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C A R B O N 4 6 ( 2 0 0 8 ) 3 4 9 – 3 5 8
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ava i lab le a t wwwjournal homepage: www.elsevier .com/ locate /carbon
The formation of dimerized molecules of C60 and their solids
Narinder Kaura,b, Shuchi Guptaa,c, Keya Dharamvira, V.K. Jindala,*aDepartment of Physics, Panjab University, Chandigarh, IndiabChandigarh College of Engineering and Technology, Chandigarh, IndiacUniversity Institute of Engineering and Technology, Panjab University, Chandigarh, India
A R T I C L E I N F O
Article history:
Received 23 April 2007
Accepted 4 December 2007
Available online 14 December 2007
0008-6223/$ - see front matter � 2007 Elsevidoi:10.1016/j.carbon.2007.12.001
* Corresponding author: Fax: +91 172 2783336E-mail address: [email protected] (V.K. Jind
A B S T R A C T
The formation of dimerized molecules of C60 was studied using the Brenner potential. Sev-
eral structures are obtained which have been classified into three categories, viz., dumb-
bells, and fused and coalesced dimers, similar to those obtained earlier, using the Tersoff
potential. However, there are differences in the binding energies obtained using these
two potentials. From these formations, we chose four forms of the composite dimer mol-
ecule as cyclo dumbbell, peanut, capped armchair (5,5) and zigzag nanotubes (10,0) to form
crystalline solids. Calculations have been performed by placing them in various crystal
structures, i.e. monoclinic, hexagonal and cubic close packed. To obtain stable dimerized
crystal structures, the dimer molecules are considered to be rigid, interacting via atom–
atom interaction of 6-exponential form. The monoclinic phase has been found to be
energetically most stable for each of the dimers. Various structural, thermodynamic and
phonon related properties of the stable dimer phases were investigated.
� 2007 Elsevier Ltd. All rights reserved.
1. Introduction
The molecules in solid C60 form a face centered cubic lattice
(fcc) at room temperature with intermolecular interactions,
reasonably well represented by Van der Waals potentials [1].
This molecular C60 solid was observed to undergo phase
transformations when doped with a suitable alkali metal
atom or exposed to visible or ultraviolet light. It also trans-
forms under heat treatment at high pressure or equivalently,
under ion irradiation of their thin film samples [2–6]. All these
phenomena result in a polymerized state in which the C60
molecules are covalently bonded to each other. Evidence for
the polymerization was obtained through infrared IR and Ra-
man measurements which indicated the loss of icosahedral
symmetry of the molecule, and through laser desorption
mass spectrometry which indicated the presence of oligo-
mers as large as (C60)20. In the experiments carried out by
Rao et al. [2], it was observed that under laser irradiation
er Ltd. All rights reserved
.al).
C60 films transformed into a toluene insoluble product [3].
This results in a polymerization reaction which is associated
with the 2 + 2 cycloaddition reaction involving the photo ex-
cited triplet state of C60 wherein the double bonds of the
neighboring molecules react to form a polymerized chain,
linked through a four-membered ring. As a result, a transfor-
mation to rhombohedral, tetragonal, monoclinic or ortho-
rhombic structures takes place which is characterized by a
short interfullerene distance of 9.2 A, compared to 10 A in
the starting fcc solid [4,5]. The solid consisting of linear chain
polymers forms a conducting phase. The dimer phase on the
other hand is considered, an insulating phase forming a
meta-stable monoclinic structure with space group P21/a [5].
The formation of polymeric chains of C60 is also seen in the
experiments on RbC60 and KC60 [6], whose quasi-one-dimen-
sional structure and electronic properties have attracted con-
siderable attention. The establishment of crystal structure of
the pressure-induced C60 dimer would be of particular interest
.
350 C A R B O N 4 6 ( 2 0 0 8 ) 3 4 9 – 3 5 8
in view of the solid-state conversion mechanisms of the
monomer into its various polymerization products, where
the dimer phase can be proposed as an intermediate.
Solids formed out of dimerized C60 molecules have also
been studied theoretically using the model potential ap-
proach [7,8]. The possible crystal packings of dimerized C60
molecules were studied by minimizing the lattice energy with
a bond charge intermolecular potential model, when the two
C60 monomers are cycloadded [7].
In our earlier paper [8], we studied the bulk structural and
thermodynamic properties of dimer C60 solid using a simple
potential model in which the molecular interactions are of
Van der Waals type. The pressure and temperature dependent
properties such as bulk modulus, lattice and orientational
structure, phonon dispersion relations, Gruneisen parame-
ters, heat capacity and entropy were studied considering
the rigid dimer molecule to be singly bonded. At that time,
only single bonded and cyclo-added dimers were known. La-
ter on, many more stable dimer configurations have been sug-
gested and experimentally observed by various groups [9,10].
In another paper,1 we investigated various possible dimer
structures with the help of Tersoff potential [11,12] and suc-
cessfully obtained the commonly referred dimer structures.
