Peas in a pod: quasi-one-dimensional C60 molecules in a nanotube

12 Pages·0.17 MB·English

by  M. Mercedes Calbi

Peas in a pod: quasi-one-dimensional C molecules in a nanotube
60
M. Mercedes Calbi, Silvina M. Gatica and Milton W. Cole
Physics Department, Pennsylvania State University, University Park,
Pennsylvania 16802
(Dated: February 2, 2008)
3
0 “How luscious lies the pea within the pod...”
0 Emily Dickinson
2

n
a We evaluate the equation of state of the quasi-one-dimensional (1D) phase of C60 molecules in
J small carbon nanotubes, nicknamed “peas in a pod”. The chemical potential and 1D pressure are
0 evaluated as functions of the temperature and density, initially with the approximation of near-
2 est neighbor interactions and classical statistical mechanics. Quantum corrections and long-range
interaction corrections are discussed, as are the effects of interactions with neighboring peapods.

] Transition phenomena involvingthe3D coupling are evaluated.
t
f
o
s I. INTRODUCTION II. MODEL CALCULATIONS
.
t
a
m Girifalco and coworkers have shown13,15 how C60
molecules within nanotubes of small radius (R ≈ 7 ˚A)
-
d experience potentials confining them to the vicinity of
Carbon nanotubes have been found to exhibit a
n the tube axis. This paper employs the potentials devel-
wide range of remarkable properties, concomitants of
o opedby thatgroupto determine the equationofstate of
c their nanometer cross-section and macroscopic length.
the quasi-1D buckyball fluid. We assume that the total
[ Extensive research has recently explored the behav-
potential energy V of the molecules satisfies
ior of gases adsorbed within, outside of and in- total
1
between such nanotubes1,2,3,4,5,6,7. Among the most
v
2 intriguing systems is that created when C60 bucky-
V (ρ ,z )= U(ρ )+ V(|z −z |) (1)
6 balls lie within a tube of small radius R, since this total i i i i j
3 is surely a quasi-one-dimensional (1D) system8. This Xi Xi<j
1 system, nicknamed “peas in a pod” has begun to
0 be studied using various theoretical and experimental Here, ri = (ρi,zi) is the position of the i-th molecule’s
3 methods9,10,11,12,13,14,15,16,17,18. Inthispaper,weexplore center,writtenintermsofitscoordinateziparalleltothe
0
theequationofstateofthissystem. Specifically,weeval- tube axis and its transverse displacement from the axis,
/
at uatethe1Dpressureandthechemicalpotential,asfunc- ρi. The potential U(ρ) is the molecule-tube interaction
m tions of temperature and density. The calculation is rel- thatdeterminesthemolecules’degreeoflocalizationnear
atively simple in the small tube case considered in this the axis, which depends also onthe temperature T. The
-
d paper(e.g. (10,10)tubes,withR≈7˚A)becausetheC otherinteractioninEq. (1), V(z),isthatbetweenapair
60
n molecules areconfinedto the vicinity of the axis andthe of C60 molecules separated a distance z along the tube
o motion perpendicular to the axis can be factored out of axis. Girifalco et al have shown that both interactions,
c the problem, as we shall see. The more general case in- U(ρ) and V(z), can be expressed in a universal way by
:
v volvingtubesofvaryingsize,withthe attendantappear- the following reduced function13:
i
X ance of 3D effects, requires a more detailed study than
is described here. Some numericalstudies and molecular
r Φ(r)
a simulations exploring this problem have been performed =
by Hodak and Girifalco14,15,16. |Φ(r0)|
4 10
5 3.41 2 3.41
= − (2)
3 3.13r˜+0.28 3 3.13r˜+0.28
(cid:18) (cid:19) (cid:18) (cid:19)
The next sectiondiscusses the model and presents the
equation of state resulting from the simplest (classical, where|Φ(r )|isthewelldepthattheequilibriumspacing
0
quasi-1D) procedure we have used. Section III assesses r , and r˜≡ (r−r )/(r −r ). The distance parameter
0 1 0 1
quantumandlong-rangecorrectionstothissimplemodel r for each system (C -C , C -tube) is chosen so that
1 60 60 60
andestimatestheregimeofapplicabilityofthoseresults. the interactions fit the universal function (2). Table I
SectionIVdescribestheeffectsofinteractionswithbuck- indicates the values of the used parameters. The func-
yballs in neighboring tubes, including phase transitions. tion in Eq. (2) is obtained using a continuum model for
2
the carbon surfaces. The carbon-carbon interatomic po-
tential is integrated over the interacting surfaces assum-
k
ingameansurfacedensityofcarbonatomsanddifferent U(ρ)=U + ρ2+γρ4 (5)
0
2
Lennard-JonesparametersforatomsinC -C andC -
