# Theoretical design of a strain$-$controlled nanoporous CN membrane for
helium separation

Yong-Chao Rao, Zhao-Qin Chu, Xiao Gu  Xiang-Mei Duan∗ Department of Physics,
Faculty of Science, Ningbo University, Ningbo-315211, P.R. China
<EMAIL_ADDRESS>
###### Abstract

Designing an efficient membrane for He purification is quite crucial in
scientific and industrial applications. Ultrathin membranes with intrinsic
pores are highly desirable for gas purification because of their controllable
aperture and homogeneous hole distribution. Based on the first$-$principles
density function theory and molecular dynamics simulations, we demonstrate
that the compressively strained graphitic carbon nitride (CN) can effectively
purify He from Ne and Ar. Under a $-6$% strain, the CN monolayer with a
suitable pore size presents an easily surmountable barrier for He ($0.11$ eV)
but formidable for Ne ($0.51$ eV) and Ar ($2.45$ eV) passing through the
membrane, and it exhibits exceptionally high selectivity of $5.17\times
10^{6}$ for He/Ne and $1.89\times 10^{39}$ for He/Ar, as well as excellent He
permeance of $1.94\times 10^{7}$ GPU at room temperature, superior to those of
porous graphene and C2N membrane. Our results confirm that strain$-$tuned CN
membrane could be potentially utilized for He separating from other noble
gases.

###### keywords:

CN membrane; Helium separation; Selectivity and permeance; First$-$principles
calculations; Molecular dynamics simulations

††journal: Journal of LaTeX Templates

##  1 Introduction

With various properties, such as lower density than air, incombustibility and
low index of refraction, the lightest noble gas, helium, has been extensively
applied in the fields of semiconductors, airships and solar telescopes.[1, 2]
However, He immediately drifts up during the extraction from the natural gas,
resulting in its irreversibly losing on earth. And the traditional physical
methods of producing He, like cryogenic distillation or pressure$-$swing
adsorption of natural gas, is hard to obtain high enough concentration to meet
the commercial utilization of He.[3] Due to the low cost and easy operation,
membrane separation technology has been increasingly used in gas separation
and sewage purification.[4] The gas permeability are quite sensitive to the
thickness of membrane materials.[5, 6, 7] Therefore, the ideal membrane should
be as thin as possible to receive maximum flux, mechanically robust to prevent
cracking, and with well$-$defined pore sizes to increase selectivity. It is
highly desirable to design the efficient He separation membrane having atomic
thickness and sub$-$nanometer pores.

The single atomic thickness makes two$-$dimensional (2D) materials ideal for
gas separation compared to conventional membranes.[8, 9] For gas purification,
the hole size of the material plays a dominated role in determining the
selectivity and permeability. Although the introduction of vacancy defects on
the pristine graphene, silicene and germanene ecane which are impermeable to
gas molecules, could provide a suitable pore size for gas separation.[10, 11,
12] However, it remains difficulties in controlling the size and number of
pores experimentally.[13] Excitingly, some 2D porous films with uniform
sub$-$nanopores, such as graphitic carbon nitride (C3N4,[14] C2N[8]) and
stanene,[7] can act as natural molecular sieves and have great potential for
future He purification. To meet the increasing demand of He, researchers are
focusing on designing more efficient and accurate molecular sieves. Based on
the characteristics that porous membranes are sensitive to mechanical strain,
the multi$-$stage gas separation approach by strain$-$modulating pore size has
attracted great interest.[15, 16, 17]

As one of the graphitic carbon nitride family, the experimentally prepared
porous CN has fascinated considerable attention owing to its potential
applications in energy storage and catalysts.[18, 19, 20] CN possesses
honeycomb$-$like structure similar to graphene but with six$-$membered
carbon$-$nitride ring in units [see Fig. 1 (a)], which makes it a candidate
for gas separation. As reported, the stability of CN was confirmed by
calculating the phonon dispersion spectrum, and CN membrane could separate H2
from mixed composition including CO, N2 and CH4.[21] To the best of our
knowledge, research on the utilization of CN for efficient He purification is
still lacking.