In the present paper, we use another short range potential,
the Brenner potential [13] to obtain the stable dimer mole-
cules, depending upon the initial configurations, i.e., inter-
molecular distance, molecular orientation and cage opening
of the two C60 molecules, in line with the earlier work done
by us using Tersoff potential.
In the next section, we describe the theoretical model used
to calculate total binding energy of a C60 dimer molecule and
also that of a dimerized C60 solid. In Section 3.1, we describe
the procedure to obtain the dimer molecules of interest using
Brenner potential. These dimer molecules, which have been
assumed to be rigid form the basis of crystalline lattice and
the proposed crystal structures, are studied in Section 3.2.
Structural and thermodynamic properties of the most stable
solid structure are presented in Section 4. We have summa-
rized the obtained results in the last section.
2. Theoretical model
First we briefly describe the model potentials which have
been used to calculate inter and intra-dimer interactions.
Tersoff as well as Brenner potential have been used to model
the bonded carbon–carbon intra-dimer interactions, while a
6-exp form of potential has been used to model non-bonded
carbon–carbon inter-dimer interactions. These potentials
have been extensively used to interpret properties of several
carbon-based systems such as carbon nanotubes [14], graph-
ite [11–13], diamond [11–13] and fullerenes [15]. These poten-
tials are also suitable for silicon and hydrocarbons [11–13].
These potentials are able to distinguish among different car-
bon environments, fourfold sp3 bond as well as threefold
sp2 bond.
1 Kaur N, Dharamvir K, Jindal VK. Dimerization and Fusion ofTwo C60 Molecules. Communicated to Chemical Physics, 2006.Also available at arXiv:0704.2502v1 [cond-mat.mtrl-sci].
2.1. Intra-dimer interaction potential
We now describe Tersoff and Brenner potentials briefly. In
these model potentials, flexibility of the C60 molecules has
been considered.
2.1.1. Tersoff potentialThe form of Tersoff potential is expressed as potential energy
between any two carbon atoms, say i and j, in the two C60
units, separated by a distance rij as
VTBij ¼ fcðrijÞ½aijVRðrijÞ þ bijVAðrijÞ�; ð1Þ
where i and j runs from 1 to 120.
VRðrijÞ ¼ Ae�k1r; VAðrijÞ ¼ Be�k2r; ð2Þ
fc(rij) is a sine function used to smooth the cutoff distance ta-
ken as 2.1 A. It varies from 1 to 0 in sine form between R � D
and R + D, D being a short distance around the range R of the
potential
fcðrÞ ¼1; r < R� D;12� 1
2 sin p2 ðr� RÞ=D� �
; R� D < r < Rþ D;
0; r > Rþ D;
8><>: ð3Þ
The state of the bonding is expressed through the term bij ex-
pressed as,
bij ¼1
ð1þ bnnnijÞ
12n
; where; ð4Þ
nij ¼Xk 6¼i;j
fcðrikÞgðhijkÞe½k33ðrij�rikÞ3 �: ð5Þ
Here hijk is the bond angle between ij and ik bonds.
gðhÞ ¼ 1þ c2
d2 �c2
½d2 þ ðh� cos hÞ2�: ð6Þ
The main feature of this potential is the presence of the bij
term. The strength of each bond depends upon the local envi-
ronment and is lowered when the number of neighbors is rel-
atively high. This dependence is expressed by bij, which can
accentuate or diminish the attractive force relative to the
repulsive force, according to the environment. The term nij de-
fines the effective coordination number of atom i, i.e., the
number of nearest neighbors taken into account, the relative
distance of two neighbors rij � rik and the bond-angle hijk. The
function g(h) has a minimum for h = cos h, the parameter d
determines how sharp the dependence on angle is, and c ex-
presses the strength of the angular effect.
Further,
aij ¼ ð1þ angijÞ� 1
2nð Þ � 1; ð7Þ
gij ¼X
fcðrikÞeðk33ðrij�rikÞ3Þ; when a and k3 are taken as 0; ð8Þ
aij 5 1 only if gij is exponentially large, which will occur for
atoms outside the first neighbor shell. Initially coordination
number of each atom (say ith) is 3 (k1, k2, k3) in a C60 molecule,
but when two neighbouring bucky balls come close to each
other for intermolecular bonding this number increases
depending upon the intermolecular distance.
Tersoff had given the parameters of this potential in order
to reproduce the diversified carbon systems, however we are
interested only in C60. Therefore, a few parameters A, B, k1,
C A R B O N 4 6 ( 2 0 0 8 ) 3 4 9 – 3 5 8 351
k2 and k3, are modified by us in order to get a better fit to the
bond lengths and binding energy of C60 molecule, keeping
others same as that given by Tersoff. Using original Tersoff
parameters, the bond lengths of C60 molecule come out to
be 1.42 A and 1.46 A for double and single bond respectively
and binding energy comes out to be �6.73 eV/atom, some-
what away from the experimental value [3]. After modifying
the parameters, we have been able to achieve the bond
lengths equal to 1.41 A and 1.45 A and binding energy of
�7.16 eV/atom, much closer to measured values. The effect
of change in k3 becomes evident only at high pressures. These
parameters are tabulated in Table 1.