60 60 60
graphene systems. Therefore, we ignore any dependence The resulting transverse chemical potential is given
of the interactions on the atomic structure of either the by perturbation theory, assuming that the value of γ is
tubes or the C60 molecules. Similarly, the neglect of any small:
interactions that depend on the C orientation means
60
that the molecular rotational problem is not affected by
2
the environment. This means that the rotational degree µ =U +h¯ω+2 ln(1−e−x)+2γ ¯h coth x (6)
t 0
of freedom may be omitted completely from the present β mω 2
analysis7. (cid:20) (cid:16) (cid:17)(cid:21)
Concerning the C -C interaction, there has been Here x = β¯hω with ω = k/m, and the regime of
60 60
some discussion in the literature of the value of the validity of this expression is that range of T for which
p
asymptotic van der Waals coefficient18,19 (C6, the coeffi- therightmostterminµt issmallcomparedtothemiddle
cientofr−6),withvaluesvaryingovertherange16.6Na2t term (i.e., kBT < kBT4 ≡ k2/γ). In the case of (10,10)
eV ˚A6 to 29.05N2 eV ˚A6. The value used in Girifalco’s nanotubes, the upper limit to the T range for which this
at
calculations13 (Eq. (2)) is 20.0 N2 eV ˚A6. expression is valid is about 13,000 K (based on our fits
at
In reality, the intermolecular interaction V ought to to the Girifalco potential U(ρ): k = 1.32 eV/ ˚A2 and
depend on the full 3D separation. By neglecting this in- γ = 1.58 ˚A−4). Thus, the expansion suffices in practice
teraction’s dependence on ρ and ρ , Eq. (1) implicitly throughout the relevant range of T.
i j
assumesthatthe molecules’rmstransversedisplacement The present case corresponds to x ≈ 32 K/T, which
d = hρ2i is small compared to the mean separation is much less than 1 at room temperature. Therefore, it
i
a=L/N along the axis. We discuss the regime of valid- is appropriate to substitute the fully classical expression
p
ityofthatapproximationbelow. AlsoimplicitinEq. (1) for µt:
is the assumption that both the nanotube and the C
60
molecules are rigid, since otherwise their shape fluctua-
1
tionswouldaffectthenanotubes’potentialenergy. While −βµ =ln d2re−βU(ρ) (7)
t λ2
conventional in this field, this assumption deserves fu- (cid:18) T Z (cid:19)
ture attention in the case of such strongly interacting Here λ = (2πβ¯h2/m)1/2 is the de Broglie thermal
molecules as C . T
60 wavelength of the molecule, which is about 0.04 ˚A at
With Eq. (1), the properties of the system may be
roomtemperature. Theresultingexpressionforµ inthe
evaluated in terms of independent transverse and longi- t
case of a harmonic potential with a small quartic term
tudinal contributions to the Hamiltonian. For example,
coincides with the classical limit (x ≪ 1) of the preced-
the partition function Z of the C molecules factorizes
60 ing quantum expressions for both µ and the transverse
into transverse and longitudinal factors, Z = Z Z , so t
t z contribution to the specific heat (per molecule), C /N:
that the Helmholtz free energy F is an additive function t
ofthese separateparts ofthe problem: F =F +F . We
t z
may therefore write the chemical potential µ of the set 2 8γ
µ =U + ln(x)+ (8)
of N molecules as a sum of a transverse part µ and a t 0 β (βω2m)2
t
longitudinal part µ :
z
C 16γ T
t
=2− =2−16 (9)
µ=µt+µz (3) NkB β(ω2m)2 T4
One may immediately evaluate µ from the transverse Thisincludesaconstantcontribution(expectedfortwo
t
partition function Z because that part of the problem harmonic motions in the classical regime of T), plus an
t
consists of independent motions of individual molecules: anharmonic term proportionalto T.
Figure 1 shows µ as a function of T, computed using
t
thedifferentaproximationsdescribedabove. Thefullline
−βµ = 1 lnZ =ln e−βEα (4) is the quantum result (Eq. (6)) obtained assuming the
t t
N ! nearly harmonic transverse potential of Eq. (5). It is
α
X interesting to note that the corresponding classical ex-
Here β−1 = k T and the sum is over all eigenvalues E pression, Eq. (8), is extremely similar to the quantum
B α
of the Schrodinger equation for transverse motion of a one (they coincide in the figure), showing that quantum
single molecule. This equationmay be evaluatedanalyt- effects are really negligible. The dashed line shows the
ically in the case when the transverse potential is nearly classical result Eq. (7) using the exact potential. For
harmonic,i.e. whenU includesjustasmallquarticterm, T >500K, the curves startto differ as the realshape of