In this work, we propose that the CN sheet, under a biaxial
compressive$-$strain of 6%, is a good candidate for He separation. The
microscopic permeation processes of He, Ne and Ar are discussed in terms of
minimum energy passway, energy profile, and electron density isosurface. In
addition, the diffusion and separation capacity are judged from permeance and
selectivity at room temperature.

##  2 Computational details

The first$-$principles density functional theory (DFT) calculations are
performed to optimize the structure of the porous CN, describe the electron
density isosurfaces for noble gas molecules interacting with the porous CN
monolayer, and carry out the energy barrier of noble gases permeating through
CN membrane by using the VASP package.[22] The perdew$-$Burke$-$Ernzerhof
(PBE) functional[23] under the generalized gradient approximation (GGA) with
van der Waals correction proposed by Grimme (DFT$-$D2)[24] is employed by the
spin$-$unrestricted all$-$electron DFT calculations. The electron
wave$-$functions are expanded by plane waves with cut$-$off energies of 500
eV, and the convergence criteria for electronic and ionic interactions during
structure relaxation are set to be 10-4 eV and 0.01 eV, respectively. The
Monkhorst$-$Pack meshes of $7\times 7\times 1$ are used in sampling the
Brillouin zone for $2\times 2$ supercells of CN, and a 20 Å vacuum thickness
is introduced to avoid interlayer interactions. For the transition state
calculations, we have performed minimum energy path profiling using the
climbing image nudged elastic band (CI$-$NEB) method[25] as implemented in the
VASP transition state tools.

At the various temperatures, the He permeance of porous CN monolayer are
investigated using MD simulations in the NVT ensemble, where the temperature
of the system is controlled by the Andersen thermostat method. A
condensed$-$phase optimized molecular potential for atomistic simulation
studies (COMPASS) in the Material Studio software is used for describing the
interatomic interaction.[26]

##  3 Results and discussion

###  3.1 Pore size and stability of CN nano$-$sheet

Figure 1(a) presents a top view of a fully relaxed $2\times 2$ CN supercell.
The C$-$C and C$-$N bond lengths are 1.51 and 1.34 Å, respectively, and the
optimized lattice parameter is 7.12 Å, consistent with previous
literature.[20] The hole size, with a diameter of 5.47 Å, are illustrated by
the dashed interior circle. The electron density isosurface plot, shown in
Fig. 1(b), indicates that the pore size of CN monolayer is characterized by
the effective diameter of the inscribed circle, 3.28 Å, which is much larger
than the kinetic diameters of He (2.6 Å) and a little bit larger than that of
Ne molecules (3.2 Å).[27] Therefore, it offers the possibility to reduce the
pore width in-between kinetic diameters of He and Ne molecules and makes CN
sheet as an efficient He$-$isolating membrane. The biaxial compressive strain
is defined as $\varepsilon=(\iota-\iota_{\rm 0})/\iota_{\rm 0}\times 100\%$ ,
where $\iota$ and $\iota_{\rm 0}$ are the strained and original lengths of the
monolayer along one of strain direction (the sample is compressed in two
distinct directions with the same strain), respectively.

We investigate the He separation performance in the absence/presence of
compressive strain on CN monolayer. Under compressive strain, the structural
stability of CN is a key issue. As presented in Fig. 1(c), CN can sustain a
biaxial compressive strain up to $-$24%, which means that the structure of the
membrane is robust under a small compression. Then, we figure out the cohesive
energies of CN membrane under different strains. The cohesive energy is
expressed by:[28]

$E_{\rm{coh}}=(n_{\rm C}E_{\rm C}+n_{\rm N}E_{\rm N}-E_{\rm CN})/(n_{\rm
C}+n_{\rm N})$ (1)

Where $E_{\rm C}$, $E_{\rm N}$ and $E_{\rm CN}$ are the energy of single C
atom, single N atom, total CN membrane, respectively; and $n_{\rm C}$, $n_{\rm
N}$ are the total number of C and N atoms. Fig. 1(c) shows that $E_{\rm coh}$
decreases with the increasing strain, while $E_{\rm coh}$ under 6% compressive
strain is 6.05 eV per atom, which is still much lower than that of silicene
(3.17 eV per atom).[29] And the strain energy of the system ($E_{\rm s}$, the
energy difference between strained and equilibrium states), which actually
equals to $E_{\rm{s}}=-[E_{\rm coh}(\varepsilon_{i})-E_{\rm
coh}(\varepsilon_{0})]\times(n_{\rm C}+n_{\rm N})$, increases with strain
monotonically until the strain reaches the critical value of $-$24%. Note that
silicene can be successfully used as gas separation membrane,[30] therefore CN
under moderate mechanical strain is stable enough for He separation.

