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Trapping of volatile ﬁssion products by C

Navaratnarajah Kuganathan

, Ashok K. Arya

, Michael J.D. Rushton

Robin W. Grimes

Department of Materials, Imperial College London, London, SW7 2AZ, UK

Material Science Division, Bhabha Atomic Research Centre, Trombay, Mumbai, 400 085, India

College of Physical and Applied Sciences, Bangor University, Bangor, Gwynedd, LL57, UK

article info

Article history:

Available online 27 February 2018

Carbon based ﬁlters provide important safety barriers that remove volatile ﬁssion products from gas

streams. The capacity and efﬁciency of a ﬁlter to trap ﬁssion products depends upon the strength of the

interaction between the ﬁssion products and the ﬁlter material. In this study, we apply density functional

theory together with a dispersion correction (DFT þ D) to predict structures and energies of volatile

ﬁssion product atoms and molecules trapped by buckminsterfullerene (C

). Endohedral encapsulation

energies and exohedral association energies show that Rb and Cs are strongly trapped as ions, each

transferring approximately one electron to C

. Kr and Xe are weakly trapped atoms with Xe showing a

preference for exohedral association and Kr for endohedral encapsulation. Br, I and Te, while strongly

trapped from atoms (and assuming charge from C

) are thermodynamically more stable as neutral

covalently bonded Br

and Te

molecules weakly trapped through van der Waals forces, exohedrally.

Heteronuclear CsBr and CsI were also considered. Both molecules were non-bonded to C

and Te

(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Gaseous efﬂuents, generated during processing of spent fuel,

contain radioactive volatile species including alkali metals, halo-

gens and noble gases [1]. Among these, species such as iodine and

strontium can be concentrated by the body, increasing the radio-

toxic impact [2]. To minimize the hazard, radioactive species should

be removed through either physical or chemical capture before the

efﬂuent is discharged to the atmosphere. Filters, in particular

activated carbon ﬁlters, are mostly used to remove gaseous iodine

species [3]. However, other porous materials including zeolites,

silica, alumina and metal organic frameworks have been studied

experimentally to improve the effectiveness of ﬁlters [3,4]. The

search for alternative ﬁlter media that can trap volatile gases con-

tinues. Such materials must be thermodynamically stable, chemi-

cally active, have adequate mechanical properties and a high

surface area, whilst still being manufacturable at scale in a cost-

effective manner.

The buckyball structured carbon fullerenes and fullerites are

candidate materials for trapping volatile ﬁssion products (FP). They

provide both inner and outer surface structures, high resistance to

external chemical attack, are light weight and exhibit mechanical

stability at high pressure and temperatures [5]. Buckminsterful-

lerene (C

), 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.

159

Gd,

161

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@imperial.ac.uk (N. Kuganathan).

Contents lists available at ScienceDirect

Carbon

journal homepage: www.elsevier.com/locate/carbon

https://doi.org/10.1016/j.carbon.2018.02.098

Carbon 132 (2018) 477e485

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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 ﬁssion products (Kr, Xe, Br, I, Rb, Cs and Te) in

order to understand whether they can be trapped by ﬁlters 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

discussed.

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 conﬁgurations to ensure that

adjacent structures do not interact (the largest linear dimension of

molecule was 7.10 Å). We deﬁne 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:

Enc/Assoc

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 ﬁssion product (the reference state)

and n is the number of ﬁssion product atoms considered in the

process.

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 ﬁeld 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 conﬁgurations

We considered six different sites for the absorption of ﬁssion

products (FP). In the ﬁrst conﬁguration, the FP occupies the center

of the cage [endohedral C

(E)], as shown in Fig. 2 (a). In terms of

surface sites, there are ﬁve possible initial positions as shown in

Fig. 2 (b) e (f). These are: (b) on top of the center hexagonal ring (H),

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 ﬁssion products interacting with

First we consider the stability of the ﬁssion products (Xe, Kr, Br, I,

Cs, Rb and Te) as single atoms occupying the centre of the C

cage.

In relaxed conﬁgurations, the positions of the ﬁssion product atoms

remain very close to the centre of the cage. The encapsulation en-

ergy for the ﬁssion products within the C

cage depends on the

relative ionization potentials and electron afﬁnities 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 afﬁnities of the ﬁssion

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 ﬁrst

ionization potentials and overestimate the electron afﬁnities. This is

because of the ionization energies and electron afﬁnities 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 afﬁnities 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 ﬁgure can be viewed online.)