2.1.2. Brenner potentialThe procedure used by Tersoff to develop classical potentials
for silicon, carbon and germanium has been to fit the pair
terms and bij, to a number of properties of the diatomic and
solid state structures. Brenner suggested that this expression
is unable to reproduce a number of properties of carbon such
as a proper description of radicals and conjugated versus
non-conjugated double bonds so he proposed a new potential
called Brenner potential. The term conjugated bond indicates
the presence of extra electrons that can be used in a bonding
process. Brenner potential is a modified form of Tersoff po-
tential and the repulsive and attractive terms in the potential
energy terms are given by,
VRðrijÞ ¼De
S� 1exp �b
ffiffiffiffiffiffi2Spðr� ReÞ
h i; ð9Þ
VAðrijÞ ¼DeS
S� 1exp �b
ffiffiffi2S
rðr� ReÞ
" #; ð10Þ
fðrijÞ ¼1; r < R1;
12 1þ cos
ðrij�R1ÞpðR2�R1Þ
� �h i; R1 < rij < R2;
0; rij > R2:
8><>: ð11Þ
The screening function f(rij) restricts the interaction to nearest
neighbors as defined by the values for R1 and R2. In addition,
the Brenner potential takes bonding topology into account
with the empirical bond order function Bij.
Bij ¼ ðBij þ BjiÞ=2þ FijðNðtÞl ;NðtÞj ;N
conjij Þ; ð12Þ
Bij ¼ 1þX
kð6¼i;jÞGiðhijkÞfikðrikÞe
aijk½ðrij�R
ðeÞijÞ�ðrik�R
ðeÞikÞ�
24
35�dl
: ð13Þ
In Eq. (12), Fij term fixes the overbinding of radicals for bonds
between pair of atoms that have different coordinations. In
our fullerene system, this term has been taken as zero and
the equation reduces to:
Bij ¼ ðBij þ BjiÞ=2; ð14ÞGcðhÞ ¼ a0f1þ c2
0=d20 � c2
0=½d20 þ ð1þ cos hÞ2�g: ð15Þ
Table 1 – Showing the original and modified parametersof the Tersoff potential
A (eV) B (eV) k1 (A�1) k2 (A�1) k3 (A�1)
Original 1393.6 346.7 3.4879 2.2119 0
Modified 1380.0 349.491 3.5679 2.2564 0, 2.2564
Brenner has defined two set of parameters and have been pre-
sented in Table 2.
Using either of these potentials, composite energy of all
the atoms of the system, given by E is written as
E ¼ 12
Xij
i 6¼j
VTBij ð16Þ
The sum in Eq. (16) includes all the atoms in each of the mol-
ecules. In our case we deal with a composite molecule con-
sisting of two C60 balls, henceforth called a dimer molecule.
The summation in Eq. (16) therefore runs from 1 to 120 atomic
indices.
2.2. Inter-dimer interaction potential
For the inter-dimer long range interactions between the non-
bonded ith and jth carbon atoms belonging to different dimer
molecules in the lattice, the interaction potential [1], of the 6-
exp form is used which is given by the expression
VLij ¼ �
A
r6ij
þB expð�arijÞ ð17Þ
where A, B and a are the interaction parameters. These
parameters have been tabulated for carbon–carbon interac-
tions in Table 3.
In this way, we calculate the inter dimer potential energy
U ‘j;‘0j0 between two such molecules, identified by j th mole-
cule in unit cell index ‘, j 0th molecule in cell ‘ 0 as:
U‘j;‘0j0 ¼X
ij
VLðrijÞ; ð18Þ
where rij is the distance between the ith atom on jth molecule
and jth atom on j 0th molecule. i and j runs from 1 to 120.
The total potential energy can be obtained by carrying out
the lattice sums, knowing the position of the lattice points, as
/ ¼ 12
X‘j;‘0j0
U‘j;‘0j0 : ð19Þ
3. Numerical calculations and results
We first obtain the dimer molecules having different types of
inter-cage bonding by using short range potentials, which
take care of the intra-dimer interactions. We have already ob-
tained the dimer structures using Tersoff potential in our ear-
lier paper.2 The same procedure has been adopted here, using
Brenner potential, yielding similar dimer structures. After
explaining the formation of C60 dimer molecule, we obtain
the minimum energy structure of dimer solid. While obtain-
ing the stable dimer solid, we assume the dimer molecules,
forming that solid, to be completely rigid and only 6-exp
interactions have been considered, as the inter-dimer interac-
tion energy is much higher than the intra-dimer interaction
energy. The phonon related and temperature dependent
properties of the stable dimer solid are also studied and the
results are presented in Sections 3.1 and 3.2.
