Radial thermal expansion of pure and Xe-saturated bundles of single-walled carbon nanotubes at low temperatures
ABSTRACT The radial thermal expansion coefficient (a)r of pure and Xe-saturated bundles of single-walled carbon nanotubes has been measured in the interval 2.2-120 K. The coefficient is positive above T = 5.5 K and negative at lower temperatures. The experiment was made using a low temperature capacitance dilatometer with a sensitivity of 2x10-9 cm and the sample was prepared by compacting a CNT powder such that the pressure applied oriented the nanotube axes perpendicular to the axis of the cylindrical sample. The data show that individual nanotubes have a negative thermal expansion while the solid compacted material has a positive expansion coefficient due to expansion of the intertube volume in the bundles. Doping the nanotubes with Xe caused a sharp increase in the magnitude of (a)r in the whole range of temperatures used, and a peak in the dependence (a)r (T) in the interval 50-65 K. A subsequent decrease in the Xe concentration lowered the peak considerably but had little effect on the thermal expansion coefficient of the sample outside the region of the peak. The features revealed have been explained qualitatively. Comment: 12 pages,6 figures
Radial thermal expansion of pure and Xe-saturated bundles of single-
walled carbon nanotubes at low temperatures
A.V. Dolbin1, V.B. Esel'son1, V.G. Gavrilko1, V.G. Manzhelii1, S.N. Popov1, N.A.Vinnikov1,
N.I. Danilenko2, B. Sundqvist3.
1 B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of
Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine
2 Frantsevich Institute for Problems of Materials Science of the National Academy of Sciences of
Ukraine, Krzhizhanovsky str., 3, Kyiv 03680, Ukraine
3 Department of Physics, Umea University, SE - 901 87 Umea, Sweden
Electronic address: firstname.lastname@example.org
Published: Fizika Nizkikh Temperatur, 2009, v. 35, No. 6, p. 613–621
The radial thermal expansion coefficient αr of pure and Xe-saturated bundles of single-
walled carbon nanotubes has been measured in the interval 2.2—120 K. The coefficient is positive
above T = 5.5 K and negative at lower temperatures. The experiment was made using a low
temperature capacitance dilatometer with a sensitivity of 2·10-9 cm and the sample was prepared by
compacting a CNT powder such that the pressure applied oriented the nanotube axes perpendicular
to the axis of the cylindrical sample. The data show that individual nanotubes have a negative
thermal expansion while the solid compacted material has a positive expansion coefficient due to
expansion of the intertube volume in the bundles. Doping the nanotubes with Xe caused a sharp
increase in the magnitude of αr in the whole range of temperatures used, and a peak in the
dependence αr (T) in the interval 50—65 K. A subsequent decrease in the Xe concentration lowered
the peak considerably but had little effect on the thermal expansion coefficient of the sample outside
the region of the peak. The features revealed have been explained qualitatively.
Since the discovery of carbon nanotubes (CNTs) in 1991 , this novel class of physical
objects has been stimulating intense experimental and theoretical research activities. The diversity
of CNT types and the problems encountered in obtaining pure CNT material in quantities needed
for experimental investigations make it rather difficult to trace the basic trends in the behavior of
carbon nanotubes (e.g., see the text and References in ). Thermal expansion is one of the least
studied properties of CNTs. The currently available experimental evidence on the thermal expansion
of single-walled nanotubes (SWNTs) and their bundles is confined to the region near and above
room temperature, whereas low temperature data are essential for understanding the CNT dynamics.
The theoretically estimated thermal expansion coefficients (TEC) of SWNTs [3—9] vary
appreciably both in magnitude and sign.
Owing to their unique geometry, CNTs can be a basis for forming novel low-dimensional
systems. For example, bundles can be used as templates to form one-dimensional chains or two-
dimensional surfaces consisting of condensed impurity molecules.
In recent years much experimental effort has been devoted to the study of structural and
thermal properties of such systems and a number of theoretical models have been advanced to
predict these properties [10—23]. However, the thermal expansion behavior of SWNT—gas
impurity systems still remains obscure.
In this study the radial thermal expansion was measured on a sample consisting of bundles
of single-walled nanotubes closed at the ends (c-SWNT) in the range T = 2.2—120 K and on
bundles of SWNTs saturated with Xe at T = 2.2—75 K. The sorption properties of bundles of
SWNTs with closed (c-SWNT) and open (o-SWNT) ends were investigated using the technique
1.Radial thermal expansion of pure single-walled carbon nanotubes
1.1.Measurement technique and investigated sample
The sample for thermal expansion measurements was prepared using a procedure for
ordering the SWNT axes by applying a pressure of 1 GPa, as described by Bendiab et al. .
