A one-dimensional Ising model for C70
molecular ordering in C70-peapods
Yutaka Maniwa1,2 , Hiromichi Kataura1 ,Kazuyuki Matsuda1 and Yutaka Okabe1
1

Department of Physics, Tokyo Metropolitan University, 1-1 Minami-osawa,Hachioji, Tokyo 192-0397, Japan
2CREST, JST, Mitsui building, Takezono 1-6-1, Tsukuba 305-0032, Japan
E-mail: maniwa@phys.metro-u.ac.jp
New Journal of Physics 5 (2003) 127.1–127.5 (http://www.njp.org/)
Received 4 August 2003
Published 3 October 2003

A description using the Ising model is proposed for the C70 molecularorientational ordering in C70 one-dimensional crystals formed inside single-wallcarbon nanotubes (SWNTs). It is shown that the observed standing and lying
alignments can be deduced naturally from the present Ising model where theSWNT diameter effectively changes the ‘effective field’ acting on C70 molecules.
The thermal expansion of the C70 intermolecular distance is well described bythis model. The possibility of antiferro-orientational ordering is also suggested.
Abstract.

It is well known that many kinds of guest atom or molecule can be encapsulated in onedimensional (1D) cavities of single-wall carbon nanotubes (SWNTs) [1]–[4]. Such materials
confined within nanometre-size spaces are expected to show novel properties that cannot beobserved in the bulk materials. In the case of C70 -peapods where the 1D arrays or crystals of
C70 molecules are formed inside SWNTs, a previous x-ray diffraction (XRD) study [5] clarified
that the molecular orientational order, related to the elongated molecular shape, is graduallydestroyed with increasing temperature. This behaviour is quite different from the cases in threedimensional (3D) C70 crystals, where C70 orientational transitions [6]–[8] occur successively at
280 and 340 K, and is assigned to the 1D nature of the C70 crystals. In the present work, we
propose an Ising model for the molecular orientational ordering in C70 -peapods.
XRD [5] and electron diffraction [9] measurements on C70 -peapods showed that the C70
molecules inside SWNTs are dominantly aligned parallel or perpendicular to the tube axis, asillustrated in figure 1. These orientations are called lying or standing, respectively. Therefore,we introduce a variable to describe the orientation of the i th C70 ; σi = 1 for the standing onesand −1 for the lying ones. The interaction energies of the nearest-neighbour C70 pair i, j  are
New Journal of Physics 5 (2003) 127.1–127.5
1367-2630/03/000127+5$30.00

PII: S1367-2630(03)67028-X
c IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
127.2
lying
MSstanding

MLC 70 molecule

Figure 1. Schematic illustrations of a C70 molecule, and standing and lying C70 s

inside SWNTs.
(a): σ =+1:σ = –1(b)

SWNT
E1,
J1E2,J2,
J3J1J2J3Figure 2. ‘Definitions’ of the orientational variable σ , inter-C70 interaction, andC70 –SWNT interactions. The ovals represent C70 molecules.

defined as in figure 2: J1 for the standing pair, J2 for the lying pair, and J3 for the standing–lyingpair. Furthermore, the interaction between the standing (or lying) C70 and the wall of the SWNTis given by E 1 (or E 2 ).
Then, the interaction energy of the nearest-neighbour pair i, j  is written asJ1 14 (σi + 1)(σ j + 1) + J2 14 (σi − 1)(σ j − 1) − J3 14 (σi + 1)(σ j − 1) − J3 14 (σi − 1)(σ j + 1)
+ E 1 12 (σi + 1) − E 2 12 (σi − 1) + E 1 12 (σ j + 1) − E 2 12 (σ j − 1).
Summing up all the pair i, j s and defining the effective interaction energies as
J1 = J1 + E 1 ,J2 = J2 + E 2 ,J3 = J3 + (E 1 + E 2 )/2,

we obtain the total Hamiltonian: −(J1 + J2 − 2J3)
1
J1 + J2 + 2J3
H =−σi +σi σ j + (J1 − J2 )N424i, jiσi σ j − Heσi + constant,≡ −Ji, j(1)iwhere N is the number of C70 molecules in the 1D crystal. This formula is equivalent to that ofthe Ising model for spin S = 1/2 with a coupling constant J in an effective field He :New Journal of Physics 5 (2003) 127.1–127.5 (http://www.njp.org/)

127.3J>0(a) standing(b) lying

J2J1

J3

J1J3J2H e >0J<0EH e <0(c) AF orstanding
J2J3
J1H e >0(d) AF orlying
J1J3J2H e <0Figure 3. Possible energy scales and ground states. The arrows show thedirections that the effective fields tend to align along the long axes of C70molecules.

