Trapping of volatile ssion products by C
Navaratnarajah Kuganathan
gens and noble gases [1]. Among these, species such as iodine and
toxic impact [2]. To minimize the hazard, radioactive species should
species [3]. However, other porous materials including zeolites,
), consists of 60 carbon atoms and has been widely
studied [6]. The interior and exterior of a C
molecule are sur-
rounded by an electron charge density of conjugate
bonds each
formed between two adjacent carbon atoms. The charge density of
the inner space is therefore higher than the outer surface. The
encapsulation of species within C
has been described in various
experimental and theoretical studies: alkali metals (e.g. Li, K) [7,8],
alkali earth metals (e.g. Ca, Sr and Ba) [7,8], radioactive isotopes (e.g.
Tb) [9,10], actinide metals (e.g. Lr) [11], non-metals (e.g. N,
P) [12,13], noble gases (e.g. Ne, Ar) [14,15], transition metals (e.g. Tc,
Fe) [16,17] and rare earth metals (e.g. La) [18]. These have all been
considered either in the form of atoms or clusters. The external
surface of C
has been studied theoretically for its interaction with
transition metal atoms and clusters, and shown to provide a good
catalyst for the activation of di-nitrogen molecules [19] and the
storage of hydrogen [20]. Furthermore, the theoretical study by
Ozdamar et al. [21] demonstrated the stability and electronic
properties of Pt and Pd in the form of atoms and dimers interacting
with the outer surface of C
. Likewise, hydrogen absorption has
* Corresponding author.
E-mail address: (N. Kuganathan).
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Carbon 132 (2018) 477e485
been considered on the surface of Pd-decorated C
by El Mahdy
[22]. Thus, C
provides different sites to capture species both
internally (endohedral) and externally (exohedral).
This study uses density functional theory (DFT) together with
dispersion corrections, to investigate the thermodynamical stabil-
ity of gaseous volatile ssion products (Kr, Xe, Br, I, Rb, Cs and Te) in
order to understand whether they can be trapped by lters that
incorporate C
. For those species that may become bound to C
calculations have been performed to establish whether there is a
preference for positions inside or on the surface of the fullerene
cages and whether they become bound as atoms or in the form of
dimers. DFT calculations, in addition to giving structural informa-
tion, provide data on electronic properties, which are also
2. Computational methods
Calculations were carried out using the spin-polarized mode of
DFT as implemented in the VASP [23,24] package. The exchange-
correlation term was modelled using the generalized gradient
approximation (GGA) parameterized by Perdew, Burke, and Ern-
zerhof (PBE) [25]. In all cases, we have used a plane-wave basis set
with a cut-off value of 500 eV. Structural optimisations were per-
formed using a conjugate gradient algorithm [26] and the forces on
the atoms were obtained via the Hellman-Feynmann theorem
including Pulay corrections. In all optimized structures, forces on
the atoms were smaller than 0.001 eV/Å and all the values in the
atomic stress tensor were less than 0.0 02 GPa. All calculations were
performed without using symmetry restrictions. All geometry op-
timisations were performed using a single k-point. To calculate the
density of states (DOS), a 4 4 4 Monkhorst-Pack [27] k point
mesh, which contains 36 k points was employed. A cubic super cell
with a length of 25 Å was used for all congurations to ensure that
adjacent structures do not interact (the largest linear dimension of
molecule was 7.10 Å). We dene encapsulation energy (for
singe atom or dimer trapping inside the cage) and association en-
ergy (for single atom or dimer trapping outside the cage) by the
following equation:
¼ E (FP-C
) eE(C
) en E (FP) (1)
where E (C
) is the total energy for the isolated C
molecule, E (FP-
) is the total energy of the gaseous atom or atoms occupying the
centre of the C
cage or interacting on the surface of C
, E (FP) is
the total energy of an isolated ssion product (the reference state)
and n is the number of ssion product atoms considered in the
The inclusion of van der Waals (vdW) forces are particularly
important for the interactions of the highly polarizable noble gases
and transition metal atoms with C
. Here, dispersion has been
included by using the pair-wise force eld as implemented by
Grimme et al. [28] in the VASP package.
3. Results and discussion
3.1. Structural and electronic properties of C
fullerene molecule
fullerene molecule is spherical with truncated icosahedral
) symmetry. It consists of 12 pentagonal rings, 20 hexagonal rings
and 32 faces as shown in Fig. 1a. There are two distinct alternate
CeC and C¼C bonds present in the molecule with the experimental
bond lengths of 1.43 Å and 1.39 Å respectively [29]. In order to
validate the computational parameters used in this study, we car-
ried out an energy minimisation calculation to optimize the
structure of a C
molecule. In the relaxed structure, we calculated
the equilibrium bond distances, energy gap between the highest
occupied molecular orbital (HOMO) and the lowest unoccupied
molecular orbital (LUMO) and the net magnetic moment, to
compare with experimental and other theoretical values.