N. Kuganathan et al. / Carbon 132 (2018) 477e485478

Fig. 2. Initial conﬁgurations considered for the FP: (a) absorbed in C

endohedrally, (b) exohedrally adjacent to a six membered ring, (c) a ﬁve 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 afﬁnity of volatile ﬁssion 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

ﬁgure 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 afﬁnity. 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

-C



). 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

exhibit

zero magnetic moment, overall complexes Rb

-C

and Cs

-C

are

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 conﬁgu-

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 afﬁnities of Br and I are reﬂected 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 afﬁnity of Br but also because the smaller Br ﬁts

better inside C

. The Bader charges on Br and I are 0.50 and 0.21

respectively. The outer electronic conﬁguration 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

60,

the overall

magnetic moment is reduced. The magnitude of the magnetic

moment is reﬂected 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

affected.

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 conﬁguration is not altered.

Next, we consider how ﬁssion products can be trapped (or not)

by the outer surface of C

(i.e. the association energy). The

behaviour of ﬁssion 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 ﬁve possible

initial exohedral conﬁgurations. These have been relaxed to give

the ﬁnal conﬁgurations 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 ﬁssion 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 conﬁgurations (identiﬁed

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

conﬁguration 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]is0.10 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 afﬁnity compared to I and the shorter bond distance (C-Br

v’s 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-

ﬁnity than C

. Its formation energy is 1.39 eV showing strong

absorption by the C

surface, in contrast to endohedral adsorption.

There is a signiﬁcant bonding interaction between Te and carbon in

the (66) position. This geometry of Te associated with (66) carbon

can be described as an icosahedral triangle with the C-Te bond

distance of 2.24 Å and C-C bond distance of 1.51 Å. The observed C-C

bond distance of 1.51 Å is 0.11 Å longer than the C-C distance pre-

dicted for the (66) positions in other parts of the relaxed C

conﬁguration. It is further evidence of the signiﬁcant interaction

with Te. The Bader charge approximation shows that Te has lost 1.11

|e| and the carbons in the (66) position have gained those electrons

with the ratio of 0.66:0.51. As the carbon in (66) position are now

four coordinated, the

interaction is reduced with the elonga-

tion in the C-C bond. However, Te is unstable inside the cage and

shows no interaction with C

as evidenced by the Bader change

and magnetic moment. This is due to the larger size of the Te and its

inability to donate charge to C-C 66 single bonds (in the relaxed

conﬁguration, Te occupies the center of the C

and the distance

between the Te and C-C 66 bond is 3.55 Å).

Rb and Cs exhibit strong association energies, essentially to the

same extent as they did for encapsulation. This is because the alkali

atoms have low ionization potentials and electron afﬁnities. Bader

analysis shows that both Rb and Cs form ~þ1 charge donating their

single s electrons to C

. The resultant supercells exhibit zero

magnetic moment. Fig. 4g shows there is a substantial Fermi energy

shift to the higher energy levels upon absorption of Cs due to

electron transfer to the C

cage. The calculated association energy

N. Kuganathan et al. / Carbon 132 (2018) 477e485480

of Cs derived by Kobayashi et al. [35]is1.85 eV including the

dispersion correction (without the dispersion correction the values

is 1.49 eV) which is in good agreement with our current DFT þ D

value of 1.96 eV, and shows the necessity of including dispersion.

Association energies calculated for of Rb and Cs by Sankar De et al.

[36]are1.59 eV and 1.70 eV respectively slightly deviating from

our corresponding values of 1.72 eV and 1.96 eV due to the

dispersion correction not being included in their calculations.

The encapsulation and association energies reported in Figs. 3

and 4 are for single atoms from single gas phase reference states.

While this is physically realistic for Xe and Kr. It is clearly not for the

other ﬁssion products that will form more complex gas phase

species or react to form solids in their particular waste streams.

