which they differ very little and the selectivity approaches unity.

328

The temperature dependent transmission values obtained from eqn (6) by numerical integration

329

are listed in Table S1 for temperatures in the range of 10-100 K, and the corresponding 3He/4He

330

transmission ratios are listed in Table S2 in Supporting Information. Figure 5 displays the

331

variation of transmission values with respect to temperature, and of the corresponding ratios for

332

pristine, hydrogen passivated, nitrogen (single and double side) and oxygen (single and double

333

side) respectively. Among the various nanostructures considered, the defective double side

334

oxygen nanopore variant shows relatively high transmission values, of about 10-3 for 3He and 10-

335

4

336

improvement in transmission is at the cost of reduced selectivity (transmission ratio), the values

for 4He, over 5 orders of magnitude higher than the other alternatives considered. Although this



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

337

of selectivity of about 4 or larger below 20 K, and about 2 and larger below 40 K, for the double

338

side oxygen case are extremely high for isotope separation, and over an order of magnitude

339

larger than that of conventional methods for such separations. Hence, this variant in defective

340

nanopore structures appears more attractive compared to the pristine nanopore and the order of

341

the transmission values for the considered structures is as follows:

Page 16 of 32

342

D-Oxygen > S-Oxygen > D-Nitrogen > S-Nitrogen > Hydrogen > Pristine

343

On increase in temperature the transmission values of 3He and 4He show large difference up to

344

about 20 K, which gradually reduces up to 60 K, and thereafter the transmissions coincide.

345

However, as a case of exception, the double side oxygen variant shows preference for3He

346

transmission up to 100K. Thus, low kinetic energy 3He transmits through the defective nanopore

347

significantly more favorably at low temperatures (up to about 20K), and on increasing the

348

temperature further up to 100K, the isotopic effect vanishes.

349

The selectivity (3He/4He ratio) value listed in SI Table S2 and plotted in Figure 6, shows that on

350

increasing temperature from 10K to 100K, the

351

exponentially and this decrease is monotonic. Among the nanopores considered, a relatively low

352

selectivity of 4.7 is obtained for the double side oxygen variant and a very high selectivity value

353

of 669 is observed for the double side nitrogen variant at 10 K. The single side nitrogen and

354

oxygen nanopore variants also provide high selectivity values of 359 and 115 respectively;

355

however, their transmission is many orders of magnitude smaller as discussed above.

356

The flux of3He, calculated using the collisions frequency of the He atoms with the pore for

357

varying temperatures from 10K to 100K and 1atm pressure, following eqn. (7) is plotted in

358

Figure 7. The lowest value of helium flux is observed for the hydrogen variant defective

3

He/4He transmission ratio decreases


Page 17 of 32

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359

nanopore, of 10-10moles.cm-2.s-1; nevertheless, in general, compared to pristine nanopore, the

360

defective nanopore provides much higher flux. The maximum helium flux is obtained for the

361

double side oxygen variant, and is of the order of 107 moles cm-2 s-1, making this the most

362

attractive from all respects.

363

364

CONCLUSION

365

It is demonstrated here that defective nanopore structures will significantly enhance the selective

366

transmission of 3He. The inclusion of STW defects and the removal of two pentagon rings in the

367

nanopore considerably reduce the tunneling barrier height of helium, without reducing the van

368

der Waals minimum. Particularly, double side functionalization decreases the barrier height more

369

than single side functionalization of the rim of a nanopore created in graphene. Among the

370

various nanopores considered, the double side oxygen variant shows the least barrier height. It is

371

evident from the transmission probability curve that selective transmission of 3He is larger than

372

that of 4Hein the low kinetic energy region, whereas 4He is marginally preferred in the high

373

kinetic energy region. The temperature dependent transmission values indicate that at low

374

temperatures the helium particles are lying on the low kinetic energy region, which leads to their

375

the selective transmission. Among the doped defective graphene nanopores investigated, the

376

double side oxygen variant provides relatively high transmission in the range of 10-3, whereas the

377

double side nitrogen variant shows very high 3He selectivity of 669. Moreover, very high flux

378

values are observed for the pore rim modified structures, of the order of 107mole.cm-2.s-1. A key

379

result from this study is that graphene nanopores accompanied by STW defects provide

380

significantly high flux and selective transmission of 3Heat low temperatures. This should prove

381

to be an efficient technique for 3He/4He separation, as the conventional methods currently used 17 ACS Paragon Plus Environment


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382

are energetically expensive, while the difference in the de Broglie wavelength of the He isotopes

383

is too small for the quantum molecular sieving concept proposed for D2/H2 separation based on

384

either kinetic28-32,55,56 or equilibriumsieving57,58.