2 See Footnote 1.
Table 2 – Parameters of the Brenner potential
De (eV) S b (A�1) Re (A) R1 (A) R2 (A) d a0 c0 d0
Set-1 6.325 1.29 1.5 1.315 1.7 2.0 0.80469 0.011304 19 2.5
Set-2 6.0 1.22 2.1 1.39 1.7 2.0 0.5 0.00020813 330 3.5
Table 3 – Atom–atom potential parameters (Kitaigorodski) [1]
A ¼ 358 kcal=mol A B ¼ 42; 000 kcal=mol a = 3.58 A�1
352 C A R B O N 4 6 ( 2 0 0 8 ) 3 4 9 – 3 5 8
3.1. Dimer molecule formation
Minimum energy structures of dimer C60 molecule have al-
ready been obtained using Tersoff potential. Now, we also
use Brenner potential with set-1 parameters to obtain stable
structures of dimer C60 molecule. With set-2 parameters,
the bond lengths and binding energy of a C60 molecule were
not well reproduced,3 so have not been reported here. Initial
configuration consists of two bucky balls with a certain mu-
tual orientation at a certain distance apart such that the
bucky balls are within chemical bonding range. The atomic
coordinates of the 120 atoms are then adjusted one by one
to obtain a configuration with lower energy. The cycle is re-
peated many times till minimum energy is obtained. In this
way, we have obtained different dimer structures. In the liter-
ature, there are five dimer structures viz., single bonded,
cycloadded, peanut, zigzag nanotube and armchair nanotube,
which have been studied extensively either theoretically or
experimentally. Out of these five structures, we have already
studied the crystalline dimer phase having single bonded di-
mer molecules [8]. The other four are now being studied in
this paper. Binding energies and intercage bond lengths of
the four stable dimer structures have been presented in
Table 4.
In Table 4, we see that under the two potentials we obtain
similar dimer structures with same type of intercage bonding
and same intercage bond length, but with different binding
energies. It is a accepted fact [13] that Tersoff potential gives
physically valid results only when the local environment of
all the C-atoms in two interacting molecules remains same.
However, it results in non-physical behaviour whenever a car-
bon atom with three nearest neighbours is bonded to a car-
bon atom with four or more neighbours, which is
happening in case of structures b, c and d where the number
of intercage bonds is more than two. To describe this situa-
tion, some non-local effects have to be included which are
incorporated in Brenner potential. In addition, by comparing
the results of Tersoff and Brenner potential given in Table 4
with that of Ref. [9,10], we find that the numerical agreement
for the binding energies, calculated by using Brenner poten-
tial is better as compared to Tersoff potential.
3 Kaur N, Gupta S, Dharamvir K, Jindal VK. Behaviour of aBucky-ball under Internal and External pressures. Communicatedto Journal of Materials Engineering and Performance, 2007. Alsoavailable at arXiv:0704.2504v1 [cond-mat.mtrl-sci].
As far as the intercage bonding style is concerned, both
model potentials give physically valid results and hence the
structures. In cyclo dumbbell, sp3 type intercage bonding also
called 2 + 2 cycloaddition is there. Pure sp2 type bonding is
seen in peanut, armchair nanotube (5,5) of length 11.84 A,
diameter 6.98 A and zigzag nanotube (10,0) of length 12.30 A,
diameter 8.08 A as shown in Fig. 1. Pressure induced dumbbell
structure is the most talked about dimer structure in the di-
mer solids. The coalesced structures i.e peanut and capped
nanotubes have been observed to form following laser abla-
tion of fullerene films, in collision between fullerene ions
and thin films of fullerenes and in fullerene–fullerene colli-
sions [16]. The bucky balls have also been observed to coa-
lesce during peapod formation reactions.
3.2. Dimer molecules in crystalline phase
The dimer molecules obtained above form the basis of the
crystalline lattice. In order to calculate the cohesive energy
for the dimer solid, the intermolecular potential energy be-
tween the dimer molecules needs to be obtained. The lattice
structures considered are
(i) monoclinic in P21/a symmetry with the dimer mole-
cules at the corners and base centers of the monoclinic
unit cell;
(ii) hexagonal close packed (hcp) with the dimer molecules
in ABABAB. . . packing and
(iii) cubical close packed (ccp) with the dimer molecules in
ABCABC. . . packing.
By using the long-range 6-exp interaction potential, inter-
dimer potential energy of the crystal is obtained. The poten-
tial energy obtained in this way is a function of the lattice
parameters and orientations of the molecules in the unit cell.
In case of monoclinic structure, the lattice parameters; a, b, c,
b the monoclinic angle and the orientation of the dimer mol-
ecules were varied till a minimum is attained in the potential
energy /, whereas in hcp and ccp structures; a(=b), c and the
orientation of the dimer molecules are varied to obtain mini-
mum energy structure. After obtaining the minimum energy
lattice and its parameters we study pressure and temperature
effects on the stable dimer solid. In Fig. 2, we show dimer
molecules in cubical close packing, hexagonal close packing
and monoclinic unit cell.
Fig. 1 – Different dimer structures obtained under Tersoff
and Brenner potentials.
Fig. 2 – Unit cell of: (a) cubical close packing; (b) hexagonal
close packing; (c) monoclinic structure.