Nevertheless, the energies reported thus are important compo-

nents in relevant thermodynamic cycles and we hope prove useful

to other studies. So, for example, it is possible that the reactive

ﬁssion products may form diatomic species that are encapsulated

or associated. The associated energies are also important

components in future thermodynamic cycles. Thus, as a ﬁrst step

towards more complex cycles we next report encapsulation and

association energies of homonuclear diatomic molecules. These

will also enable us to determine if encapsulation or association of

atoms or molecules is the more favourable.

3.4. Formation of homonucler dimers

Calculations were performed to investigate the energetics

associated with homonuclear dimers both inside and outside C

with respect to isolated atoms and with respect to isolated dimer

molecules (i.e. isolated atoms and molecules as reference states but

normalised per atom). Only the reactive ﬁssion products were

considered (i.e. those that form gas phase molecules). We consid-

ered Br, Rb and Cs for the formation of a dimer inside C

(see Fig. 5)

whereas Br, I, Te, Rb and Cs dimers were considered as exohedral

species (see Fig. 6).

The relaxed structures of dimers introduced inside the C

cage,

Table 1

Initial and ﬁnal conﬁgurations (see Fig. 2 for deﬁnitions) of volatile ﬁssion products absorbed on C

exohedrally. The most stable conﬁgurations are highlighted in bold.

Initial Conﬁguration Final conﬁguration and relative association energy (eV)

Xe Kr Br I Te Rb Cs

H H, 0.00 H, 0.00 H, þ0.52 H, þ0.30 H, þ1.19 H, 0.00 H, þ0.02

PP,þ0.07 P, þ0.03 P,þ0.58 P, þ0.32 P, þ1.42 P, þ0.07 P, þ0.10

(66) 66, þ0.02 66, þ0.03 66, þ0.23 66, þ0.06 66, 0.00 66, þ0.03 66, 0.00

(65) 65, þ0.05 65, þ0.03 C, 0.00 65, þ0.13 65, þ0.62 65, þ0.18 65, þ0.20

CC,þ0.06 C, þ0.04 C, 0.00 C, 0.00 66, 0.00 C, þ0.20 C, þ0.22

Fig. 4. (a) Shortest distance between a C atom in C

and the exohedral FP, (b) association energy, (c) Bader charge on FP, (d) magnetic moment of the supercell and (e), (f) and (g)

the total DOS of the most stable conﬁgurations of Br-C

, Te-C

and Cs-C

respectively. (A colour version of this ﬁgure can be viewed online.)

N. Kuganathan et al. / Carbon 132 (2018) 477e485 481

their encapsulation energies, and the diameter of the host C

molecules upon encapsulation, the dimer bond distances and the

Bader charges on the dimers are shown in Fig. 5. Calculations show

that all three dimers cause signiﬁcant distortion of C

; the degree

of distortion is reﬂected in the large positive encapsulation energies

and the diameters of the dimers containing C

(d) compared to

dimer free C

(7.10 Å) (see Fig. 5d). The encapsulation energies for

Br, Rb and Cs are positive meaning that they are not stable with

respect to gas phase atoms or molecules. Endohedral encapsulation

will therefore not be observed.

A dimer conﬁguration for each species was allowed to absorb on

the surface of C

. Fig. 6 shows: the relaxed structures of Br

,Te

and Cs

and reports association energies (both from isolated atoms

and isolated dimer reference states), the Bader charges on the di-

mers, shortest FP-C distance and the dimer bond length (D). The

relaxed structures for I

and Rb

are similar to Br

and Cs

respectively.

Rb and Cs exhibit slightly smaller association energies per atom

if absorbed as a dimer to C

(Fig. 6) compared to single atoms

(Fig. 4). Further, the separations of Rb-Rb (4.81 Å) and Cs-Cs (5.10 Å)

(see Fig. 6d), are longer than in the corresponding gas phase

diatomic molecules (4.30 Å for Rb

and 4.83 Å for Cs

) consistent

with these alkali metal species being non-bonded. This is because

Rb and Cs interact strongly with C

with each atom donating ~1

electron (so that the Bader charge, reported in Fig. 6 for pairs, is

much greater than 1). The association energies per atom for Rb and

Cs pairs are 1.20 eV and 1.48 eV respectively, less favourable

than the values calculated for the corresponding single

atoms, 1.72 eV and 1.95 eV respectively. Thus, while there is a

driving force for a second alkali metal to join an alkali metal

Fig. 5. Relaxed structures of (a) Br

(b) Rb

encapsulated in C

, (d) encapsulation energies from atoms, encapsulation energies of gas phase dimer (per atom), diameters of

relaxed C

molecule, D, dimer bond lengths, B, and the Bader charge on dimers. (A colour version of this ﬁgure can be viewed online.)