involving a parameter γ: the potential deviates from the quartic aproximation.
3
The solution of the remaining 1D problem of interact- The rightmostintegralmay alternativelybe expressed
ing buckyballs requires a numerical calculation, in gen- intermsofthe1Dcompressibility,κ=−(∂lna/∂P) as
T
eral. Quantum calculations have been carried out for follows:
some 1D fluids (helium and hydrogen2,3 in their ground
states), but for classical systems at finite T the deter-
a k T 1
mination of the equation of state is more straightfor- ∆µ (T,a)= B − da′ (13)
ward. Specifically, the equation of a 1D classical system int ∞ a′ κ(a′,T)
Z (cid:18) (cid:19)
assumesananalyticforminthecasewhennearestneigh-
Because the attraction dominates the effect of the in-
bor interactions adequately represent the total potential
teractionovermostofthedensityrangeshowninFigures
energy20. We assume that to be the case here (and jus-
2to4,thechemicalpotentialshiftduetothe interaction
tify that in Section III) so that we may write down an
is negative until one reaches high density.
expression relating the 1D density, N/L=1/a, in terms
of the 1D pressure P:
III. CORRECTIONS TO THE MODEL
1 ∞dz e−β[V(z)+zP]
a = ∞0 dz z e−β[V(z)+zP] (10) We assess three corrections to the preceding section’s
R0
results. One is the long range correction to the thermo-
where V(z) is the CR60-C60 interaction given by Eq. (2). dynamic properties. This is the energy shift associated
Similar results were reported recently by Hodak and withtheneglectofinteractionsotherthannearestneigh-
Girifalco16,17. borsandcouldbewrittenintermsoftheprobabilityper
Figure 2 shows the “compression ratio”, i.e. the ratio unit length P (x) that a second, third or other (than
>
of P to the pressure of an ideal 1D gas (kBT/a); the in- first) nearest neighbor is present at distance x. Since we
teractioneffects canbe clearlyobserved. The pressure is seek only anestimate of its effect, we take P (x) to be a
>
evidently reduced by interactions over an extended den- Heaviside step function for |x|>2σ, where σ is the hard
sity range until the point when the molecules start to core diameter (7.1 ˚A) of the C molecule. Then, our
60
come nearly into contact (a ≈ 12.5 ˚A at T = 300 K), approximate energy correction per molecule due to this
abovewhichdensitythe pressurerisesextremelyrapidly. neglect is given by
Oneobservesthatthe C gasbehaviorisnearlyidealat
60
T >1000 K while the effects of the interaction are quite
∞
significantbelowroomtemperature. TheBoyletempera- dz
∆E /N = V(z) (14)
>
tureis around1400K.The calculatedbehaviorandhigh a
Z2σ
characteristic temperature are not surprising in view of
Note that a “missing” factor of (1/2) is cancelled by
thefactthatthewelldepthofthepairpotentialisabout
the omission of the region z < 0. If the potential V(z)
3200 K.
werethe familiarLennard-Jones6-12interaction,the re-
The 1D pressure can be integrated (according to the
sultof this integrationwouldbe ∆E /(Nǫ)=−σ/(40a)
Gibbs-Duhem relation) to obtain the total chemical po- >
(where ǫ is the welldepth), which representsa negligible
tentialoftheC fluidattemperatureT anddensity1/a:
60
correction of the energy.
The second correction we assess is the quantum cor-
P
′ ′ rection. To do this, we introduce a phonon model as a
µ(T,a)=µ (T)+µ (T,a )+ a(P ,T)dP (11)
t id 0
means of characterizing the dynamics of the fluid. As
ZP0
was shown recently5, for a Lennard-Jones system in its
Here µ (T,a ) = k T ln(λ /a ) is the chemical po-
id 0 B T 0 ground state, the ratio of the zero-point energy E to
zp
tentialofa1Didealgasatareferencedensity(1/a )that
0 the pair potential well depth ǫ is
is low enoughfor the fluid to be ideal21. The integration
domainextendsfromthereferencepressureP =k T/a
0 B 0
to the pressure corresponding to the specified spacing a. Ezp =24/3 3 Λ∗ (15)
TheresultofthisanalysisappearsinFigure3forarange ǫ π2
of temperatures.
Here Λ∗ = h/[σ(mǫ)1/2] is the de Boer quantum pa-
Figure 4 exhibits the effects of the interaction on the
rameter. In the case of a C modeled by such an
chemical potential, expressed in terms of the difference 60
interaction22, Λ∗ ≈ 0.003, so this ratio is 0.002. This
between the chemical potential of the (fully interacting)
indicates that the quantum effects are small even in
systemandthatofanidealsystemwhichhasthemutual
the ground state. They become even smaller (relatively
interactions “turned off”:
speaking) with increasing T and can be neglected at