###  3.2 Diffusion energy barrier of the noble molecule

Before exploring the transition state and energy barrier, we first optimize
the energetically most stable state (SS) of gas molecules on the surface of CN
monolayer. The atomic structures of He atom adsorbed on CN surface with and
without strain are presented in Fig. 2. The interaction energy between a noble
gas molecule and CN membrane is defined by:

$E_{\rm{int}}=E_{\rm{gas/CN}}-E_{\rm gas}-E_{\rm CN}$ (2)

Where $E_{\rm{gas/CN}}$, $E_{\rm gas}$ and $E_{\rm CN}$ represent the total
enegy of gas/CN system, isolated gas molecule and pure CN monolayer,
respectively. The interaction energy and adsorption height of the SS under
different compressive strains are summarized in Table 1. The interaction
energy and adsorption height are in the range of $-$6.9 to $-$25.5 meV and
1.05 to 2.61 Å, respectively, indicating the gas molecules are all physically
adsorbed on the CN monolayer via weak van der Waals interaction. The magnitude
of adsorption energy is comparable to those of the noble gases adsorbed on
C3N4 ($-$17 to $-$99 meV) and C2N sheet ($-$60 to $-$90 meV). The general
trend is that as the compressive strain increases, the reduction in pore size
would enhance the repulsion interaction between the gas molecules and the CN
membrane, leading to an increase in molecular adsorption height. Specifically,
Ar has the highest adsorption height, which is due to the relatively large
kinetic diameter.

When the gas migrates from the most stable adsorption site to the CN film, due
to the symmetry, the transition state (TS) should be which the gas molecules
are at the center of the cavity and in the same plane as the film. The TS
state is confirmed using the NEB approach. The energy barrier (Eb) is defined
as:

$E_{\rm{b}}=E_{\rm TS}-E_{\rm SS}$ (3)

Where $E_{\rm TS}$ and $E_{\rm SS}$ represents the energy between gas
molecules and CN membrane at the TS and SS, respectively. The energy profiles
and barriers of He, Ne and Ar passing through the pore under different
compressive strain are shown in Fig. 3. The barrier for He, Ne and Ar passing
through the pristine CN is 0.03, 0.18 and 1.13 eV, respectively. Clearly, the
relatively low penetration barrier for both He and Ne make it impracticable to
purify He via unstrained CN membrane. Fortunately, the energy barrier
increases as the compressive strain is exerted. In particular, when the strain
is reached to $-$6%, the barrier for He, Ne and Ar is further increased to
0.11, 0.51 and 2.45 eV, respectively. The unexpectedly lower penetration
barrier for He is within the threshold barrier for gas penetration (0.50
eV),[31] while the values for the other two noble gases are lager than 0.50
eV, suggesting that the He separation performance of CN can be significantly
improved by inducing a 6% compressive strain.

To deeply understand of the change of energy barrier when the molecules pass
strain$-$modified CN film, we show the electron density isosurface of noble
gases interacting with CN at TS under $-$6% strain. Intuitively, large
electron density overlap would induce strong repulsion which hinder the
movement of gas molecules through the pore. From Fig. 4(a), it can be seen
that there is no overlap between the electron density of He and the CN,
therefore He could easily move through the membrane. However, the electron
density overlap increases for Ne and Ar, as presented in Fig. 4(b) and (c),
the Ar molecule and the porous CN sheet completely overlap in electron
density, so the Ar gas molecules cannot pass through the film. The effective
diameter of strained CN [shown in the inscribed circle of Fig. 4(a)] is now
2.99 Å, which is between the kinetic diameter of He and Ne. Hence the
compressed membrane only allows He to penetrate.