Fig. 6. Relaxed structures of (a) Br

(b) Te

on the surface of C

and (d) as-

sociation energies of FP dimers, shortest FP-C distances, FP-FP distances and the Bader

charges on dimers. (A colour version of this ﬁgure can be viewed online.)

N. Kuganathan et al. / Carbon 132 (2018) 477e485482

exohedrally (i.e. 0.68 eV for the second Rb and 1.01 eV for the

second Cs) if there are other unoccupied C

molecules available,

the second alkali metal will become associated with an unoccupied

molecule.

Br, I and Te all form covalently bonded dimer molecules with

bond lengths values when exohedrally absorbed (Fig. 6), that are

very similar to data from experimental (Br

2.28 Å, I

2.66 Å and Te

2.56 Å) [37] and calculated (Br

2.31 Å,I

2.68 Å and Te

2.58 Å) gas

phase species. This indicates these molecules are not bonded to C

The long FP-C distances are further evidence of this (see Fig. 6).

While there are strong association energies for Br

and Te

per

atom, from the reference states of atoms, for the dimers to asso-

ciate, the association energies per dimer from an existing dimer are

small (i.e. from the dimer reference state). The dimer association is

a result of the induced-induced polarization observed on each of

the atoms of the dimer, though the overall Bader charge is zero,

consistent with a non-bonded interaction. The rationale for this is

that a single atom has considerable electronegativity based on

gaining charge that will occupy a p-orbital, whereas additional

negative charge to a dimer has to occupy a higher energy

mo-

lecular orbital. Thus, while Br, I and Te demonstrate a preference to

form exohedral dimers rather than forming two single atoms on the

surfaces of two different C

molecules, that driving force is a

consequence of the covalent bond between the two ﬁssion product

species. Thus, the association of these dimers is quite different to

that of their atoms, in contrast to the situation for the alkali metals.

Table 2 reports the energetics for taking homonuclear diatomic

gas phase molecules and associating them with C

either as atoms

or molecules, assuming an abundance of C

molecules. These

values show that Rb

and Cs

dissociate and associate as single

positive ions. Conversely, Br

and Te

stay as strong covalently

bonded molecules, weakly associating via vdW interactions.

3.5. Heteronuclear dimer association and endohedral/exohedral

pairs

Finally, we have considered interactions from heteronuclear

diatomic molecule reference states. Consider ﬁrst CsBr: there are

two different ﬁssion atoms (Cs and Br) interacting with C

in four

different ways. The four conﬁgurations for Cs and Br are: Fig. 7 (a)

with Br encapsulated by C

while Cs is associated exohedrally,

Fig. 7b where Cs is encapsulated by C

and Br associated exohe-

drally, Fig. 7c where both Cs and Br are associated as single atoms

exohedrally and Fig. 7 (d) Cs and Br are associated exohedrally as a

pair. This enables an investigation of whether Cs encapsulation

facilitates Br association or vice versa. We have also considered Cs

and I as a pair but only associating with C

exohedrally since

neither Cs nor I show any preference for endohedral encapsulation.

The energies to encapsulate and associate CsBr and CsI are re-

ported in Table 3. Total energies for conﬁgurations (geometries) as

in Fig. 7d are always more favourable than those as in Fig. 7c. Thus,

exohedral molecule energies discussed will be with respect only to

Fig. 7d. The values with respect to single atoms can be used to ﬁrst

assess how much more strongly a Cs, Br or I atom will become

trapped if an opposite heteronuclear species is already present

(note: energies with respect to single gas atom reference states).

For example, Table 3 reports Br is encapsulated to C

with an en-

ergy of 0.98 eV but if Cs is already present as an exohedral species

the Br encapsulation energy increases to 2.29 eV. Similarly for the

Cs exohedral association energy raises from 1.96 eV to 3.28 eV if

Br is already present as an endohedral species. Equivalent combi-

nations show similar gains in energy. Table 3 also shows that Bader

charges on Br and Cs increase when both species are present. The

same effect is reported for Cs and I.

