A dvances in Physics, 1999, Vol. 48, No. 1, 1 134
Fullerenes under high pressuresB. Sundqvist Department of
Experimental Physics, Umea University, S-90187 Umea, Sweden
[Revision received 2 December 1997; accepted 11 December 1997]
Abstract This paper reviews the properties and phases of fullerenes
and their derivatives and compounds under high pressures. For
obvious reasons most of the paper deals with C60 but the materials
reviewed also include C70, simple derivatives of C60 , carbon
nanotubes, and intercalation compounds of C60 with both acceptors
and donors, mainly alkali metals. After a brief overview of
high-pressure techniques and the structures and properties of C60
at atmospheric pressure, the structural phase diagram of C60 from
atmospheric pressure to above 40GPa (400kbar) is reviewed. The
evolution with pressure of the orientational and translational
structure of `normal molecular C60 in the range up to 1 5 GPa
(depending on temperature) is discussed in some detail, as is the
appearance of a large number of polymeric phases at higher
pressures and temperatures, some of them known to have extreme
mechanical properties. At very high static (or shock) pressures or
temperatures, C60 transforms into ordered or disordered forms of
diamond or graphite. The phase diagramis reasonably well
investigated up to near 10GPa, but at higher pressures there are
still large gaps in our knowledge. Available experimental data for
the physical properties of both monomeric and polymeric C60 under
high pressures are reviewed as far as possible. The compression
behaviour of C60 has been well investigated and is discussed in
detail because of its basic importance, but optical, electrical and
lattice properties have also been studied for several of the many
structural phases of C60 . Whenever possible, experimental data are
compared with the results of theoretical calculations. The phase
diagram and properties of C70 are much less known because of the
larger complexity caused by the anisotropy of the molecule, and
very little is known about most compounds of C60 . However,
noble-gas intercalation in C60 has been reasonably well
investigated. Finally, the high-pressure properties of
superconducting alkali-metalintercalated C60 are brie y reviewed.
Contents 1. 2. 3. 4. Introduction Structures and phases of
fullerenes at zero pressure Some comments on high-pressure
techniques Structures and properties of molecular (monomeric) C60
4.1. Structural phase diagram 4.1.1. High-temperature behaviour and
the fcc phase 4.1.2. Fcc-to-sc transition boundary 4.1.3. The
orientational structure in the sc phase 4.1.4. The glassy crystal
transition and the properties of the glassy phase 4.1.5.
Low-pressure phase diagram of C60 4.1.6. Does liquid C60 exist?
4.2. Physical properties of monomeric C60 under pressure 4.2.1.
Compressibility and other elastic properties0001 8732/99 $1200
page2 4 7 9 9 10 11 14 19 21 23 24 24
1999 Taylor & Francis Ltd
2
B. Sundqvist4.2.2. Lattice vibrations 4.2.3. Thermal properties
4.2.4. Electronic band structure 4.2.4.1. Optical studies 4.2.4.2.
Positron annihilation 4.2.4.3. Other studies 4.2.5. Electrical
resistivity 4.2.6. Nuclear magnetic resonance 5. Structures and
properties of polymeric C60 5.1. Photopolymerization 5.2. Early
very-high-pressure studies 5.3. Low-pressure polymerized material (
p < 2 GPa) 5.4. Intermediate-pressure range (2 8 GPa) 5.5.
High-pressure range ( p > 8 GPa) 5.6. Transformations into other
forms of carbon 5.6.1. High static pressures and/or temperatures
5.6.2. Catalytic conversion to diamond 5.6.3. Shock wave
experiments 5.6.4. Other studies 5.7. Pressure temperature phase
diagram of C60 6. Structures and properties of C70 6.1. Phase
diagram 6.2. Physical properties of C70 6.2.1. Thermophysical
properties 6.2.2. Lattice vibrations 6.2.3. Electronic properties
7. Other fullerenes and fullerene compounds 7.1. Nanotubes 7.2.
C60O, C61 H2, C60Hx and ( C59N) 2 7.3. Other C60 complexes and
compounds 7.4. Intercalation of gases into C60 7.5. Endohedral
fullerenes 8. Alkali-metal- and alkaline-earth-metal-doped C60 8.1.