T >E /k , which is of order 10 K for C .
zp B 60
P ′ kBFTinally, ′we estimate the effect on the mutual interac-
∆µ (T,a)=µ(T,a)−[µ (T)+µ (T,a)]= a(P ,T)− dP
int t id
tioPnenergyofthemolecules’motionperpendiculartothe
ZP0 (cid:18) (cid:19)
(12) axis. To do so, we evaluate the expectation value of the
4
differenceinpotentialenergyduetochangingtheapprox-
imate separation of a pair of molecules (zij) used in Eq. Nν ∞
(lo1w)einsttoortdheeririfnultlhye3tDrasnespvaerrasteiosnepraijra=tio(nρ,2ijt+hezi2dji)ff1e/2re.nTcoe ∆Finter = 2a Z−∞ dz V[(b2+z2)1/2] (18)
in potential energy is ∆V(z) = V′(z)[ρ2ij/(2zij)], where Here b is the separation between tubes and ν is the
the prime means a derivative. Substituting the mean number of nearest neighbor tubes. This expression as-
value hρ2iji = 2d2, the energy shift per molecule due to sumes a simple form in the case when b is sufficiently
this substitution is obtained from the radialdistribution large that the interaction may be approximated by the
function g1D(z): asymptotic form V(r)≈−C6/r6, in which case the inte-
gration yields
∆E V′(z)d2
t
N = dz g1D(z) z (16) ∆F =−νN 3π C6 1 (19)
Z inter 16 b5 a
Forthisestimate,wesubstituteforg1D(z)astepfunc- Including this term in the equation of state yields a
tion at z = σ. The result for a Lennard-Jones potential relation for the shift of the 1D pressure proportional to
is 1/a2
∆E 24 d2 α 3π C6
t P =P − ; α=ν (20)
=− (17) 1D a2 16 b5
Nǫ 91 aσ
Note that α = (3νπ/16)bV(b); thus it is proportional
Fora(10,10)tubeinahighdensitycase,witha=σ ≈
to the product of the strength of the transverse inter-
9.5 ˚A, for example, at room temperature (d = 0.5 ˚A),
action and the number of neighbors interacting with a
weobtain∆E /(Nǫ)=6x10−4,whichisnegligiblysmall
t molecule (≈b/a).
(using ǫ ≈ 3200 K). Although this is a relatively crude
Figure5showstheeffectonthe isothermsofincluding
estimate, the order of magnitude is plausible in view of
the intertube interaction term. One observes that van
the smallratio ofd/a. Evidently,the smallvalue of ∆E
t der Waals loops develop, indicating the appearance of a
implies that the 1D approximation is adequate for the
phasetransition. Figure6showshowthecriticaltemper-
present circumstances.
atureT varieswiththeparameterα. Foratypicallattice
c
of (10,10) tubes with b = 17 ˚A and ν = 6, α ≈ 2180 K
˚Awhich yields T ≈ 530 K. There is a singular rise of
c
IV. TRANSITIONS DUE TO INTERACTIONS T for small α, followed by a region of weaker depen-
c
WITH NEIGHBORING TUBES dence. We may understand this behavior by examining
analternativemodel,theanisotropicIsingmodelapplied
In this section we evaluate an interesting possibility- to the lattice gas version of this problem. As shown by
transitionsofthe C fluidas aresultofmolecularinter- Fisher25,forthecaseofsuchamodelwithnearestneigh-
60
actions between buckyballs in neighboring tubes. In our bor interactions Jl along the 1D chains and transverse
analysis,weassumeaninfinite arrayofperfectly parallel interactions Jt, the critical temperature is given by the
and identical tubes and chemical equilibrium among the expression
C molecules. Under these circumstances, molecules in
60
nearbytubescanundergoacooperativetransition,acon-
2J
l
densationtoa3Dliquid. Aqualitativelysimilarproblem kBTc = (21)
ln(1/c)−ln[ln(1/c)]
(condensationofheliumconfinedwithininterstitialchan-
nelsofacarbonnanotubebundle)wasstudiedpreviously where c = J /J. This is an asymptotic formula valid
t l
byourgroup,usingaMonteCarloevaluationofalattice when c < 0.1 (weak transverse interaction). Figure 6
gas model6. showsthe dependence derivedfromthis expressionwhen
The present analysis is carried out first by perturba- the values J = 800 K and J = 18.2 K are substituted
l t
tion theory; then, a comparison is made with an ex- in the Fisher formula. The result is seen to track the
actly solved model that captures the essential physics perturbationtheorybehaviorwell. Thissimilarityissur-
but employs a number of simplifications. Although nei- prising, at first sight, because the perturbation theory
thermodeliscompletelysatisfactory,wewillseethatthe is a mean field theory, neglecting density fluctuations in
two sets of results are quite similar because the key pre- adjacent tubes. We rationalize this agreement by noting
dictions (parameters of the criticalpoint) turn out to be that the exact calculation yields critical exponents (3D