Table 3 also reports energies to trap CsBr and CsI molecules. For

CsBr both Br endohedrally and exohedrally are considered with Cs

considered exohedrally. In both cases CsBr is trapped but not

strongly and there is a preference for both species being exohedral.

The energy for Cs encapsulated and Br associated exohedrally is

also reported in Table 3. In this case the energy positive (i.e.

unfavourable) reﬂecting the strong preference for Cs to be an

exohedral species rather than endohedral. The association energy

of CsI (both exohedral species) is similar to that of CsBr and small

consistent with these molecules being trapped by vdW forces. The

net Bader charge on the CsBr and CsI molecules exohedral to C

almost zero, again consistent with a non-bonded interaction. The

energies are similar to those observed for homonuclear dimers (Br

and Te

). For CsI, the adsorption energy with respect to the dimer

derived by Kobayashi et al. [35]is0.12 eV without dispersion

and 0.62 eV with dispersion (though we assume only for Cs). The

current calculation shows that the association of CsI is 0.42 eV

though we include dispersion for Cs and I thus importantly be-

tween Cs and I.

Finally, we calculate the energy difference (reaction) between ½

(and ½ I

) exohedrally trapped and Cs (and Rb) trapped as ions

to form CsBr (and CsI) trapped exohedrally. In both cases the het-

eronuclear molecules are more stable; by 1.24 eV for CsBr

and 0.98 eV CsI.

4. Conclusions

is a candidate active material for ﬁlters used in nuclear

plants where volatile ﬁssion products must be separated from

efﬂuent gases. In this context it has good mechanical and chemical

properties. Using density functional theory together with a

dispersion correction and vdW forces, we have investigated the

capacity of C

to trap key volatile ﬁssion products.

Consider ﬁrst trapping of atoms from the reference state of

atoms. Xe and Kr are weakly trapped via vdW forces with exohedral

association energies of about 0.05 eV. Kr is endohedrally trapped

with a stronger encapsulation energy of 0.20 eV but Xe, being

much larger, is not trapped endohedrally. Single gaseous Rb and Cs

atoms exhibit endohedral encapsulation energies of 1.82 eV

and 1.86 eV respectively (i.e. strongly exothermic). The outer

surface of the C

also exothermically absorbs Rb and Cs as single

gaseous atoms with similar energies of 1.72 eV for Rb

and 1.96 eV for Cs. Thus endohedral and exohedral behaviour is

similar. Calculated trapping energies for Br, I and Te as single atom

Table 2

Exohedral association energies of FP atoms and homonuclear diatomic molecules from molecule reference states.

Most stable position for atoms/dimers associating with C

exohedrally

Rb Cs Br I Te

Most stable site H (66) C C (66)

Association energy of an atom (eV/atom) with respect to dimer 1.44 1.70 þ0.34 þ0.56 þ0.38

Association energy of dimer (eV/atom) with respect to dimer 1.20 1.48 0.06 0.07 0.13

N. Kuganathan et al. / Carbon 132 (2018) 477e485 483

species from the gas phase showed different behaviours. Br is

exohedrally and endohedrally trapped with energies of 0.93 eV

and 0.98 eV respectively. I exhibits an encapsulation energy

of  0.2 eV but an association energy of 0.6 eV. For Te the differ-

ence is even greater with an endothermic encapsulation energy and

an exothermic association energy of 1.4 eV.

A dimer will not form inside the cage of C

for any species

because there is insufﬁcient volume, as evidenced by the distortion

introduced by the dimer (see Fig. 5). However, the outer surface has

the ability to trap Br

or Te

dimers via non-bonding van-der

Waals interactions with association energies per atom of only

around 0.1 eV per atom. Conversely, Rb

and Cs

are strongly

absorbed but as single atoms, forming positive charged ions as

these ions are highly stable.

Finally, trapping of CsBr and CsI molecules was considered. Both

showed a preference to be endohedral molecules with encapsula-

tion energies of just under 0.2 eV per atom.

Acknowledgements

We thank the EPSRC for funding as part of the “PACIFIC” pro-

gramme (grant code EP/L018616/1). Computational facilities and

support were provided by High Performance Computing Centre at

Imperial College London. Mr. Conor O. T. Galvin is acknowledged for

helpful discussions.

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