Table 4 – Comparison of relative binding energy and bond length of the dimer structures under the two potentials
S. No. Structure Orientation Tersoff potential Brenner potential with set-1 parameters
Binding energy (eV) Bond length (A) Binding energy (eV) Bond length (A)
a Cyclo dumbbell (2) DB–DB �1.51 1.54 �1.3 1.56
b Peanut (6) Open HF–closed HF �16.3 1.39 �8.3 1.39
c Armchair nanotube (6) Open PF–open PF �33.6 1.41 �15.2 1.41
d Zigzag nanotube (6) C120 isomer �35.1 1.41 �16.9 1.41
Numbers in parenthesis in column 2 represent the number of intercage bonds.
C A R B O N 4 6 ( 2 0 0 8 ) 3 4 9 – 3 5 8 353
Some of the numerical results for lattice parameters, total
potential energy (/) thus obtained in minimized energy con-
figuration in hcp, ccp and monoclinic lattice are presented
in Table 5–7.
From Table 5–7 it is observed that minimum energy config-
uration is that when the dimer molecules are arranged in
monoclinic lattice. The plots of potential energy as a function
of lattice parameter for structure 1 in monoclinic lattice are
presented in Fig. 3, similarly we have obtained the equilib-
rium lattice parameters of the remaining three structures
and the values are shown in Table 7.
3.2.1. Phonon frequenciesDimer C60 solid is a molecular crystal, having vibrational and
librational modes. The total potential energy of the crystal is
dependent upon the intermolecular separation as well as on
the orientation of the molecules. Its lattice dynamics is best de-
scribed by diagonalization of the dynamical matrix defined as
Ml1l2ðj;j0qÞ ¼1
ðml1ðjÞml2
ðj0ÞÞ1=2X
l0/l1l2
ðlj; l0j0Þexp½iq
� ðRðl0j0Þ � RðljÞÞ� ð20Þ
which leads to the calculation of phonon frequencies and
eigenvectors for values of wave vector q in the Brillouin zone.
3.2.1.1. Phonon dispersion relation. The phonon frequen-
cies for the four structures are presented in Table 8 where R
and T represent rotational and translational modes. First
three frequencies correspond to acoustic modes, which van-
ish at zero wave vector. Phonon dispersion curves are pre-
sented in Fig. 4.
3.2.1.2. Phonon density of states. We also calculate phonon
density of states, g(x), defined as:
gðxÞ ¼ 2x6rN
Xqj
dðx2 � x2qjÞ
0@
1A; ð21Þ
where 6r are the number of modes for a given k vector. In our
case this is 12. N is the total number of q-vectors taken in the
first Brillouin zone.
The Dirac-delta function in Eq. (21) has been represented
by a Lorentzian function whose width function has been
appropriately fixed. The density of states thus calculated is
presented in Fig. 5.
Phonon density of states of dimer C60 solid has been calcu-
lated in the energy range from 0 meV to 10 meV. The intermo-
lecular phonon density of states shows peaks corresponding
to librational mode at around 4.5 meV in case of dumbbell
and peanut whereas in case of capped nanotubes these peaks
are around 3 meV, 4.5 meV and 6 meV. Peaks corresponding to
translational mode are at around 1.5 meV in case of dumbbell
and peanut solid, whereas in case of capped nanotubes these
peaks are around 1 meV.
In Fig. 6, the experimental density-of-states for the RbC60
dimer is shown [17]. The modes corresponding to energy
more than 10 meV are the twisting and bending modes within
the dimer molecule involving the intercage bond, which we
are not considering in our model as we have assumed a rigid
dimer model. RbC60 resembles dumbbell structure. By com-
paring Fig. 5a with Fig. 6, we find that the four peak experi-
mental curve below 10 meV is reflected in our dumbbell
solid DOS data. INS investigation of the dimer state of C60 ob-
tained upon high pressure high temperature treatment have
been reported by Rols et al [18] and the calculated phonon
spectrum agrees well with these inelastic neutron-scattering
experiment results [17,18].