Normal-state properties 8.1.1. Bulk modulus 8.1.2. Electrical
resistivity 8.1.3. Other studies 8.2. Superconducting-state
properties 9. Comments, speculations and conclusions
Acknowledgments References 1. Introduction 37 44 46 46 48 49 50 53
54 54 54 56 64 73 79 79 81 81 82 82 83 83 89 89 91 92 93 93 94 97
98 100 101 102 102 104 107 108 113 115 115
It is hardly original or controversial to state that the identi
cation of the C60 molecule by Kroto et al. [1]and the subsequent
discovery by Kratschmer et al. [2]of a simple and inexpensive
method to produce large amounts of this and related materials must
be counted among the most important scienti c discoveries during
the last two decades of this century. Over the last 6 years,
research in this eld has developed at a breathtaking rate, exceeded
only by that on ceramic high-transitiontemperature superconductors,
and the discoverers were recently honoured by being awarded the
1996 Nobel Prize in chemistry. Not surprisingly, a large number of
books, reviews and conference proceedings have already been
published on this subject. A number of these will be referred to
below and it should be stated here that
Fullerenes under high pressures
3
the selection to some extent re ects the authors own
preferences, but in particular the fact that these particular
publications happened to be readily available at the time of
writing. For a particularly complete and up-to-date general review
of the eld the reader is referred to recent excellent book by
Dresselhaus et al. [3] to which , reference will often be made
below, but several other very general reviews [4 7] or collections
of reviews [8 10] have appeared over the last few years. However,
although the whole fullerene eld, various sub elds, and particular
properties of various fullerenes have been reviewed in di erent
publications, I am not aware of any previous general review of the
properties of C60 under high pressures. Only a small number of
short, rather specialized reviews [11 15]and some sections on
highpressure e ects in general reviews [3, 4, 7, 16] have so far
been published. This is somewhat surprising, since high-pressure
studies have given very important information on both pure and
doped fullerenes and contributed signi cantly to the rapid
developments in this eld, and many recent high-pressure studies
hint at possibilities for commercial exploitation of the structural
phases existing at high pressures. In this paper, I have tried to
collect as much as possible of the information available on the
high-pressure properties and phases of both pure and doped solid
fullerenes, as well as compounds and derivatives. Fullerenes are
here de ned in a broad sense to include both quasispherical
molecular species such as C60, C70 and C76 and larger complexes
such as `buckytubes and `buckyonions. Most high-pressure work so
far has of course been carried out on C60 because of the easy
availability of this material in pure form, but some information is
also available on other fullerenes. `High pressure is also very
loosely de ned, but as a general rule I have arbitrarily selected
100MPa (= 1 kbar) as the lower limit for inclusion, although in
some cases reference is made to work at lower pressures. The paper
will also concentrate on experimental results and discuss
theoretical work only brie y, partly because I am myself an
experimentalist, partly because only a small number of theoretical
papers have explicitly discussed the pressure or volume dependence
and partly because I feel that at the present level of
understanding the most important task is to collect and evaluate
available experimental data to build a reference platform on which
to build future work, experimental or theoretical. The rst version
of this review attempted to give the reader a view of the state of
our knowledge in the eld up to the autumn of 1996. During the
refereeing process the eld has developed rapidly in some areas and
the Editor has been kind enough to permit me to update the paper.
In this nal version I have therefore extended the text, in
particular in section 5, to take into account also the most
important developments between the initial submission of the paper
and the summer of 1997. I make no claim that the paper is complete
and I apologize for any oversights or errors that have succeeded in
creeping in, but I hope that I have included references to most of
the important work that has been carried out under high pressures.
The structure of the paper is as follows. A very brief and
incomplete overview is rst given of the ambient-pressure structural
phases of undoped fullerenes to set the stage for subsequent
sections. Since excellent reviews already exist in this area,
reference will be made either to original work or to available
reviews rather than presenting too many details. Before starting
with the actual review, some brief comments on high-pressure
studies and techniques are also given in section 3. In sections 4
and 5 the high-pressure phases, structures and properties of C60
will then be discussed in some detail since these have now been
extensively investigated by many groups. The discussion will be
divided into two main parts, with section 4
4
B. Sundqvist
discussing rst the rather well known structures of the phases
based on molecular (or monomeric) C60 and then various physical
properties of these phases under pressure, and section 5 discussing
the structure and properties of the less well studied polymeric
forms of C60 . It should be noted that most physical properties of
these phases have only been measured on metastable material at
`zero pressure, here de ned as any pressure near or below
atmospheric. The short sections 6 and 7 then discuss available
information on the phase diagrams and physical properties of C70
and other fullerenes, derivatives and compounds respectively,
before we turn to doped fullerenes and their properties under high
pressures in section 8. Finally, the paper is rounded o by a short
section 9 containing some general comments and conclusions.2.