These authors showed that in SWNT plates of up to 0.4 mm thickness, such a pressure aligned the
CNT axes in the sample such that their average angular deviation from a plane normal to the
pressure vector was ~4º.
The starting material was a CNT powder (CCVD method, Cheap Tubes, USA) which
according to the manufacturer contained over 90% of SWNTs. The main characteristics are given in
Table I. Characteristics of carbon nanotube powder given by the manufacturer.
Diameter l-2 nm
Length 5-30 µm
SWNT fraction > 90 wt %
Amorphous carbon fraction < 1.5 wt %
Co catalyst fraction 2,9 wt %
> 407 m2/g Specific surface
Electrical conduction > 102 S/cm
The quality of the powder was confirmed by Raman analysis performed both by the
supplying company and at Umeå University, Sweden. According to the manufacturer, the average
outer diameter of the tubes was 1.1 nm but no information is available about the chirality
distribution. From our own Raman data, obtained using four different excitation lasers with
wavelengths in the range 541-830 nm, we find that the radial breathing modes indicate a wide range
of tube diameters, 0.8 – 2.1 nm. All samples studied show typical SWNT G-bands and only weak
disorder bands. Although multi-wall tubes may also be present, judging from the spread in
diameters, the Raman spectra are completely dominated by the response from single (or possibly
few-) wall nanotubes. However, a small fraction of MWNTs might be invisible due to their large
diameters and possibly lower Raman cross sections.
The starting SWNT powder was also investigated by high-resolution transmission electron
microscopy (HRTEM) at both the Institute of Problems of Material Science, NAS of Ukraine (Fig.
1a) and at Umea University, Sweden (Fig. 1b). The pictures show that large sample fractions
contain little amorphous carbon or residual catalyst. By measuring the bundle diameters we
estimate that in the starting powder each bundle contains 7 to 600 SWNTs.
Fig. 1 TEM images of the starting SWNT powder.
The compacted sample used was prepared at Umea University (Sweden) by first
compacting pressure-oriented (P = 1.1 GPa) SWNT plates (an individual plate was up to 0.4 mm
thick), then pressing several stacked plates together at a ten percent higher pressure to form a
cylinder 7.2 mm high and 10 mm in diameter with a density of 1.2 g/cm3. The sample was made in
a special cylindrical segmented die designed for compacting CNT powder under effective pressures
0.5—2 GPa, consisting of a ring with a cylindrical channel and a conical outer surface, which was
inserted into a hardened-steel cylinder supported inside a larger pressure vessel. The structure so
arranged was resistant to internal stresses. The 10 mm in diameter piston was made from sintered
tungsten carbide (WC). The pressures used were high enough to consolidate the powder into a solid
with well oriented tubes , but still low enough to keep the integrity and structure of the tubes
and avoid tube collapse, and Raman spectra taken on pressed plates showed no systematic changes
relative to spectra taken on the pristine powder. The sample prepared by this technology has a
pronounced anisotropy of properties in the directions perpendicular and parallel to the sample axis.
In the direction perpendicular to the applied pressure the axes of the SWNT bundles are disordered.
The compaction aligns the axes of the SWNT bundles in the plane perpendicular to the sample axis
. As a result the radial component of the expansion of the SWNT bundles makes a dominant
contribution to the thermal expansion of the sample in the directional parallel to the sample axis. If
the axial component of the thermal expansion coefficient has a magnitude comparable to that of the
radial one, an angle of typically 4o implies that the typical contribution to the total coefficient from
the axial component is about 7 % of the magnitude of the radial component.
The radial thermal expansion of the sample was investigated using a capacitance dilatometer
(its design and the measurement technique are described in ). The linear thermal expansion
coefficient (LTEC) was measured in the direction of the applied compacting pressure, i.e. radially to
the SWNT bundles. Prior to measurement, the gas impurities were removed from the sample by
dynamic evacuation for 72 hours at 10-3 mm Hg and room temperature. Immediately before
dilatometric investigation, the measuring cell with the sample was cooled slowly (for 8 hours) down
to liquid helium temperature (4.2 K) and the sample was held at this temperature for about 4 hours.
The cooling and investigation were made in vacuum down to 10-5 mm Hg.
1.2.Experimental results and discussion
The temperature dependence of the LTEC in the interval 2.2—120 K is shown in Fig. 2. The
curves were obtained by least-square averaging over several series of measurement.