The calculated bond lengths of CeC and C¼C were 1.45 Å and
1.40 Å respectively, which agree well with experimental values [29]
and other theoretical calculations [22]. The calculated HOMO-
LUMO gap of the C
molecule is 1.55 eV, in good agreement with
the value of 1.64 eV calculated by the DFT calculations of Goclon
et al. [30]. Our calculation shows that the C
molecule is non-
magnetic (see Fig. 1b) as predicted in previous studies [31].
3.2. Initial congurations
We considered six different sites for the absorption of ssion
products (FP). In the rst conguration, the FP occupies the center
of the cage [endohedral C
(E)], as shown in Fig. 2 (a). In terms of
surface sites, there are ve possible initial positions as shown in
Fig. 2 (b) e (f). These are: (b) on top of the center hexagonal ring (H),
(c) on top of the center pentagonal ring (P), (d) on top of a C-C bond
between two hexagonal rings (66), (e) on top of a C-C bond be-
tween an hexagonal ring and a pentagonal ring (65) and (f) on top
of a C atom on the C
cage (C).
3.3. Formation of single gaseous ssion products interacting with
First we consider the stability of the ssion products (Xe, Kr, Br, I,
Cs, Rb and Te) as single atoms occupying the centre of the C
In relaxed congurations, the positions of the ssion product atoms
remain very close to the centre of the cage. The encapsulation en-
ergy for the ssion products within the C
cage depends on the
relative ionization potentials and electron afnities of the atom and
the C
molecule but also the size of the atom or ion relative to the
size of the internal volume of C
. Fig. 3 (a) shows the calculated
ionization potentials and the electron afnities of the ssion
product atoms and C
together with values reported in the data
book [32]. Though the trend agrees well with the values reported in
the data book, the current DFT calculations underestimate the rst
ionization potentials and overestimate the electron afnities. This is
because of the ionization energies and electron afnities calculated
from DFT orbital energies are usually poorer than those of Koop-
mans theorem, depending on the exchange-correlation approxi-
mation employed [33,34].
Both Rb and Cs have lower ionization potentials and lower
electron afnities than the C
molecule. A lower ionization po-
tential enables the (easy) removal of the outermost electron from
Fig. 1. (a) Optimised structure of a C
molecule and (b) its density of state spectra. The
vertical dotted line, at ~ 4.10 eV, corresponds to the Fermi energy. (A colour version of
this gure can be viewed online.)
N. Kuganathan et al. / Carbon 132 (2018) 477e485478
Fig. 2. Initial congurations considered for the FP: (a) absorbed in C
endohedrally, (b) exohedrally adjacent to a six membered ring, (c) a ve membered ring, (d) a C-C bond
between two hexagons, (e) a C-C bond between an hexagon and a pentagon and (f) a C on the fullerene cage.
Fig. 3. (a) Calculated ionisation potential and electron afnity of volatile ssion product atoms and C
together with the values reported in the data book [32], (b) encapsulation
energies, (c) Bader charge on atoms, (d) magnetic moment of the supercell and (e), (f) and (g) the total DOS of Br, Rb and Cs encapsulated in C
respectively. (A colour version of this
gure can be viewed online.)
N. Kuganathan et al. / Carbon 132 (2018) 477e485 479
Rb and Cs which is gained by the C
molecule as it has a higher
electron afnity. The calculations show that the encapsulation en-
ergies of Rb and Cs are highly exothermic [see Fig. 3b] meaning that
they are more stable in the inner cage of C
than as isolated atoms.
Bader analysis also shows that both Rb and Cs gain a charge close
to þ1 by donating their outer s-electron to the C
cage. Cs and Rb
are therefore ions, which exhibit smaller radii than their corre-
sponding atoms. The resultant complex can be considered as a
donor-acceptor endohedral complex (M
). Wang et al. [7]have
calculated the formation energies for the endohedral encapsulation
of Rb and Cs using Hartree-Fock calculations together with the
Born-Haber cycle. In their calculations, they assumed that both
metals lose one electron to form a þ1 charge and C
gains that lost
electron to form 1 charge. The calculated formation energies by
Wang et al. [7] for Rb and Cs are 1.91 eV and 1.64 eV respectively,
in good agreement with our corresponding calculated values
of 1.82 eV and 1.86 eV though the trend is slightly reversed.