Structures and phases of the fullerenes at zero pressure
This review is focused on solid fullerenes only and, since
little information is available for materials other than C60 and
C70 under high pressures, most of the text deals with these. In
this section a very brief introduction will be given to the
structures of the zero-pressure phases as a help to understand the
modi cations brought about by the application of a high pressure.
Information on the structural and dynamic properties of C60 at zero
pressure can be found in a number of reviews [3, 6, 16 21]which
will also be referred to where necessary below. The most convenient
method to produce fullerenes such as C60 in large amounts is to
evaporate graphite electrodes to soot under a low-pressure He gas
[2] usually , using an electric arc discharge [22] The soluble
fullerenes are then washed out of the . soot using toluene which on
evaporation leaves solid fullerene deposits behind. The production,
separation and puri cation of fullerenes and related materials have
recently been discussed in some detail by Dresselhaus et al. [3]
Fullerenes can also be . produced in ames [23], although normally
most molecules burn before leaving the ame, and claims have been
made for the detection of natural fullerenes in carbonrich minerals
(Shungite) [24], in fulgurites (rocks produced by melting minerals
during lightning discharges) [25], in sediments containing traces
of intense burning from the Cretaceous Tertiary boundary layer [26]
and even in trace amounts at , minuscule impact craters on
spacecrafts [27]. Whatever the source, solid fullerenes form black
or brownish powders or crystals. As discussed recently by
Dresselhaus et al. [3] and many others, the most stable crystal
structure at high temperatures is the same for all quasispherical
fullerenes (C60 , C70, C76 , etc. ). Because the molecules rotate
almost completely freely they are e ectively spherical and the
molecules form a close-packed face-centred cubic (fcc) structure.
The room-temperature ( RT) structure of C60 has been investigated
in very close detail by X-ray and neutron di raction. These studies
show [3, 21, 28 31] that the molecular rotation is not, in fact,
completely free, but that there is a strong intermolecular
orientational correlation even in this phase. Close to the
orientational transition these correlated clusters may reach 40 A
in size [30, 31]and thus consist of several dozen molecules. For
our purposes, however, the description of the molecules as freely
rotating is good enough. On cooling the rotation slows down and,
below an orientational transition temperature [3, 16 19, 29, 32,
33] To = 260K, the rotational motion is replaced by a combination
of rapid jumping motion between a number of well de ned
orientational states and a librational motion about the equilibrium
orientation in each such state. The
Fullerenes under high pressures
5
realization by Heiney et al. [32]that in this low-temperature
phase the molecules are orientationally ordered was one of the rst
of the many surprising and exciting discoveries concerning solid
C60 . The basic structure of the low-temperature phase is identical
with that of the fcc phase except that the four molecules in the
cubic unit cell have di erent orientations, changing the fcc
symmetry into a simple cubic (sc) structure. However, there are two
possible orientations [3, 29, 33] obtained by rotating the
molecules by about 38 and 98 respectively around the [111]axis of
the crystal from the `standard orientation. (A recent study
[34]showed that the actual angles may in fact be closer to 42 and
102 , values optimizing nearest-neighbour orientations rather than
long-range orientational order. ) In the resulting ordered
structures, electron-rich double bonds on one molecule face the
electron poor centres of hexagons (H orientation) and pentagons (P
orientation) respectively on its neighbour, as shown in the simpli
ed sketch in gure 1 (A). Since the latter description is easier to
visualize for non-specialists (including the author), I shall refer
in the following to H- or P-oriented molecules. The di erence in
energy is very small and the fraction of molecules in the more
stable P orientation is only about 60% near 260K ( gure 1 (B)),
increasing to about 84% near Tg = 90 K [33]. Below this the thermal
energy becomes too small compared with the energy threshold between
the two states for further reorientation to be possible, and below
a glass transition (or glassy crystal transition) temperature Tg
the remaining orientational disorder becomes frozen in creating an
orientational glass. Interestingly, the lattice parameter of
H-oriented sc C60 is slightly smaller than for the P-oriented
state. This can be observed in gure 2 as an anomalously low thermal
expansion coe cient a in the sc phase [29, 35] since for every
increase in temperature a certain number of , molecules reorient
from the majority P orientation to the minority H orientation,
giving a small volume decrease which adds to the normal volume
increase due to the increase in temperature. Most physical
properties of C60 have already been well studied in these three