Fig. 2. LTECs of pressure-oriented SWNT compacted sample in the direction perpendicular
to the SWNT bundle axes: a) T = 2.2—120 K; b) T = 2.2—25 K (curve 1 — heating and cooling,
curve 2 — first heating from T = 2.2 K).
Curve 2 was taken on the first heating of the sample from T = 2.2 K. Curve 1 data were
measured in the subsequent heating-cooling process. The non-equilibrium LTECs obtained on the
first heating from T = 2.2 K may account for heating-induced alignment and ordering of the bundles
and the nanotubes in them, which causes a compression of bundles and, as a result, negative thermal
The equilibrium radial LTEC αr (curve 1) is positive above 5.5 K and negative at lower
Assuming that the impurity effect is negligible, αr comprises two components αd and αg
accounting for temperature-induced changes in the CNT diameters and the intertube gap. From a
simple Grüneisen-type model it might be expected that αd should be similar to the in-plane thermal
expansion of graphite, and thus probably small and negative well below room temperature. Because
the sample is a mixture of all chiralities, the average αd should also be very similar to the average
axial expansion coefficient of the tubes. The thermal expansion of a bundle should thus probably be
dominated by αg, which should be similar to the out-of-plane thermal expansion of graphite or,
considering the curvature, to the thermal expansion of fullerenes or linear fullerene polymers .
So far there has been only one study  in which both αd and αg were measured by the X-
ray diffraction method in the interval 300—950 K. At T = 300 K αr = (0.75 ± 0.25) · 10-5 K-1, αd =
(-0.15 ± 0.2) · 10-5 K-1 and αg = (4.2 ± 1.4) · 10-5 K-1. Another measurement, of αr only, by the same
method  arrived at negative values in the whole range of measurement temperatures (200—
1600 K). We are not aware of further experimental attempts to directly investigate the thermal
expansion of SWNT bundles, but some experiments have been made to estimate the thermal
expansion from the temperature dependence of the radial breathing Raman modes of nanotubes.
Although these modes shift down rapidly with increasing temperature, indicating a large strong
positive thermal expansion coefficient, it was concluded by Raravikar et al.  that this effect is
almost completely caused by changes in intra- and intertube interactions, and that αd is very small.
It is rather problematic to compare our results with theoretical data quantitatively, mainly
because the available theoretical studies are concerned with the radial and axial thermal expansion
of individual CNTs. Some of them offer general conjectures on how thermal expansion can be
affected by the interaction of nanotubes in a bundle (e.g., see ). Also, there is little agreement
between the theoretical conclusions from different groups about the TEC magnitude, sign and
temperature dependence, about the effect of chirality and CNT diameter upon thermal expansion,
and about the correlation between the radial and axial components of the thermal expansion of
nanotubes. For example, the thermal expansion is negative in a wide temperature interval
(0—800 K) in , changes from negative magnitudes at low temperatures to positive ones at
moderate and high temperatures in  or is positive at all the temperatures investigated in .
The qualitative interpretation of our results is based on the Grüneisen coefficients calculated
 for carbon modifications — diamond, graphene, graphite and nanotubes. It is found  that the
Grüneisen coefficients and the radial thermal expansion of CNTs are negative at relatively low
temperatures, an effect caused mainly by the contribution from transverse acoustic vibrations
perpendicular to the CNT surface. However, our measurements show that a negative thermal
expansion coefficient exists only in a temperature interval much more narrow than found in the
calculations . We believe that the main reason for this is that the calculations were performed for
individual nanotubes only . Our sample is clearly dominated by CNT bundles (Fig. 1), and in this
case additional factors contributing to the thermal expansion come into play. Firstly, there appears a
positive contribution αg caused by variations of the intertube gaps with temperature. Secondly, the
nanotube interaction in the bundles suppresses the negative contribution of the transverse acoustic
vibrations perpendicular to the nanotube surfaces . These two positive contributions to the
thermal expansion of SWNT bundles decrease both the magnitude and the temperature region of the
total negative thermal expansion. If we use this model and assume αd to vary slowly with
temperature over a wide temperature interval we can use the data shown in Fig. 2b to estimate αd =
(-4 ± 1) ⋅ 10-8 K-1 at T = 2.2 K. Assuming further that the temperature dependent part of α at low
temperatures is dominated by a positive coefficient αg, we see from Fig. 2b that a polynomial of the
third order in T is a good approximation to αg(T) up to about 25 K. Although the scatter in the data
is somewhat high it is clear that to get a good fit it is necessary to include one term in T3 and one
term linear (or, with a less good fit, quadratic) in T. In a Grüneisen model, the thermal expansion
coefficient of a bundle is closely related to its specific heat capacity, and it is well known that the
experimentally found low-temperature specific heat of nanotube bundles shows a similar behaviour
above 2 K . In that case the experimental behaviour cp(T) = aT + bT3 could be fitted by an
anisotropic two-band Debye model with weak coupling between tubes in the bundle by adding a
contribution from the first optic branch. It thus seems quite reasonable to attribute the strongly
temperature dependent positive component of the total thermal expansion to αg. The data in Fig. 2a
also shows a noticeable plateau-like structure between 40 and 60 K. We point out that the
intermolecular interaction in C60, which should be similar in magnitude to the inter-tube interaction,
corresponds to an effective Debye temperature near 50-60 K which gives rise to a plateau in the
specific heat in this range for both molecular and polymeric C60 . The plateau structure
observed here might thus indicate the cross-over between the acoustic modes and the lowest
optical/molecular 3D modes in the bundle lattice.