The He separation efficiency of the porous CN membrane can be quantitatively
examined through the Arrhenius$-$equation.[32] The selectivity (S) for He over
other two noble gas molecules is estimated by:

$S_{\rm{He/gas}}=\frac{{\gamma}_{\rm He}}{{\gamma}_{\rm gas}}=\frac{A_{\rm
He}exp(-E_{\rm He}/RT)}{A_{\rm gas}exp(-E_{\rm gas}/RT)}$ (4)

Where $\gamma$ is the diffusion rate, the diffusion prefactor, $A$, is set to
$10^{11}~{}s^{-1}$,[27] $R$ and $T$ are the Boltzmann constant and the
absolute temperature, respectively. $E$ is the diffusion energy barrier. The
temperature$-$dependent diffusion rates of noble gas molecules and He
selectivity relative to Ne and Ar are depicted in Fig. 5(a) and (b),
respectively. Generally, with the temperature increasing, the diffusion rate
of the molecules increases, while the He selectivity of CN membrane relative
to Ne and Ar decreases. The main reason is that, when the temperature rises,
the kinetic energy of the inert gas molecule is greatly increased, make it
possible to overcome the energy barriers and pass the CN film. Significantly,
at a certain temperature, the enhanced compressive strain reduces down the He
diffusion rate [Fig. 5(a)] but greatly increases the selectivity for He/Ne and
He/Ar [Fig. 5(b)], which demonstrates the appropriate compressive strain can
effectively accelerate the separation of He from Ne and Ar. As summarized in
Table 2, at room temperature, the selectivity for He/Ne and He/Ar are about
$5.17\times 10^{6}$ and $1.89\times 10^{39}$ by subjecting to a $-$6% strain,
respectively. The values are much superior than that of other theoretical
results and surely acceptable for the industrial application.[8, 30, 33]

###  3.3 He separation by MD simulations

Permeability is another critical criterion to characterize the separation
performance of CN membrane. Based on MD simulations, we investigate the gas
flow to estimate the permeability of CN membrane quantitatively at room
temperature, which is defined as:[34]

$F=\frac{\nu}{S\times{t}\times{\Delta{P}}}$ (5)

Where $\nu$ and $S$ represent the moles of gas molecules in the permeate side
and the area of CN membrane, $t$ is the time duration, and the pressure drop
($\Delta{P}$) is set to 1 bar across the pore.

As shown in the Fig. 6, the $4\times 4$ supercell with a height of 90 Å is
divided into three parts of the same capacity. Periodic boundary conditions
along the lateral directions are imposed to mimic the infinite film. At the
beginning of each simulation, the gas reservoir, consisting of 40 He
molecules, 40 Ne molecules and 40 Ar molecules, is initially located in the
middle part of the supercell, that is, between the two monolayers of CN. The
van der Waals interactions and Ewald electrostatic interactions are applied
with a cutoff distance of 9.5 Å. Each simulation is carried out for a time
period of 10 $ns$ with a time$-$step of 1 $fs$. In the process of simulations,
due to the existing of difference in gas concentration among different spaces,
the molecules are pushed cross the sheet into the vacuum region. After 10 $ns$
simulation, the final configurations under 0%, $-$3% and $-$6% strain at room
temperature are shown in Fig. 6. Some of Ne atoms pass through the pristine CN
membrane [see Fig. 6(a)], and there are still several Ne atoms penetrate the
membrane under a small strain of $-$3% [see Fig. 6(b)], which are consistent
with the low penetration barriers in Fig. 3(a) and (b). However, under a $-$6%
strain [see Fig. 6(c)], 27 He in 40 molecules pass through CN film to the
vacuum space, without any Ne and Ar penetrating. The MD simulations further
confirm that the strain$-$controlled nanoporous CN membrane can be applied in
the purification of He from other noble gas molecules.