phases at zero pressure and data can be found in the reviews given
above. In addition to the fcc, sc and `glassy states, a fourth
state, photopolymerized C60, insoluble in common solvents, can be
obtained by submitting thin lms of fcc C60 to high-intensity
visible or ultraviolet (UV) light [36]. This treatment is believed
to break up the sp2 double bonds joining adjacent hexagons on the
molecules and to reform the bonds to form four-membered carbon
rings linking neighbouring molecules by two single carbon bonds
with a strong sp3 character. Because C60 is a strong absorber of
light, only very thin lms can be completely polymerized in this way
and this has seriously hampered the study of this interesting
material, but it is believed that the polymer chains formed in this
way are probably quite disordered and that chain length is highly
variable with polymerization conditions. The situation regarding
the structure of C70 is much more complicated. Although molecular
dynamics calculations [37 40] have shown that the fcc structure
should theoretically be the most stable high-temperature structure,
in real crystals the other close-packed structure, hexagonal close
packed (hcp), very often appears. Because of the small di erence in
energy, most crystals contain enough stacking faults to be
considered as mixtures of the fcc and hcp phases. Molecular
rotation in C70 can also give a much more complicated phase diagram
because of the anisotropy of the molecule. `Free molecular rotation
(with the same caveats as above) seems to be the rule at high
temperatures, where the fcc and hcp structures are both based on
the resulting quasispherical molecular shape. Below about 350K,
rotation around the
6
B. Sundqvist
(A)
(B) Figure 1. (A) Molecules may be oriented with the double bond
on the black molecule facing either a pentagon (top) or hexagon
(bottom) on the grey neighbouring molecule. (B) Temperature
dependence of the fraction of C60 molecules in the more stable
Poriented state at atmospheric pressure. Insets (a) and (b) show
projections of molecules in the P- and H-oriented states
respectively. (Reprinted with permission from David et al. [33])
.
short molecular axes is believed to stop and the molecules
rotate around the long axis only. This rotation persists at least
down to about 280K. Taking the fcc phase as an example, most
workers agree that this phase is stable above 350K. Below this, the
axes of the uniaxially rotating molecules can either remain in
random directions or line up along random [111]directions, in both
cases giving a fcc structure, or line up
Fullerenes under high pressures
7
Figure 2. Cubic lattice parameter of C60 plotted against
temperature, showing clearly the glass transition temperature Tg
near 90K and the orientational transition temperature To near 260K.
( Reprinted with permission from David et al. [33]) .
in parallel along one particular [111] direction giving a
rhombohedral (rh) crystal structure. When the uniaxial rotation
nally stops below 280K, any of these structures may remain, or
molecules in the rh phase may order orientationally relative to
their nearest neighbours in much the same way as in C60 to form a
monoclinic (mc) phase. However, there are many other possibilities.
For example, it has been suggested that in the uniaxially rotating
phase (280 350K) the molecules behave as rotating tops, carrying
out a nutational movement where the long axis rotates around its
equilibrium orientation [40, 41] and two hcp phases have been ,
reported to occur at RT [42] Recent studies also show that uniaxial
rotation . continues down to about 150K [43 46] but it is still
unclear whether this , temperature corresponds to the arrest of
`free uniaxial rotation or to a glass transition similar to that
observed near 90 K in C60. Starting with the less stable hcp
structure at high temperatures a di erent structural evolution not
discussed here is believed to occur on cooling. It is thus obvious
that the structural properties of C70 are much less well known than
those of C60 , mainly because most samples are probably mixtures of
initially fcc and hcp materials and transitions belonging to both
structural sequences are often observed [47]. In fact, it may well
be thermodynamically impossible to create a pure single-phase
crystal of C70. As will be seen below, this is also re ected in the
present status regarding the knowledge of the phase diagram and the
physical properties of C70 .3. Some comments on high-pressure
techniques
Before discussing the phase diagrams and properties of
fullerenes under high pressures a few remarks should be made about
high-pressure experiments and techniques. Many experimental
techniques have been used in studies of fullerenes and it has
sometimes been suggested that the technique used has a large e ect
on the
8
B. Sundqvist
results obtained. The most obvious problem that may occur is
deformation of the sample. If the pressure is transmitted by a uid
medium (gas or liquid) with zero shear strength, the deformation
will be homogeneous and a cubic single crystal such as C60 will
simply su er a decrease in volume without any change in shape.