2. Xe sorption in the powder of carbon nanotubes with closed and
2.1.Measurement technique and investigated samples
Carbon nanotubes (CNT) prepared by standard methods (electric-arc, laser evaporation of
carbon, or CCVD method) are arranged into bundles. Inside a bundle the CNTs form a close-packed
two-dimensional (2D) triangular lattice . Normally, CNTs have fullerene-like semispheres at the
ends (CNTs with closed ends, or c-SWNT). The final CNT product can contain large amounts of
amorphous carbon, fullerenes and other carbon modifications [2, 31—37]. The currently used
methods of cleaning CNT materials involve oxidative treatment with acid-oxidant mixtures, ozone
, etc. They lead to partial or complete opening of the CNT ends and produce defects at the
The possible sites for sorption of gas impurity molecules in bundles of infinite, open and
equal-diameter SWNTs are shown in Fig. 3. However, in practice such SWNT systems can have
additional zones of impurity sorption. For example, nanotubes of different diameters form
rather large channels parallel to the nanotube axes, which can be occupied by impurity molecules
. Besides, oxidation can produce interstices between the nanotubes in a bundle .
Fig. 3. Sites of possible sorption of gas impurity molecules in bundles of infinite, open and equal-
We investigated Xe sorption in c-SWNT and o-SWNT powders at T = 78—200 K. The
choice of the temperature interval and the impurity was dictated by the following considerations.
The interaction of gas impurities with different parts of the CNT surface is most evident at low
temperatures. Owing to their geometric configuration, SWNT bundles ideally (Fig. 3) have
favorable sites where sorption of impurity molecules is energy-advantageous. A number of
theoretical models were proposed [39—43] to describe the physical sorption and dynamics of
admixed gas molecules at the surface and in the interstitial channels of SWNT bundles. According
to mathematical simulations , the inner CNT surfaces and the interstices between the
neighboring tubes at the surface of SWNT bundles (the grooves — G, Fig. 3) are the most energy-
advantageous sites for sorbing impurity gas molecules. Xe was used because the SWNT—Xe
system is already a well-studied “model” system [44—47]. A Xe atom is too large to penetrate into
the interstitial channels (IC) of close-packed bundles of identical nanotubes whose energy of
binding to impurity molecules is comparable to that at the inner surface . Therefore, the Xe
impurity is sorbed inside a nanotube (I), in a groove between two neighboring tubes at the outer
surface of a bundle (G) and at the surface of the individual tubes forming the outer surface of a
bundle (S) (see Fig. 3).
To obtain the necessary information about gas impurity desorption from CNT materials, a
laboratory test bench (Fig. 4) was constructed for investigating the process of Xe sorption and
desorption in a CNT powder at T = 78—200 K.
The measuring cell V1 containing a CNT sample was filled with Xe at 12 torr and cooled
slowly to T = 78 K. At this temperature the xenon available in the cell was sorbed by the CNT
powder and condensed on the cell walls. The cell temperature was then increased in steps of 5 K.
The Xe evaporated from the cell surface and was desorbed from different sites of the SWNT bundle
surface. The evaporated Xe was condensed in the vessel V2 cooled with liquid nitrogen. When the
stepwise heating brought the pressure in the V1—V2 system to a constant value, the cell V1 with the
sample was separated from the vessel V2. The Xe condensed in the vessel V2 was evaporated and its
pressure in the system was measured with the capacitive pressure transducer 5. With the volume of
the system known, we could estimate the quantity of Xe desorbed from the sample at a particular
temperature. To reduce the error due to the temperature gradient over the vessel V2, the vessels V2
and V3 were minimized to the form of capillaries 1 mm in diameter. After each measurement run,
Xe was recondensed from vessel V2 to vessel V3.