The calculated He permeance of porous CN monolayer under $-$6% strain with
area of $28.48\times 28.48$ Å2 together with that of the previously proposed
porous monolayer are summarized in Table 3. The strain$-$modified CN membrane
exhibits a He permeability as high as $1.94\times 10^{7}$ GPU at 300 $K$,
which is almost twice that of C2N, and 270 times greater than that of the
porous graphene (PG), far much higher than the industrially acceptable gas
separation value of 20.[33]

##  4 Conclusions

Combining DFT$-$D2 calculations and classical MD simulation, we predict that
strained porous CN monolayer can be used for He separation with high
selectivity and permeability. The pristine CN is permeable for both He and Ne
because of the lower barriers than threshold value. The energy barriers of gas
molecules can be effectively modulated by biaxial compressive strain to the
membrane. Subjecting the CN membrane to a $-$6% strain results in
non$-$passability of Ne and exceptionally high selectivity for He/Ne (Ar) at
300 K. Meanwhile, the MD simulations further verify that the strained CN
membrane exhibits a high He permeance of $1.94\times 10^{7}$ GPU, which is
superior to those of PG and C2N sheets. Our results demonstrate that strain is
an efficient strategy to tune the separation performance of low$-$dimensional
materials and reveal that strain$-$controlled CN monolayer has great potential
application in He separation.

## Acknowledgments

This research is supported by the Natural Science Foundation of China (grant
No. 11574167), the New Century 151 Talents Project of Zhejiang Province and
the KC Wong Magna Foundation in Ningbo University.

## Conflict of interest

The authors declare they have no conflict of interest

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Figure 1: (a) Fully optimized $2\times 2$ supercell of CN sheet. (b) Electron density isosurface of CN sheet with an isovalue of 0.015 e/Å3. The brown and blue balls represent the C and N atoms, respectively. (c) The stress (black triangles) and the cohesive energy (blue squares) as a function of strain. The long dashed lines between the symbols are guides to the eyes. The vertical short dashed line indicates the critical strain. Figure 2: The geometrical structures of He adsorbed on CN sheet in the absence and presence of strain. The yellow, brown and blue balls represent He, C and N atoms, respectively. Figure 3: Energy profiles for He, Ne and Ar penetrating through CN membrane under (a) 0%, (b) $-$3% and (c) $-$6% strain. Figure 4: Electron density isosurface for (a) He, (b) Ne and (c) Ar at transition state under $-$6% strain. The isovalue is 0.015 e/Å3. Figure 5: (a) Diffusion rate for the noble gas molecules, and (b) the selectivity of He relative to Ne and Ar molecules, as a function of temperature. Figure 6: Final configuration of the mixed gases permeating through the CN membrane at room temperature under a strain of (a) 0%, (b) $-$3% and (c) $-$6%, respectively. Color code: yellow, He; pink, Ne; green, Ar.  Table 1:  Kinetic diameter D0 (Å) of the gas molecules, the interaction energy $E_{\rm int}$ (meV) and the adsorption height H0 ( Å) of CN membrane with gas molecules. |  D0 |  Strain |  $E_{\rm{int}}$ |  H0
---|---|---|---|---
|  |  0% |  $-$9.5 |  1.05
He |  2.6 |  $-$3% |  $-$18.0 |  1.26
|  |  $-$6% |  $-$25.5 |  1.33
|  |  0% |  $-$7.1 |  1.64
Ne |  3.2 |  $-$3% |  $-$17.6 |  1.81
|  |  $-$6% |  $-$24.8 |  1.92
|  |  0% |  $-$6.9 |  2.42
Ar |  3.4 |  $-$3% |  $-$16.3 |  2.56
|  |  $-$6% |  $-$24.0 |  2.61
Table 2:  The selectivity (S) of He toward Ne and Ar penetrating through CN membrane under $-$6% strain at room temperature. The results of previously studied membranes C2N and silicene are also included for comparison. |  CN |  C2N (Ref.[8]) |  Silicene (Ref.[30])
---|---|---|---
S(He/Ne) |  $5.17\times 10^{6}$ |  $3\times 10^{3}$ |  $2\times 10^{3}$
S(He/Ar) |  $1.89\times 10^{39}$ |  $4\times 10^{18}$ |  $1\times 10^{18}$
Table 3:  The He permeance (GPU) of CN monolayer under $-$6% strain at 300 K, and the comparison with those of porous graphene (PG) and C2N. $\left[1~{}{\rm GPU}=3.35\times 10^{-10}~{}mol/(m^{2}\cdot s\cdot Pa)\right]$ |  CN |  C2N (Ref.[8]) |  PG (Ref.[35])
---|---|---|---
Permeance |  $1.94\times 10^{7}$ |  $1\times 10^{7}$ |  $7\times 10^{4}$