Page 18 of 32

385

386

387

SUPPORTING INFORMATION

388

Tables containing the thermally weighted transmission values and selectivity ratio, illustration of

389

pore size of the nanopores obtained from the electron density isosurface, f(E,T) curves for

390

varying energies at different temperatures of defective nanopores, and f(E,T)t(E) curve for

391

double side oxygen passivated defective nanopore at 100 K are reported. This material is

392

available free of charge via the Internet at http://pubs.acs.org.

393 394

ACKNOWLEDGEMENT

395

This research has been supported by a grant (DP1092437) from the Australian Research Council

396

under the Discovery Scheme. One of us (LS) gratefully acknowledges an Indo-Australia Early

397

Career Science and Technology Visiting Fellowship supported by Indian National Science

398

Academy and Australian Academy of Sciences, enabling his visit to The University of

399

Queensland (UQ) during 2013-2014. M. Lalitha acknowledges financial support from UQ,

400

enabling her visit to UQ during the conduct of this research.

401

Notes

402

The authors declare no competing financial interest.


Page 19 of 32

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403

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TABLES

538 539 540

Page 22 of 32

Table 1: Pore size of the graphene nanostructures measured from the electron density isosurface. Pore radius (Å) 1.46 With STW defect Hydrogen passivated 1.48 Single side Nitrogen 1.60 Double side Nitrogen 1.72 Single side Oxygen 1.61 Double side Oxygen 1.64 Single side Fluorine 1.30 Double side Fluorine 1.16 Structure Pristine

541 542 543

Table 2: Helium tunneling barrier of graphene defective nanopores, and corresponding van der Waals minimum Structure Pristine Hydrogen passivated Single side Nitrogen Double side Nitrogen Single side Oxygen Double side Oxygen Single side Fluorine Double side Fluorine

Helium tunneling barrier x 1021 (J/atom) 9.0252 With STW defect 3.3572 4.8396 3.6667 3.3572 2.0056 20.8844 25.2444

544 545 546 547 548 549 550 551


van der Waals minimum x 1021(J/atom) -4.4036 -3.4880 -4.0548 -4.1550 -3.4008 -4.6652 -3.8368 -4.6652

Page 23 of 32

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552

List of Figure Captions

553 554

Figure 1. Graphene sheet with three STW defects. Carbon atoms within the encircled region are removed to create a nanopore.

555 556 557 558

Figure 2. Defective graphene nanopore functionalized with hydrogen, nitrogen and oxygen atoms. (a) Pristine, (b) Hydrogen passivated, (c) Single side nitrogen passivated (d)Double side nitrogen passivated, (e) Single side oxygen passivated (f) Double side oxygen passivated (g) Single side fluorine passivated and (h) Double side fluorine passivated.

559 560

Figure 3. Potentialbarrier curves for a single helium atom passing through defective graphene nanopore functionalized with (a) hydrogen, nitrogen and oxygen atoms, and (b) fluorine atoms.

561 562 563

Figure 4. Transmission probability curve for 3He and 4He, for (a) pristine nanopore, and for defective graphene nanopore passivated with (b) hydrogen, and pore rim passivated with (c) single side nitrogen, (d) double side nitrogen, (e) single side oxygen, and (f) double side oxygen.

564 565

Figure 5. Temperature variation of thermally weighted transmission of 3He and 4He for pristine and functionalized defective nanoporous graphene.

566 567

Figure 6. Temperature variation of 3He/4He selectivity for pristine and functionalized defective nanoporous graphene.

568 569

Figure 7. Temperature variation of 3He flux for pristine and functionalized defective nanoporous graphene.

570 571 572 573 574 575 576 577 578



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Page 24 of 32

579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596

Figure 1. Graphene sheet with three STW defects. Carbon atoms within the encircled region are removed to create a nanopore.

597


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598 599 600 601 602

603

(a) Pristine

(b) Hydrogen passivated

604 605 606 607 608 609 610

(c) Single sidenitrogen passivated

(d)Double side nitrogen passivated

611 612 613 614

615

(e) Single side oxygen passivated (f) Double side oxygen passivated



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Page 26 of 32

616 617 618 619 620 621

(g) Single side fluorine passivated (h) Double side fluorine passivated Figure 2. Defective graphene nanopore functionalized with hydrogen, nitrogen and oxygen atoms. (a) Pristine, (b) Hydrogen passivated, (c) Single side nitrogen passivated (d)Double side nitrogen passivated, (e) Single side oxygen passivated (f) Double side oxygen passivated (g) Single side fluorine passivated and (h) Double side fluorine passivated.