Fig. 4. Schematic view of the laboratory test bench for investigation of gas sorption-desorption in
CNT samples at low temperatures.
1 – Sample of nanotubes
2, 3, 10 – Heaters
4, 11 – Temperature sensor (silicone diode DT – 470)
5 – Pressure transducer (capacitance manometer MKS Baratron 627B)
6 – Gas input
7 – Digital multimeter (Keithley 2700)
8 – Temperature controller (Cryo-Con model 34)
9 – Matching device (Advantech PCI – 1670)
2.2.Results and discussion
The sorption properties of the starting pure c-SWNT powder (0.0416 g) were investigated
through thermo-programmed desorption (see above). Fig. 5 illustrates the temperature distribution
of the desorbed impurity. The greatest quantities of Xe were desorbed at T = 125—135 K. In the
case of close-packed bundles of infinite equal-diameter SWNTs (Fig. 3), the highest desorption of
Xe in this temperature interval is expected from the grooves at the outer bundle surface (G) and
from the interior channels of some nanotubes (I) because Xe atoms have the highest and nearly
equal binding energies at these sites . In our powder the desorption can be enhanced
considerably by removal of Xe atoms from the axial large-diameter channels (IC). Such channels
are possible in bundles of nanotubes of varying diameters. Xenon can penetrate into interior
channels through defects at the ends or the lateral surfaces that can be present in some nanotubes of
the starting powder. A rather small quantity of Xe was also desorbed at T = 100—105 K, which may
be due to removal of the layers (S) of Xe molecules that form at the outer surface of SWNT
To open the nanotube ends, a portion (0.0705 g) of the starting powder was placed into a
capsule which was then evacuated for 8 hours and heated to 450 ºC. At this temperature the capsule
was filled with air for 12 min. under atmospheric pressure. According to the literature data, the ends
of over 90% of CNTs can be opened through this procedure . Thereafter, the capsule was
evacuated again to about 10-3 mm Hg, heated to 750 ºC and held at this temperature for an hour to
remove the gaseous oxidation products. The post-treatment weighting showed a loss of ~ 5% in the
The sorption properties of the nanotubes with the opened ends were then investigated using
the same thermoprogrammed desorption technique (see above). The oxidation-induced
opening of the CNT ends made the inner CNT surfaces and the intertube interstice in the bundles
accessible to Xe sorption , which enhanced the sorption capacity of the SWNT powder almost
fivefold as compared to the starting material (see Fig. 5).
Fig. 5. Temperature distribution of Xe impurity (mole per mole and mole per gram)
desorbed from powder samples of c-SWNTs (dark columns) and o-SWNTs (light columns).
3. Radial thermal expansion of xenon-saturated single-walled carbon
The radial thermal expansion of Xe-saturated SWNTs was also investigated on the
compacted sample used previously to measure the LTECs of pure SWNTs. The measurement
technique is described in Section 1. Immediately before measurement, the cell with a pure CNT
sample was evacuated at room temperature for 96 hours and then filled with Xe at 760 mm Hg. The
evacuated measuring cell of the dilatometer with the sample in the Xe atmosphere was cooled to 90
K. At this temperature it was evacuated again and then cooled to liquid helium temperature. The
thermal expansion was measured in vacuum down to 1·10-5 mm Hg.
The temperature dependence of the LTEC taken on a Xe—SWNT sample in the interval
2.2—75 K is shown in Fig. 6 (curve 1). The sharp increase in the LTECs of the Xe-saturated sample
(cf. curves 1, 3) can reasonably be attributed to the heavy Xe atoms affecting the transverse
vibrations of the nanotubes in the direction perpendicular to their surface. At low temperatures the
Grüneisen coefficients of such vibrations are negative in two-dimensional (graphene) or quasi-two-
dimensional (graphite, nanotubes) carbon systems [8, 49] and positive in a three-dimensional
carbon modification (diamond). The formation of SWNT bundles and the sorption of impurity
atoms at the surface or inside the nanotubes generate three-dimensional features in the system. As a
result, the negative Grüneisen coefficients of such system decrease in magnitude or become
positive. The thermal expansion coefficients are expected to behave in a similar way. That is why
the negative contribution to the radial thermal expansion of Xe-saturated SWNT bundles decreases
and shifts towards lower temperatures (see Fig. 6; cf. curves 1, 2 and 3).