Table 5 – Lattice parameters of the four hexagonal close packed structures with (a = b = 90�) and (c = 120�)
Dimer lattices a (10�10 m) b (10�10 m) c (10�10 m) Orientation h(x, y, z) (rad) / (kcal/mol)
Cyclo dumbbell (C60)2 11.17 11.17 34.96 0.45, �0.03, 0.03 �35.73
Peanut (C60)2 11.16 11.16 34.70 0.39, 0.05, 0.19 �36.113
Zigzag nanotube 12.61 12.61 25.68 0, 0, 0.14 �41.26
Armchair nanotube 11.22 11.22 30.36 0.09, 0.45, 0.24 �47.80
Table 6 – Lattice parameters of the four cubical close packed structures (a = b = 90�) and (c = 120�)
Dimer lattices a (10�10 m) b (10�10 m) c (10�10 m) Orientation h(x, y, z) (rad) / (kcal/mol)
Cyclo dumbbell (C60)2 11.43 11.43 34.47 0.02, 0.18, 0.08 �32.17
Peanut (C60)2 11.28 11.28 34.08 �0.01, 0.21, �0.025 �35.05
Zigzag nanotube 12.61 12.61 25.60 0.07, 0.50, 0.14 �44.63
Armchair nanotube 11.22 11.23 30.68 0.11, 0.44, 0.26 �49.74
Table 7 – Lattice parameters of the four monoclinic structures (a = c = 90�)
Dimer lattices a (10�10 m) b (10�10 m) c (10�10 m) b (�) Orientation h(x, y, z) (rad) / (kcal/mol)
Cyclo dumbbell (C60)2 16.92 9.89 19.57 125.56 0.52, 0.20, 0.15 �70.39
Peanut (C60)2 17.23 9.84 19.26 121.53 0.43, 0.17, 0.49 �66.21
Zigzag nanotube 19.13 11.00 17.73 133.77 �0.00, 0.10, 0.14 �74.44
Armchair nanotube 17.30 9.91 18.99 125.29 �0.05, 0.1, 0.14 �77.07
Cyclo dumbbell (Theor.) (KC60)2 [7] 17.04 9.67 19.04 123.8 – �76.46
Single bonded (C60)2 (Theor.) [8] 16.99 9.75 20.99 123.1 – �64.69
Single bonded dumbbell (Exp.) (KC60)2 [6] 17.15 9.793 19.224 124.10 – –
Single bonded dumbbell (Exp.) (RbC60)2 [6] 17.14 9.929 19.277 124.4 – –
4 See Footnote 1.
354 C A R B O N 4 6 ( 2 0 0 8 ) 3 4 9 – 3 5 8
3.2.2. Anisotropic compressionAn important feature of this solid evidently concerns its
anisotropic compression. We calculate these directional com-
pressibilities and find that the coefficient along these three
directions is highly anisotropic.
An application of a hydrostatic pressure P alters the total
potential energy such that
E ¼ /þ PDV; ð22Þ
where, DV is the change in volume due to an application of
pressure P. Therefore, a minimization of the new potential en-
ergy leads to the P–V curve. P–V curve for various structures
has been shown in Fig. 7.
Application of pressure leads to directional compressibili-
ties, i.e. an information about P–a, P–b and P–c curve. The
average compressions along a, b and c for a pressure of
0–10 kbar, for the four dimer solids are given in Table 9. From
Table 9, we find that the dimer crystals are highly anisotropic,
unlike a C60 solid and this anisotropy can be useful for various
applications (see Fig. 8).
3.2.3. Temperature effectsThe thermodynamic properties are easily calculable from the
Helmholtz free energy function. Its quasi-harmonic part is gi-
ven by
FQH ¼X
qj
ln 2 sinh�hxqj
2kBT
� �þ /ðVÞ; ð23Þ
where we include the static lattice energy as the second term,
written as a function of volume. The sum over q runs over
first Brillouin zone and j sum runs over the phonon branches.
Knowledge of the phonon frequencies, as discussed in the
previous section, enables us to calculate the quantities like
heat capacity at constant volume CV, and entropy S, at various
temperatures. Entropy is given by
S ¼ � oFoT
� �V
: ð24Þ
Internal energy of the system given by Eq. (25) has been used
to calculate the heat capacity of the solid at constant volume
is given by Eq. (26).
E ¼ F� ToFoT
� �V
; ð25Þ
CV ¼oEoT
� �V
: ð26Þ
The values of heat capacity CV and entropy S thus calculated
are shown in Figs. 9 and 10 respectively.
4. Discussion
In order to confirm theoretically the formation of the dimer
structures, obtained already using Tersoff potential,4 we have
used another model potential, i.e Brenner set-1 potential, in
16.7 16.8 16.9 17.0 17.1 17.2
0100200300400500600700
a
9.76 9.84 9.92 10.00 10.08-120
0120240360480600
b
dumbbell
19.4 19.5 19.6 19.7 19.8-100
-100
0100200300400500600700
φ(kc
al/m
ole)
φ(kc
al/m
ole)
φ(kc
al/m
ole)
φ(kc
al/m
ole)
c
125.1 125.2 125.3 125.4 125.5 125.6-100
-500
50100150200250300
β
dumbbell
dumbbell
dumbbell
Fig. 3 – The four curves show the minimum energy
configuration w.r.t. a, b, c, the monoclinic angle b for the
monoclinic lattice.