(However, relative changes in atomic positions within the molecule
or unit cell may still occur. ) In practice, low-pressure
experiments are often carried out under such hydrostatic conditions
but, at su ciently high pressures (greater than 12 GPa), only
non-hydrostatic solid media are available, and in this case shear
stress is unavoidable. The question is then: how important is shear
stress? Let us take the compressibility of C60, discussed in detail
in section 4.2.1, as an example. Many methods have been devised for
such studies and the most common method today is to measure
directly the lattice constant using X-ray or neutron di raction
methods which give information on the lattice structure in the same
experiment. Di raction methods can be used with or without a
pressure-transmitting medium, uid or solid. In very high-pressure
studies the sample is usually compressed without the use of an
additional medium, even if a `soft quasihydrostatic material such
as NaCl can be added in the cell to work as a combined
pressuretransmitting medium and in-situ pressure calibrant, but at
lower pressures a uid pressure medium is often used. The
compression properties can also be obtained by measuring the volume
of a sample mechanically at known pressures. In the
pistonand-cylinder geometry the piston displacement is measured
during compression and the maximum pressure is limited to 3 4 GPa.
Usually no pressure-transmitting medium is used since the presence
of such media makes it necessary to carry out additional
corrections for their often not very well de ned compression
properties. The compressibility of C60 has been investigated
experimentally by a large number of groups using all the above
methods and a variety of pressure-transmitting media, ranging from
noble gases to relying simply on the very small shear strength of
C60 itself [48] to produce quasihydrostatic conditions in the
experiment without a pressure-transmitting medium. As will be shown
below, measurements on the sc phase below 1 GPa give almost
identical results independent of whether noble gases, Freons, NaCl
or the C60 sample itself are used as pressure-transmitting medium,
showing that in this pressure range this property does not depend
on whether the pressure is hydrostatic or not. However,
surprisingly, at the lowest pressures, in the fcc phase, the
results from the same experiments di er signi cantly, such that the
resulting bulk moduli di er by a factor of up to two. Turning to
the higher-pressure range, it was observed already in the rst
very-high-pressure compression experiments on C60 by Duclos et al.
[49] that the compression properties were strongly dependent on the
pressure medium used, and it has been veri ed in many experiments
that the crystal structures observed can be very di erent under
hydrostatic and non-hydrostatic conditions since high
non-hydrostatic pressures seem to promote the formation of the
polymeric high-pressure phases discussed in section 5. The
evaluation of data measured under non-hydrostatic conditions thus
needs some care, since shear stress sometimes has a large e ect on
the measured properties and sometimes little or no e ect at all.
The most important e ect of nonhydrostatic pressure seems to be to
promote the formation of the high-pressure polymeric phases of C60
, and the di erences observed between data from experiments under
di erent conditions at very high pressures can usually be ascribed
to this e ect. The di erences observed between results obtained
under di erent
Fullerenes under high pressures
9
conditions in the fcc phase have not yet been explained but
possible causes are discussed below in section 4.2.1. The upper
pressure limits for hydrostatic conditions depend on temperature.
At room temperature, most light hydrocarbons, alcohols and oils
give hydrostatic conditions only below 0.5 2 GPa. Mixtures of light
hydrocarbons or alcohols do not crystallize easily and can be used
up to higher pressures. Common examples are the 50 50 mixture of
isopentane and n-pentane which can be used up to 5 7 GPa, depending
on how we de ne `hydrostatic, and the 4 : 1 mixture of methanol and
ethanol which is hydrostatic up to greater than 10 GPa [50].