only weakly dependent on the assumptions. Ising model) that are, in fact, different from those of the
Theperturbationtheoryisbasedontheshiftinfreeen- mean field perturbation theory. On the other hand, in
ergyofthesystemduetointeractionsbetweenmolecules 3D, one does not expect a large error in the mean field
in neighboring tubes, ∆F : theory value of the critical temperature23.
inter
5
System |Φ(r0)| (eV) r0 (˚A) r1 (˚A) where δ = b/a. The difference would be identically zero
iftheEuler-MacLaurintheoremwereanaliticallyusedto
C60-C60 0.278 10.05 7.10
evaluate it. The integral can be analitically evaluated
C60-(10,10) tube 0.537 13.28 10.12 yielding 3π/(8δ5). Hence, we compute the sum numeri-
cally and find the difference. It is negligible small when
TABLE I: Parameters of the universal function Eq. (2) for
b/a>2 and increases as b/a goes to zero. This is shown
thecarbon structures investigated.
in Figure 7, expressed as a freezing temperature
Finally, we note that Carraro has described the possi-
bilityofacrystallizationtransitionforsuchaproblem24. Tf =∆Ef/kB (23)
To estimate the onset temperature for this freezing, we
simply evaluate the energy difference ∆Ef between the In the case of C60 inside (10,10) tubes, b/a = 1.7 and
ground state crystal and the energy of the alternative Tf ≈ 0.5K. Such small values of Tf imply that such a
fluid state. This is determined by comparing the energy transition will be difficult to observe in the laboratory.
computed above (∆F ) for the intertube interaction
inter
energy. The difference is just the difference between the
integralandthe sumthatariseswhenthe buckyballsare Acknowledgments
located in lattice sites, separated by a:
WearegratefultoVictorBakaev,MaryJ.Bojan,Karl
∞ Johnson,CarloCarraro,LouisGirifalco,MiroHodakand
νC 1 dz
∆E =− 6 − DavidLuzziforstimulatingandhelpfulcommentsandto
f 2a6 " (δ2+n2)3 −∞ (δ2+z2)3# NSF and the Petroleum Research Fund of the American
n Z
X
(22) Chemical Society for their support.
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Nanotubes”, toappearintheEncyclopediaofNanoscience 15 M. Hodak, L.A. Girifalco, Chem. Phys. Lett. 350, 405
and Nanotechnology to be published by American Scien- (2001).
tific Publishers, editor H.S. Nalwa. 16 M.Hodak,PhDthesis,UniversityofPennsylvania,unpub-
2 M.Boninsegni,S.Y.Lee,V.H.Crespi,Phys.Rev.Lett.86, lished (2002).
3360 (2001). 17 L.A. Girifalco and Miroslav Hodak, Appl. Phys. A, in
3 J.Boronat,M.C.Gordillo,andJ.Casulleras,J.LowTemp. press.
Phys. 126, 199 (2002); M.C. Gordillo, J. Boronat, J. Ca- 18 L.A. Girifalco and Miroslav Hodak, Phys. Rev. B 65,
sulleras, Phys. Rev. B 65, 014503 (2002); M.C. Gordillo, 125404 (2002).
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(2000); M.C. Gordillo, J. Boronat, J. Casulleras, Phys. Phys. Rev.B 59, 4708 (1999).
Rev.Lett. 85, 2348 (2000). 20 H. Takahashi, Mathematical Physics in One Dimension,
4 M.M. Calbi, M.W. Cole, S.M. Gatica, M.J. Bojan and G. edited by E.H. Lieb and D.C. Mattis (Academic Press,
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(2002). 21 The choice of this reference is arbitrary because the a0
6 M.W. Cole, V.H. Crespi, G. Stan, C. Ebner, J.M. Hart- term cancels out after the integral is performed.
man, S. Moroni, and M. Boninsegni, Phys. Rev. Lett. 84, 22 If a Lennard Jones type of potential was used to charac-
3883 (2000). terizetheC60−C60 interaction,theparameterswouldbe:
7 M.K. Kostov, H. Cheng, R.M. Herman, M.W. Cole, J.C. ǫ≈ 3200 K and σ≈ 9.5 ˚A.
Lewis, J. Chem Phys. 116, 1720 (2002). 23 Forthe3DIsingmodel,themeanfieldtheory’serrorinTc
8 B.W. Smith, M. Monthioux and D.E. Luzzi, Nature 296, is ≈ 33%.
323 (1998). 24 C.Carraro,Phys.Rev.Lett.89,115702(2002);Phys.Rev.
9 M. Monthioux,Carbon 40, 1809 (2002). B 61, R16351 (2000).
10 J. Vavro, M.C. Llaguno, B.C. Satishkumar, D.E. Luzzi, 25 M.E. Fisher, Phys. Rev. 162, 480 (1967); T. Graim and
and J.E. Fisher, App. Phys. Lett. 80, 1450 (2002); D.E. D.P. Landau, Phys.Rev.B 24, 5156 (1981).
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(2000).
6
-3
-3.5
-4
)
V
e -4.5
(
t
µ
-5
-5.5
-6
0 500 1000 1500 2000 2500 3000
T(K)
FIG.1: Thetransversechemicalpotentialµt asfunctionoftemperature. Thefulllinecorrespondstothequarticaproximation
of thepotential (Eq.5); quantumand classical results (Eqs. (6) and (8)) are so similar that they cannot be distinguished over
thewhole range of T. Thedashed line shows theclassical result using thefull potential, Eq. (7).
7
T=1300 K
1
T=1000 K
l
a
e
d T=700 K
i
P