Table 8 – Phonon frequencies, at zero wave-vector for thefour dimer structures in monoclinic dimer solid
Phonon frequencies (THz)
Branchnumber
Dumbbell Peanut Armchairnanotube
Zigzagnanotube
4 0.350 (R) 0.380 (R) 0.189 (R) 0.189 (R)
5 0.421 (R) 0.444 (R) 0.202 (R) 0.243 (R)
6 0.689 (R) 0.541 (R) 0.438 (R) 0.381 (R)
7 0.897 (T) 0.750 (R) 0.652 (R) 0.460 (R)
8 1.000 (T) 0.778 (R) 0.727 (R) 0.587 (R)
9 1.113 (R) 0.852 (T) 0.775 (T) 0.801 (R)
10 1.117 (R) 0.919 (T) 0.973 (R) 1.023 (T)
11 1.229 (R) 1.126 (R) 1.314 (T) 1.189 (R)
12 1.614 (T) 1.658 (T) 1.736 (T) 1.243 (R)
C A R B O N 4 6 ( 2 0 0 8 ) 3 4 9 – 3 5 8 355
this paper and obtained the similar structures. With Brenner
set-2 parameters, the obtained single and double bond
lengths of a C60 molecule were 1.37% and 3.6% larger respec-
tively, so we have not used these parameters for further cal-
culations and used only set-1 parameters. The resulting
dimer structures by using the two model potentials are simi-
lar in the intercage bonding style although having different
binding energies. This shows the validity of these two model
potentials to describe the intercage bonding of the dimer
molecules. These four types of dimer molecules have also
been experimentally observed.
For crystalline structure studies, at moderate pressures
and temperatures, the flexibility of the molecules do not play
any role. Therefore, the structures obtained by using any of
these two model potentials give similar results under a long
range interaction potential. To determine the structure of
the dimer C60 solid, the dimer molecules were placed in ccp,
hcp and base centered monoclinic lattices and the minimum
energy structure was obtained in each case, using 6-exp po-
tential, already used successfully for various molecular crys-
tals, including C60 in the past. The dimer molecules in the
crystalline phase are assumed to be rigid (see Table 4–7)
which is a good approximation as the inter dimer and intra di-
mer potential energies are of different order. From the com-
parison of the results for these structures, monoclinic
structure is shown as the most stable structure of the dimer
C60 solid. This structure, as noticed from Table 7, compares
well with the experimental data for RbC60 dimer solid and
pressure dimerized C60 solid.
In our calculations, we show that, the monoclinic phase of
dimerized C60 molecules is about 18% higher in cohesive en-
ergy per carbon atom for dumbbells and buckytubes and 6%
higher for peanut as compared to fcc phase of pure C60 solid.
But at suitable temperature or pressure, C60 molecules in fcc
phase, bind together, resulting in a transformation in mono-
clinic, hcp or ccp phase of dimer C60 solid. Hexagonal close
packing and cubical close packing were expected to be more
stable as compared to monoclinic structure due to larger
number of interactions between adjacent dimers in each
layer. But, due to the larger interlayer spacing in comparison
to that in monoclinic lattice, the interlayer interactions be-
come weak in ccp and hcp structures. Since, monoclinic
phase comes out to be the most stable, so subsequent calcu-
lations of frequency etc. have been performed by putting
dimer molecules in this configuration.
The present study also reports several, structural and ther-
modynamical properties of the dimer C60 solid, in the mono-
clinic phase using a simple, rigid molecule model. We provide
detailed phonon dispersion curves for external modes in the
dimer C60 solid in Fig. 4. Phonon density of states, arising
from external modes (Fig. 5) have also been calculated, and
can be suitably compared with inelastic neutron scattering
measurements (Fig. 6). External modes of this solid lie below
(0.5,ξ,0)(ξ,0,0)(0,ξ,0)
Dumbbells in monoclinic lattice
B C D E F G H I J K L M
Peanuts in monoclinic lattice
(.5,ζ,0)(ζ,0,0)(0,ζ,0) B C D E F G H I J K L M
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6 (.5,ζ,0)(ζ,0,0)(0,ζ,0)
.1.1.1 0 .3.3.3.5 .5
Zigzag Nanotubes in monoclinic lattice
Phon
on fr
eque
ncie
s(TH
z)
q-vector
.1 0.1.1 .3.3.3.5 .5q-vector
q-vector
B C D E F G H I J K L M
0.00.20.40.60.81.01.21.41.61.8 (.5,ζ,0)(ζ,0,0)(0,ζ,0)
.1 0.1.1 0 .3.3.3.5 .5
Armchair nanotubes in monoclinic lattice
Phon
on fr
eque
ncie
s(TH
z)
0.00.20.40.60.81.01.21.41.61.8
Phon
on fr
eque
ncie
s(TH
z)
0.00.20.40.60.81.01.21.41.61.8
Phon
on fr
eque
ncie
s(TH
z)
B C D E F G H I J K L M
0
.1 0.1.1 .3.3.3.5 .5q-vector
0
0
Fig. 4 – Phonon dispersion curves for C60 dimer solids in
various symmetry directions. B, C, D refer to acoustic
modes, E, F, G, H, I, J refer to librational modes and K, L, M
refer to optic modes.
0.00
0.04
0.08
0.12
0.16
0.20
0.24
(a) Dumbbell
Phon
on d
ensi
ty o
f sta
tes(
meV
-1)
Phon
on d
ensi
ty o
f sta
tes(
meV
-1)
Phon
on d
ensi
ty o
f sta
tes(
meV
-1)
Phon
on d
ensi
ty o
f sta
tes(
meV
-1)
TranslationalLibrational Combined
0.000.020.040.060.080.100.120.140.160.180.20
(b) Peanut
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
(c) Armchair nanotube
Translational Librational Combined
Translational Librational Combined
0 2 4 6 8 100.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
(d) Zigzag naotubeEnergy(meV)
0 2 4 6 8 10
0 2 4 6 8 10
Energy(meV)
Energy(meV)
0 2 4 6 8 10Energy(meV)
Translational Librational Combined
Fig. 5 – Calculated phonon density of states for external
modes in monoclinic dimer solid.