In contrast to pure CNTs the thermal expansion of Xe-saturated SWNTs is similar during the
first heating and in the subsequent heating and cooling runs. It is possible that the first heating of
pure SWNT bundles with xenon can make the system more rigid and its geometry insensitive to
heating at low temperatures.
Fig. 6. The radial LTECs of SWNT bundles: 1 – Xe-saturated bundles, 2 – after partial Xe
removal at T = 110 K, 3 – pure SWNTs (αr) at T = 2.2—75 K (a) and 2.2—7 K (b), compacted
It is interesting that the LTECs have maximum values in the interval 50—65 K, which may
be a manifestation of spatial redistribution of the Xe atoms in the SWNT bundles. The simulation
(by Wang-Landau algorithm) [23, 50] of the potential energy for a system of SWNT bundles
saturated with inert gases predicted peaks in the temperature dependence of the heat capacity at T =
50—100 K, attributed to reordering of the impurity atoms.
To test the prediction, it was necessary to remove the Xe impurity from the surface of the
SWNT bundles. For this purpose, the sample was heated to T = 110 K. This temperature causes
intensive desorption of Xe from the sample surface but leaves it undisturbed in the grooves of the
SWNT bundles (G) and the inner interstices (I) of the nanotubes having surface defects (Fig. 5).
The sample was kept at T = 110 K until the desorbed Xe was entirely removed and the pressure in
the measuring cell reached ~ 1·10-5 mm Hg. The sample was then cooled to T = 2.2 K and the
thermal expansion was measured again (Fig. 6, curve 2). It is seen that the LTEC peak is much
lower after Xe was removed from the SWNT bundle surfaces. However, this partial Xe desorption
leaves the temperature dependence of the LTEC practically unaffected outside the interval of the
peak. This suggests that the Xe atoms residing on the bundle surface have only a small effect upon
the thermal expansion of SWNT bundles when the process of spatial redistribution of atoms are
This is the first time that the temperature dependences of the radial thermal expansion
coefficients αr (T) of pure and Xe-saturated SWNT bundles have been investigated experimentally
at low temperatures. The measurements were made on heating and cooling the samples in the
interval 2.2—120 K using a capacitance dilatometer.
The dependence αr (T) measured on the first heating showed very strong nonequilibrium
effects, and in the interval 3.2—120 K it differed significantly from the well reproducible
equilibrium dependences αr (T) that were found on subsequent heating and cooling runs in this
The equilibrium coefficients of the radial thermal expansion αr (Fig. 2, curve 1) are positive
above 5.5 K and negative at lower temperatures. The nonequilibrium coefficients of the radial
thermal expansion αr (Fig. 2, curve 2) are negative in the interval 2.2—82 K. It is assumed that the
non-equilibrium αr-values measured on the first heating of the sample are due to the irreversible
alignment and ordering of the bundle positions and the nanotubes in the bundles at rising
temperature. As this occurs, the density of the system increases, and the thermal
expansion becomes negative.
The qualitative interpretation of the equilibrium dependence αr (T) was based on the
theoretical conclusions about the Grüneisen coefficients for carbon modifications . The
Grüneisen coefficient and the radial thermal expansion of nanotubes are negative at reasonably low
temperatures , which is determined mainly by the contribution of the transverse acoustic
vibrations perpendicular to the nanotube surfaces. However, in the experiment the temperature
interval of the negative thermal expansion is much narrower in comparison with the theoretical
predictions. This is most likely because the cited theory  investigated individual nanotubes.
Additional contributions to the thermal expansion come into play in SWNT bundles. First, there is a
positive contribution αg, generated by the variations of the intertube gaps with temperature. In
addition, the nanotube interaction in the bundles suppresses the negative contribution from the
transverse acoustic vibrations perpendicular to the nanotube surfaces . These two positive
contributions to the thermal expansion of the SWNT bundles decrease both the magnitude and the
temperature interval of the negative thermal expansion.
The saturation of SWNT bundles with xenon brings about new features in their thermal
1) The magnitude of αr increases sharply in the whole range of temperature investigated. This is
because the Xe impurity suppresses the negative contribution to the thermal expansion from the
transverse acoustic vibrations perpendicular to the nanotube surfaces .