Figs. 1 and 3d report that the magnetic moment of C
is zero
before and after the incorporation of Rb and Cs, indicating that
encapsulation does not affect the magnetism. Calculations indicate
that C
exhibits zero magnetic moment. As Rb
and Cs
zero magnetic moment, overall complexes Rb
and Cs
thus also non-magnetic. The total DOS for the Rb and Cs encapsu-
lated in C
are shown in Fig. 3f and (g) respectively. The encap-
sulation of Rb and Cs shift their DOS towards higher energies and
the lowest unoccupied band of C
becomes occupied by the s
electron transferred to C
Turning now to the inert gasses, the outer electronic congu-
rations of Xe and Kr are unaffected by encapsulation. This is a
consequence of their higher ionization potentials. This is further
supported by the very low encapsulation energies and the Bader
charge. The encapsulation energies are a balance between attrac-
tive van der Waals forces of the polarisable atoms and Pauli
repulsion of the electron clouds. Since Xe is large the Pauli repul-
sion is greater than exhibited by Kr and so Xe exhibits a slightly
positive (unfavourable) encapsulation energy whereas for Kr the
energy is slightly favourable.
The electron afnities of Br and I are reected in the encapsu-
lation energies and the Bader charges shown in Fig. 3; the encap-
sulation energy of Br is much more favourable than that of I due to
the high electron afnity of Br but also because the smaller Br ts
better inside C
. The Bader charges on Br and I are 0.50 and 0.21
respectively. The outer electronic conguration of both Br and I are
with one unpaired electron. As C
and C
are non-magnetic
and both Br and I gain electronic charge from C
the overall
magnetic moment is reduced. The magnitude of the magnetic
moment is reected in the degree of the Bader charge. The total
DOS (see Fig. 3e) shows that incorporation of Br introduces addi-
tional bands in the HOMO of C
but the band gap is not much
The positive encapsulation energy for Te reveals that it is un-
stable inside the cage due to its larger size and its inability to donate
charge to C-C 66 single bonds. A small Bader charge and magnetic
moment show that its outer electronic conguration is not altered.
Next, we consider how ssion products can be trapped (or not)
by the outer surface of C
(i.e. the association energy). The
behaviour of ssion products occupying positions on the inner and
outer faces of the C
cage are different as the faces are convex and
concave respectively. As explained earlier, we consider ve possible
initial exohedral congurations. These have been relaxed to give
the nal congurations and association energies. The relative as-
sociation energies of each atom, reported in Table 1, are energies
relative to the most stable site (which therefore has zero relative
energy but not zero absolute association energy (see Fig. 4)).
The most stable exohedral position for Xe and Kr is on top of the
hexagonal ring (H). However, differences in energy compared with
the other sites are very small, a consequence of how unreactive are
the noble ssion product gas atoms. The shortest FP-C distance,
formation energies, Bader charge, magnetic moment of the super-
cell and the total DOS for the most stable congurations (identied
in Table 1) are plotted in Fig. 4. While the (absolute) association
energy is very small, it is negative due to the vdW interaction (see
Fig. 4 a). Both a zero Bader charge and magnetic moment validate
this conclusion. This is why the two noble gas atoms are almost
4.00 Å away from the surface of C
. Interestingly the exohedral
association energy for Kr is smaller than the encapsulation energy
because Kr only interacts on the surface with six C atoms while
inside C
it interacts with the whole molecule. For Xe the situation
is reversed: larger Xe is stable on the surface while its encapsula-
tion energy is positive due to the large Pauli repulsion (compare
values in Figs. 3 and 4).
Br and I show a preference to be on top of the C-C bond, site (66),
and gain a small charge (see Fig. 4c). That their magnetic moments
are closer to 1 and Bader charge small means that their s
conguration with one spin up electron is unaltered. The formation
energy for Br is 0.93 eV, a more favourable value than that for I
(0.56 eV). The association energy calculated for I by Kobayashi
et al. [35]is0.10 eV (without dispersion correction). This deviates
from our calculated value of 0.56 eV where a dispersion correc-
tion is included. The stronger absorption of Br is due to its higher
electron afnity compared to I and the shorter bond distance (C-Br
vs C-I). Compared to Br and I inside C
, less charge is transferred.
This can be due to the halide atoms inside the cage being adjacent
to and thus able to gain electron density from all the C atoms of C
However, inside the cage the bonding process is in opposition to
Pauli repulsion; this is enough that I is more stable outside C
Conversely exohedral absorption of smaller Br is less favourable
than inside the cage. Furthermore, with Br the Fermi level is shifted
slightly to higher energies and the DOS spectrum of C
is now
dispersed with additional peaks. This is due to the chemical inter-
action of Br with the C atoms.
The (66) position is the preferred site for the Te, which loses ~1
electron (see Bader charge in Fig. 4 c) due to its lower electron af-
nity than C
. Its formation energy is 1.39 eV showing strong
absorption by the C
surface, in contrast to