/
0.5

P
T=500 K
T=300 K
0
0 0.02 0.04 0.06 0.08 0.1
-1
1/a (Å )
FIG. 2: Compression ratio at various temperatures of the 1D pressure P to that of an ideal 1D gas, Pideal = kBT/a, as a
function of the density1/a.
8
-3.2
-3.4 T=300 K
-3.6
T=500 K
) -3.8
V T=700 K
e
( -4

µ
-4.2
T=1000 K
-4.4
-4.6
-4.8
0 0.02 0.04 0.06 0.08 0.1
-1
1/a (Å )
FIG. 3: Total chemical potential µ(T,a) = µt(T)+µz(T,a) of the C60 fluid as a function of molecular density 1/a and
temperature T.
9
-3.1
-3.2
T=300 K
-3.3
)
V
e T=500 K
-3.4
(

t
n
T=700 K
i
µ -3.5

∆

-3.6
T=1000 K
-3.7
-3.8
0 0.02 0.04 0.06 0.08 0.1
-1
1/a (Å )
FIG. 4: Chemical potential shift dueto theintermolecular interactions ∆µint(T,a) as a function of 1/a.
10
30
1300 K
1000 K
20
700 K
)
Å 10
500 K
/
K
(

P 0
300 K
-10
100 K
-20
0 0.02 0.04 0.06 0.08 0.1
-1
1/a (Å )
FIG.5: EffectsoftheinteractionbetweenC60 moleculesinnearestneighbortubes(α=2180K˚A).Theisothermsdevelopvan
derWaals loops as a function of the temperature, with a critical temperaturearound 530 K.