2) The dependence αr (T) has a peak in the interval 50—65 K, which appears to be due to the spatial
redistribution of the Xe atoms over the SWNT bundle surfaces. Removal of the Xe impurity from
these surfaces decreases the peak significantly but leaves the temperature dependence of the LTEC
practically unchanged outside the interval of the peak. This suggests that the Xe atoms atoms
located at the bundle surfaces have little effect on the thermal expansion of SWNT bundles when
the processes of their spatial redistribution are inoperative.
3) For the Xe saturated material there is no non-equilibrium thermal expansion behaviour such as
was observed during the first heating of the sample and attributed to irreversible alignment and
ordering of the bundle positions and the nanotubes in the bundles at rising temperature. It is likely
that the saturation with Xe makes the system of SWNT bundles more rigid and its geometry
insensitive to heating in a low temperature interval.
Finally, the employed technique of thermoprogrammed desorption has also enabled us to
measure the temperature dependence of Xe desorption from both open and closed SWNT bundles.
We wish to thank Prof. V.M. Loktev for valuable discussion.
The authors are indebted to the Science and Technology Center of Ukraine (STCU) for the
financial support of this study (project No 4266).
1. S. Iijima, Nature 354, 56 (1991)
2. A. V. Eletskii, Phys. Usp. 47, 1119 (2004)
3. H. Jiang, B. Liu and Y. Huang, J. Eng. Mater. Technol. 126, 265 (2004)
4. Y. Kwon, S. Berber and D. Tomanek, Phys. Rev. Lett. 92, 015901 (2004)
5. N. M. Prakash. Determination of coefficient of thermal expansion of single-walled carbon
nanotubes using molecular dynamics simulation. The Florida State University. Dissertation
Master of Science, (2005) p. 54.
6. C. Li and T. Chou, Phys. Rev. B 71, 235414 (2005)
7. N. R. Raravikar, P. Keblinski, A. M. Rao, M. S. Dresselhaus, L. S. Schadler and P.
M. Ajayan, Phys. Rev. B 66, 235424 (2002)
8. P. K. Schelling and P. Keblinski, Phys. Rev. B 68, 035425 (2003)
9. G. Cao, X. Chen and J. W. Kysar, Phys. Rev. B 72, 235404 (2005)
10. B. K. Pradhan, A. R. Harutyunyan, D. Stojkovic, J. C. Grossman, P. Zhang, M. W. Cole, V.
Crespi, H. Goto, J. Fujiwara and P. C. Eklund, J. Mater. Res. 17, 2209 (2002)
11. M. R. Johnson, S. Rols, P. Wass, M. Muris, M. Bienfait, P. Zeppenfeld and N. Dupont-
Pavlovsky, Chem. Phys. 293, 217 (2003)
12. S. Ramachandran, T. A. Wilson, D. Vandervelde, D. K. Holmes and O. E. Vilches, J. Low
Temp. Phys. 134, 115 (2004)
13. Y. H. Kahng, R. B. Hallock and E. Dujardin, Physica B: Cond. Matter 329, 280 (2003)
14. T. Wilson, A. Tyburski, M. R. DePies, O. E. Vilches, D. Becquet and M. Bienfait, J. Low
Temp.e Phys. 126, 403 (2002)
15. F. R. Hung, K. E. Gubbins, R. Radhakrishnan, K. Szostak, F. Beguin, G. Dudziak and M.
Sliwinska-Bartkowiak, Appl. Phys. Lett. 86, 103110 (2005)
16. H. Chen, J. K. Johnson and D. S. Sholl, J. Phys. Chem. B 110, 1971 (2006)
17. C. Matranga, L. Chen, B. Bockrath and J. K. Johnson, Phys. Rev. B 70, 165416 (2004)
18. A. Kuznetsova, J. T. J. Yates, V. V. Simonyan, J. K. Johnson, C. B. Huffman and R. E.
Smalley, J. Chem. Phys. 115, 6691 (2001)
19. Adsorption by carbons, Eds. E. Bottani and J. Tascón, ELSEVIER, Amsterdam (2008)
20. D. G. Narehood, J. V. Pearce, P. C. Eklund, P. E. Sokol, R. E. Lechner, J. Pieper, J. R.
Copley and J. C. Cook, Phys. Rev. B 67, 205409 (2003)