−(J1 + J2 − 2J3 )−(J1 + J2 − 2J3 )=,44J  − J1 + E 2 − E 1(J2 − J1)
= 2.
He ≡22J≡(2)

From the above equations (1) and (2), we find that there are several cases for the C70 ordering,as summarized in figure 3:(i) For J > 0, ‘ferro-orientational (FO) ordering’ appears at T = 0 K, with standing FO
ordering for J2 > J1 and lying FO ordering for J2 < J1.
(ii) On the other hand, for J < 0 antiferro-orientational (AFO) ordering is favourable, but theeffective field tends to stabilize the standing or lying orientation.
Here, it should be emphasized that He is determined by both the C70 –C70 interaction and theC70 –SWNT interaction, whereas J is primarily determined by the C70 intermolecular interaction
and independent of the tube diameter within the present model. Therefore, He can be changed
by changing the tube diameter, keeping J constant. Thus, the standing or lying alignments canbe determined by the tube diameter, as observed. Tentatively we assume J > 0 and J1 < J2 ,for which the standing FO ordering is energetically favourable. Even in this case, however, theinteraction between the C70 and the wall of the SWNT can also stabilize the lying FO ordering,as well as the standing ordering. The lying ordering would occur for thin and too thick SWNTs.
In these tubes, E 1 may be much higher than E 2 , because the potential energy is minimized at theappropriate distance between the C70 molecule and the wall of the SWNT. Similar comments
are also possible for J < 0. However, another interesting situation appears for He ≈ 0 in thiscase; the ‘AFO’ state, which has not yet been observed, could be observed.
Now, we discuss the thermal expansion of the C70 intermolecular distance in the C70 standingalignment. The average lattice expansion a is given bya = αg T g(0) + (ML − MS )N− /N ,(3)where MS , ML , and g(0) are the mean lengths of the short and long axes of the molecules, andthe intermolecular gap at T = 0 K, respectively. The first term describes the thermal expansion
New Journal of Physics 5 (2003) 127.1–127.5 (http://www.njp.org/)

127.4
1.03
standing
1.02

C 70 -peapods

a(T)/a(RT)

lying
1.01

10.99

0.98

200

400

600

800

1000

1200

T (K)

Figure 4. Thermal expansion of the C70 intermolecular distance. The solid curve
is a fit with equation (5). The data were taken from [5].

of the intermolecular distance and the coefficient αg is approximated as the value for the solidC60 , ∼5.4 × 10−5 (K −1 ). N− is the number of the lying C70 molecules, and is given by 
1N−σi=1−.
(4)
N2NThus, we have 
1σi
a = g(0)αg T +1−(ML − MS ).
2
N

For this 1D Ising model the exact solution is well known:
e2K sinh L
Ji σi=√,with K ≡2NkB T
1 + e4K sinh L(5)

andL≡

He.
kB T

(6)

Inserting this equation into equation (5), we have the thermal expansion of the C70 intermoleculardistance. Fitting the equation to the observed lattice expansion [5] makes it possible to estimate
the magnitudes of J and He . However, equation (5) must be slightly modified before fitting tothe observed data; the factor 1/2 is replaced by 1/3 in order to explain the high temperaturethermal expansion (T → ∞), where N− /N should be 1/3. This is due to the limitation of thepresent Ising model. In the actual system, there are two possible standing directions with respect
to the tube axis.
The result of the fitting is fairly good, as demonstrated in figure 4 with the parameters
J /kB = −21 ± 29 K

andHe /kB = 282 ± 53 K.

These parameters imply that the observed standing FO alignment is dominantly stabilized by
the effective field He . However, because the effective coupling J may be negative, the AFO
ordering may be realized by choosing the tube diameter such that He ≈ 0.
New Journal of Physics 5 (2003) 127.1–127.5 (http://www.njp.org/)

127.5Here, we should compare the present fit with the previous one [5] with a phenomenologicalequation with an activation energy E 0 . The results of the two fits are almost identical in spiteof the different parameters and functions. This is because the present fitting function is moregeneral than the previous one, and it is reduced to a function similar to the previous one for theparameters obtained. Therefore, the present analysis can be taken as the microscopic grounds
for the previous phenomenological treatment.
In summary, it was shown that the 1D Ising model successfully describes the observed
behaviours of the C70 orientational alignment inside SWNTs. The C70 orientational order cannot
be realized at finite temperatures owing to the 1D nature of the C70 crystals inside SWNTs asexperimentally observed, although the ‘effective field’ tends to stabilize the standing or lying C70 .
Numerical estimations of the interaction energies from a microscopic point of view and using amore sophisticated treatment taking account of the continuous rotational freedom have not yet
been performed. Although these works are left for the future, we believe that the Ising model isthe simplest and most useful model to use to extract the essence of the orientational behaviours
inside SWNTs. The present method or similar treatments, therefore, can be applied to otherinteresting systems, such as metal–fullerene peapods where 1D crystals of metal-encapsulating
fullerene molecules with magnetic and/or electric dipole moments are formed inside SWNTs.
Acknowledgments

The XRD data were taken at BL02B2 of SPring-8 in collaboration with M Abe, A Fujiwara,R Fujiwara, H Kira, H Tou, S Suzuki, Y Achiba, E Nishibori, M Takata, M Sakata andH Suematsu.
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New Journal of Physics 5 (2003) 127.1–127.5 (http://www.njp.org/)