Although even He solidi es already near 11.5 GPa [51] the use of
liquid or solid noble gases allows pressures to remain
approximately hydrostatic up to about 60 GPa [52]. However, as will
be shown in section 7.4 such media cannot be used in the case of
fullerenes. A number of good books on experimental high-pressure
techniques have been published recently [53] To give a brief
overview, at low pressures (less than 3 GPa), . either
piston-and-cylinder or directly pumped cylinder devices are usually
used. Depending on the design and material such devices can be used
for almost any type of scienti c study, from optical spectroscopy
(using quartz, sapphire or other windows) and electrical transport
studies to mechanical compressibility measurements and neutron
scattering studies, over a temperature range from below 4 K to
2000K. If a higher pressure is desired, large-volume belt-type
devices [53], often made from tungsten carbide, can be used over
similar ranges in temperature. These can also be used for neutron
and X-ray scattering studies and for resistivity measurements. The
upper pressure limit is usually 8 10 GPa, but maximum pressures
approaching 20 GPa have been achieved [54]. Finally, for the
highest pressures, anvil devices are used. Most present-day studies
are carried out in diamond anvil cells (DACs) [52, 53] consisting
of two natural diamonds, about 1 carat in size, between the
surfaces of which the sample is squeezed. These devices have a very
convenient size and can usually be compressed by hand using a
screw. The sample is usually placed in a small hole (up to 0.5 mm
in diameter) in a simple metal gasket between the diamonds, and
this hole also gives the possibility of using a uid
pressure-transmitting medium. Because DACs in many cases are
capable of generating pressures well in excess of 100GPa (1Mbar),
techniques have been developed to use liquid noble gases, such as
liquid He, as pressure media even at RT and above. If a larger
sample but a lower maximum pressure is desired, anvil devices
[53]based on sintered diamond, sapphire, tungsten carbide or even
steel [55] can be used. Although resistance or neutron scattering
measurements are sometimes carried out in anvil devices, most
anvil-type devices are constructed from sapphire or diamond and
mainly designed for optical and/or X-ray di raction studies.4.
Structure and properties of molecular (monomeric) C60
4.1. Structural phase diagram As described in section 2 above,
solid C60 forms two crystal phases at zero pressure, a fcc phase
with quasifree molecular rotation above To = 260K and a nominally
orientationally ordered sc phase with a strongly
temperature-dependent orientational disorder below. Near Tg < 90
K, molecular reorientation becomes su ciently slow that the
orientational disorder appears `frozen in most experiments, and the
resulting low-temperature structure is usually described as an
orientational glass. The application of pressure changes the
intermolecular distances and thus the
10
B. Sundqvist
intermolecular interactions, leading to large changes in the
transition temperatures and the orientational order. This section
will discuss the e ects of pressure on the translational and
orientational structure, starting in the fcc phase and working down
towards low temperatures, and map the low-pressure part of the
presssure temperature phase diagram. At high pressures, polymeric
phases are formed but, in this section, only monomeric C60 , that
is materials built up from free molecules without covalent
intermolecular bonds, will be discussed. A small number of reviews
of the structural evolution with pressure in this range have been
published, dealing mainly with the fcc-to-sc transition [13] and
the orientational structure in the sc phase [14, 15] . 4.1.1.
High-temperature behaviour and the fcc phase A simple phase
diagram, drawn on logarithmic scales to bring out clearly the
low-pressure behaviour, has been given by Poirier et al. [56]and is
shown in gure 3. The diagram shows that solid fcc C60 sublimes on
heating without forming a liquid phase at normal pressures. (The
possible existence of a liquid phase under pressure will be
discussed in section 4.1.6.) The vapour pressure of C60 has been
measured by several groups [56 60] and may be extrapolated as in
the gure to nd an e ective upper phase boundary for the fcc phase.
Vorobev and Eletskii [61, 62]have shown that this curve can be
redrawn in reduced coordinates to coincide with that for noble
gases. A molecular dynamics simulation of the sublimation of C60 at
2700K and
Figure 3. Phase diagram of C60 drawn on a logarithmic pressure
scale to show the lowpressure range: ( ), data from experiment;
(---), theoretical curves or extrapolations from experiment. The
liquid phase indicated has not yet been observed (see text).
(Reprinted with permission from Poirier et al. [56]) .