21. S. M. Gatica, M. J. Bojan, G. Stan, and M. W. Cole, J. Chem. Phys. 114, 3765 (2001)
22. M. M. Calbi, S. M. Gatica, M. J. Bojan, and M. W. Cole, J. Chem. Phys. 115, 9975 (2001)
23. N. M. Urban, S. M. Gatica, M. W. Cole and J. L. Riccardo, Phys. Rev. B 71, 245410 (2005)
24. N. Bendiab, R. Almairac, J. Sauvajol and S. Rols, J. Appl. Phys. 93, 1769 (2002)
25. A. N. Aleksandrovskii, V. B. Esel'son, V. G. Manzhelii, B. G. Udovidchenko, A. V.
Soldatov and B. Sundqvist, Fiz. Nizk. Temp. 23, 1256 (1997) [Low Temp. Phys. 23, 943
26. P. Nagel, V. Pasler, S. Lebedkin, A. Soldatov, C. Meingast, B. Sundqvist, P.-A. Persson, T.
Tanaka, K. Komatsu, S. Buga and A. Inaba, Phys. Rev. B 60, 16920 (1999).
27. Y. Maniwa, R. Fujiwara, H. Kira, H. Tou, H. Kataura, S. Suzuki, Y. Achiba, E. Nishibori,
M. Takata, M. Sakata, A. Fujiwara and H. Suematsu, Phys. Rev. B 64, 241402 (2001)
28. Y. Yosida, J. Appl. Phys. 87, 3338 (2000)
29. J. Hone, B. Batlogg, Z. Benes, A.T. Johnson and J.E. Fischer, Science 289, 1730 (2000).
30. A. Inaba, T. Matsuo, Å. Fransson and B. Sundqvist, J. Chem. Phys. 110, 12226 (1999).
31. A. Thess, R. Lee, P. Nikolaev, H. Dai, P. Petit, J. Robert, C. Xu, Y. H. Lee, S. G. Kim, A.
G. Rinzler, D. T. Colbert, G. E. Scuseria, D. Tománek, J. E. Fischer and R. E. Smalley,
Science 273, 483 (1996)
32. A. G. Rinzler, J. Liu, H. Dai, P. Nikolaev, C. B. Human, F. J. Rodriguez-Macias, P. J. Boul,
A. H. Lu, D. Heymann, D. T. Colbert, R. S. Lee, J. E. Fischer, A. M. Rao, P. C. Eklund and
R. E. Smalley, Appl. Phys. A 67, 29 (1998)
33. A. C. Dillon, T. Gennett, K. M. Jones, J. L. Alleman, P. A. Parilla and M. J.
Heben, Adv. Mater. 11, 1354 (1999)
34. А. В. Елецкий, Б. М. Смирнов, УФН 163, 33 (1993)
35. A. V. Eletskii and B. M. Smirnov, Phys. Usp. 38, 935 (1995)
36. A. V. Eletskii, Phys. Usp. 40, 899 (1997)
37. B. P. Tarasov, N. F. Goldshleger and A. P. Moravsky, Russ. Chem. Rev. 70, 131 (2001)
38. D. B. Mawhinney, V. Naumenko, A. Kuznetsova, J. T. J. Yates, J. Liu and R. E. Smalley, J.
Am. Chem. Soc. 122, 2383 (2000)
39. W. Shi and J. K. Johnson, Phys. Rev. Lett. 91, 015504 (2003)
40. M. W. Cole, V. H. Crespi, G. Stan, C. Ebner, J. M. Hartman, S. Moroni and M. Boninsegni,
Phys. Rev. Lett. 84, 3883 (2000)
41. M. T. Cvitas and A. Šiber, Phys. Rev. B 67, 193401 (2003)
42. A. Šiber, Phys. Rev. B 66, 235414 (2002)
43. G. Stan, M. Bojan, S. Curtarolo, S. M. Gatica and M. W. Cole, Phys. Rev. B 62, 2173 (2000)
44. A. Kuznetsova, D. B. Mawhinney, V. Naumenko, J. T. J. Yates, J. Liu and R. E. Smalley,
Chem. Phys. Lett. 321, 292 (2000)
45. A. Kuznetsova, J. T. J. Yates, J. Liu and R. E. Smalley, J. Chem. Phys. 112, 9590 (2000)
46. V. V. Simonyan, J. K. Johnson, A. Kuznetsova and J. T. J. Yates, J. Chem. Phys. 114, 4180
47. H. Ulbricht, J. Kriebel, G. Moos and T. Hertel, Chem. Phys. Lett. 363, 252 (2002)
48. P. M. Ajayan, T. W. Ebbesen, T. Ichihashi and S. Iijima, Nature 362, 522 (1993)
49. A. C. Bailey and B. Yates, J. Appl. Phys. 41, 5088 (1970)
50. E. S. Daniel, M. U. Nathan and W. C. Milton, Phys. Rev. B 77, 205427 (2008)