622 623


Page 27 of 32

10 energy x 10 (J/He)

Pristine Hydrogen Single side nitrogen Double side nitrogen Single side oxygen Double side oxygen

(a)

8 6

21

4 2 0 -2 -4 -6 0

2

4

6 8 10 12 14 16 18 distance (Å)

624

energy x 10 (J/He)

30

21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


25

(b)

Single side fluorine Double side fluorine

20 15 10 5 0 -5

-10 0 625 626 627

2

4

6 8 10 12 14 16 18 20 distance (Å)

Figure 3. Potentialbarrier curves for a single helium atom passing through defective graphene nanopore functionalized with (a) hydrogen, nitrogen and oxygen atoms, and (b) fluorine atoms.

628 6 ACS Paragon Plus Environment


1.0

1.0

629

(b) Hydrogen passivated (a) Pristine

0.8

631

0.6

0.6

632

t(E)

0.8 t(E)

630

3He 4He

0.4

3

0.4

633

4

0.2

0.2

0.0

0.0 3.5

He He

634 635

7

636

8

9 10 11 energy x 1021(J/He)

12

4.0

4.5 5.0 5.5 21 energy x 10 (J/He)

6.0

1.0

1.0

(d) D-nitrogen passivated

(c) S-nitrogen passivated

637

0.8

0.8 0.6

3He 4He

0.4

640 641

0.2

642

0.0

t(E)

639

t(E)

638

0.4

7

4

644

10

(f) D-Oxygen passivated

(e) S-oxygen passivated

0.8

0.8 0.6

t(E)

3He 4He

0.6

647

0.4

0.4

648

0.2

0.2

649

0.0

650

6 8 21 energy x 10 (J/He)

1.0

1.0 645

3He 4 He

0.0 4 5 6 21 energy x 10 (J/He)

643

646

0.6

0.2

3

t(E)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 32

3He 4He

0.0 2

3

4 5 6 21 energy x 10 (J/He)

7

1

2 3 4 21 energy x10 (J/He)

651 652 653 654

Figure 4. Transmission probability curve for 3He and 4He, for (a) pristine nanopore, and for defective graphene nanopore passivated with (b) hydrogen, and pore rim passivated with (c) single side nitrogen, (d) double side nitrogen, (e) single side oxygen, and (f) double side oxygen. 7 ACS Paragon Plus Environment

5

Page 29 of 32

655 656 657 658 659 660 661 662 663 664 665 666 667 668 669

transmission

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 10-13 10-14 10-15 10-16 10-17 10-18 10-19 10-20 10-21 10-22 10-23 10-24 10-25 10

3

He: D-oxygen He: D-oxygen 4 He: S-oxygen 4 He: S-oxygen 3 He: D-nitrogen 4 He: D-nitrogen 3 He: S-nitrogen 4 He: S-nitrogen 3 He: Pristine 4 He: Pristine 3 He: Hydrogen 4 He: Hydrogen 4

15

20

25

30

35

40

temperature (K)

670 671 672

Figure 5. Temperature variation of thermally weighted transmission of 3He and 4He for pristine and functionalized defective nanoporous graphene.

673 674



675 676

1000

677 678

Pristine Hydrogen S-Nitrogen D-Nitrogen S-Oxygen D-Oxygen

679 680 681 682 683

selectivity

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 32

100

10

684 685 686 687 688

1 10

20

30

40

50

temperature (K)

689 690 691 692 693

Figure 6. Temperature variation of 3He/4He selectivity for pristine and functionalized defective nanoporous graphene.

694 695 696 697 698 699 700 9 ACS Paragon Plus Environment

Page 31 of 32

701 702 703 704

706

-1

707

-2

He flux (moles.cm .s )

705

708 709 710 711 712 713 714 715 716 717 718

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1010 109 108 107 106 105 104 103 102 101 100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 0

Pristine Hydrogen S-Nitrogen D-Nitrogen S-Oxygen D-Oxygen

20

40

60

80

100

temperature (K) Figure 7. Temperature variation of 3He flux for pristine and functionalized defective nanoporous graphene.

719 720 721 722 723 724 725 726 10 ACS Paragon Plus Environment


727

TABLE OF CONTENTS GRAPHIC

728 729 100 730 10-1 10-2 10-3 731 10-4 10-5 10-6 732 10-7 10-8 10-9 733 10-10 10-11 10-12 734 10-13 10-14 10-15 10-16735 10-17 10-18 10-19736 10-20 10-21 10-22737 10-23 10-24 10-25738 10

transmission

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 32

d 2ψ 2m + [ E − V ( x )]ψ = 0 dx 2 h 2

V(x)

3

He: He: He: 4 He: 3 He: 4 He: 3 He: 4 He: 3 He: 4 He: 3 He: 4 He: 4 4

15

739 740 741 742 743 744 745 746 747


20

25

30

temperature (K)

D-oxygen D-oxygen S-oxygen S-oxygen D-nitrogen D-nitrogen S-nitrogen S-nitrogen Pristine Pristine Hydrogen Hydrogen

35

40

Defect-Mediated Reduction in Barrier for Helium Tunneling through

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