356 C A R B O N 4 6 ( 2 0 0 8 ) 3 4 9 – 3 5 8
Fig. 6 – Experimental curve for the generalized density of
states GDOS for Rb1C60 dimer solid.
-5 -4 -3 -2 -1 0 1 2 3 4 5
2600
2650
2700
2750
2800
2850
2900
2950
O
O
OO
V(10
-30 m
3 )
P(kbar)
armchair zigzag dumbbell peanut
Fig. 7 – Calculated P–V curve for the four structures in a
monoclinic dimer solid.
Table 9 – Compression of the crystalline structures
Compression at P = 10 kbar
Lattice a (%) b (%) c (%)
Dumbbell 2.22 2.21 1.10
Peanut 2.04 2.17 1.19
Zigzag nanotube 1.25 1.72 1.12
Armchair nanotube 1.59 1.89 1.16
C60 solid 1.27 1.27 1.27
16.56
16.64
16.72
16.80
16.88
16.96
a
Dumbbell
9.68
9.76
9.84
9.92
b
Dumbbell
0 2 4 6 8 10
19.36
19.44
19.52
19.60
c
P(kbar)
0 2 4 6 8 10P(kbar)
0 2 4 6 8 10P(kbar)
Dumbbell
a
b
c
Fig. 8 – Variation of lattice constants of monoclinic dimer
solid having dimers as dumbbells. (a), (b) and (c) shows the
pressure dependence of lattice constants a, b and c (in A).
0 50 100 150 2000
2
4
6
8
10
12
14
16
0 10 20 30 40 500
2
4
6
8
10
12
14
16
dumbbellpeanut armchair nanotubezigzag nanotube
hea
t cap
acity
(10-1
6 )erg
/K
hea
t cap
acity
(10-1
6 )erg
/K
T(K)
T(K)
dumbbellpeanutarmchair nanotubezigzag nanotube
Fig. 9 – Calculated heat capacity for the dimer solid at
various temperatures.
C A R B O N 4 6 ( 2 0 0 8 ) 3 4 9 – 3 5 8 357
8 meV and the lowest internal mode lie around 10 meV [5],
they are well separated so that our assumption that the di-
mers are rigid in the crystalline phase is again validated.
The modes corresponding to energy more than 8 meV are
the twisting and bending modes, involving the intercage
bond, which we are not considering in our model as we have
assumed a rigid dimer model. In case of buckytubes, the
twisting and bending modes would be absent due to more
number of intercage bonds.
The pressure dependent structure calculation also shows
that nanotube crystalline structures are harder than those
of dumbbell and peanut, shown in Fig. 7. The pressure depen-
dent structure calculations show that a hydrostatically
0 50 100 150 200T(K)
0
10
20
30
40
50
entr
op
y(10
-16 )e
rg/K
dumbbellpeanutarmchair nanotubezigzag nanotube
Fig. 10 – Calculated entropy for the dimer solid at various
temperatures.
358 C A R B O N 4 6 ( 2 0 0 8 ) 3 4 9 – 3 5 8
applied pressure results in a significant anisotropy in com-
pression, a and b directions getting compressed nearly twice
in comparison to c direction for the dumbbell and peanut sol-
ids; whereas for the two C120 carbon nanotube solids, again
the anisotropy is not so pronounced. The non-cubic nature
of these solids is very special along the c axis. Along this
direction, the crystal is much harder to compress. The mole-
cules are packed much more along this axis. Because of this,
the solid is much less compressible under isotropic pressure
as compared to pure C60 solid (Table 9).
Finally, heat capacity and entropy of the solid have also
been calculated at various temperatures and presented in
Figs. 9 and 10. Value of heat capacity at room temperature,
which comes out to be 16.56 · 10�16 erg/K, agrees well with
the expected value, i.e. 6kB. Heat capacity, which is directly
proportional to the binding strength of the solid, has been
estimated here to be more in case of capped nanotubes as
compared to the dumbbells as expected. Entropy, at room
temperature has been calculated to be �50 · 10�16 erg/K, it
is more in case of the capped nanotubes.
We want to point out that with these potentials, we are
able to understand the basic physics involved in the forma-
tion of dimerized molecules of C60, in a C60 solid, when C60
molecules are pushed against each other, with different ini-
tial configurations. However, in order to get more accurate
quantitative results, which will be valid even in high pressure
regime, we next propose to use classical molecular dynamics
on this system and this work is in progress.
Our calculation presented here motivates more experi-
mental work on structure and dynamics of dimerized mole-
cules and their solids. The structure of several solids
presented here and phonons in these are interesting input
for planning experiments for their structure and phonon
dispersions.
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