Fullerenes under high pressures
11
50MPa has also been published [63] but it is uncertain to what
extent this is realistic , since the C60 molecules have been
reported by Sundar et al. [64]to start to break up into amorphous
carbon near 700 C (973K). However, Kolodney et al. [65]reported
that free molecules are thermally stable to much higher
temperatures. It may be speculated that material produced by the
standard solvent extraction method has a lower amorphization
temperature than sublimed C60 because of the unavoidable presence
of remaining solvent molecules in the lattice. Little is known
about the detailed e ects of pressure on the dynamic (rotational)
properties in the fcc phase, but it is reasonable to assume that
the increase in intermolecular interaction with pressure will slow
down the molecular rotation and increase the rotational anisotropy
observed even at zero pressure [21, 28 31] At high . pressures the
rotational motion will probably approach the ratcheting motion
observed in the sc phase, where the molecules instead of rotating
quasifreely will jump between a number of discrete orientational
states. The decrease in the intermolecular distance with increasing
pressure and the continuation of molecular rotation increase the
probability that double bonds on neighbouring molecules can
interact to form intermolecular covalent bonds, and as a result the
application of pressures greater than 1 GPa at high temperatures
usually leads to the formation of polymeric phases, to be discussed
in some detail in section 5. At lower temperatures, the application
of pressure and/or cooling at any pressure rst gives a transition
into the well known sc phase. 4.1.2. Fcc-to-sc transition boundary
The orientational ordering or fcc-to-sc transition was initially
suspected [32]to be a continuous transition but thermal and
structural studies soon showed it to be rst order with a small
hysteresis and connected with rather large volume changes D V <
1% [29, 33, 66] as well as entropy changes D S [67] The transition
can thus . be detected fairly easily by a variety of methods,
although both the transition temperature and the size and shape of
the transition anomalies are very sensitive to deformation and
impurities. Since the rst measurements of the compressibility of
C60 [49, 68] were carried out almost simultaneously with the
discovery of the orientationally ordered sc phase [32] it was
immediately realized that the application , of pressure to this
soft material should lead to signi cant changes in the
intermolecular interactions, which in turn might give signi cant e
ects on the transition temperature To. To investigate this idea,
Samara et al. [69] and almost simul, taneously Kriza et al. [70]
used di erential thermal analysis (DTA) to measure the , fcc-to-sc
transition temperature as a function of pressure. The two groups
used virtually identical techniques and found similar results for
the slope of the phase line, dTo /dp = 104KGPa- 1 [69] and 117KGPa-
1 [70] the very large values re ecting , the large e ects of
pressure on the intermolecular potential. However, later
investigations have shown that these values are not, in fact,
intrinsic to pure C60 for which the true slope of the phase line is
even larger. Both groups applied pressure using helium gas which
rapidly di uses into the interstitial cavities of solid C60 ,
changing the physical properties of the material and also the phase
diagram. Such e ects are discussed in some detail in section 7.4.
The slope of the fcc-to-sc phase line has later been reinvestigated
by a large number of techniques, including DTA, nuclear magnetic
resonance (NMR), electron paramagnetic resonance (EPR), compression
measurements and resistivity measurements, using several di erent
pressure-transmitting media. The results from a
12
B. Sundqvist
Table 1. Experimental data for the slope dTo /dp of the
fcc-to-sc phase boundary. If only one pressure value is given, the
experiment was carried out at a single temperature or pressure and
a literature value for the zero-pressure value taken as a second
reference point. The error limits given are estimates by the
original authors Pressure range Pressure (GPa) medium 0 0.8 0 0.5 0
0.8 0.6 0 0.5 0.6 0.6 0 0.3 0 0.5 0 0.8 0 1.4 0 0.5 0 0.5 0 0.5 He
He He He He Ne Ar Ar Ar N2 Pentane Isopentane Oil b dTc /dp (K GPa-
1) 104 6 2 117 109 6 4 111 119.5 118 141 174.2 165 164 6 2 159 6 3
160 132 6 3 163
Technique used DTAa DTA DTAa Compressibility Thermal expansion
Compressibility Compressibility Thermal expansion Compressibility
DTAa DTAa NMR Resistance Compressibility
Reference Samara et al. [69] Kriza et al. [70] Samara et al.
[71] Schirber et al. [72] Grube [73] Schirber et al. [72] Schirber
et al. [72] Grube [73] Pintschovius et al. [74] Samara et al. [71,
75] Samara et al. [71, 75] Kerkoud and co-workers [76, 77] Matsuura
et al. [78] Lundin and Sundqvist [79, 80]
0.4 Alcohol 0 0.2 Te on 0 1.26 Talc epoxy 0.5 (Solid) 0.4 ?a
b
90 Raman Meletov et al. [81 83] 120 Thermal conductivity
Andersson et al. [84] 99 Resistancea Ramasesha and Singh [85] 6 5
Positron annihilation Jean et al. [86] 65 100 6 10 EPR Kempinski et
al. [87]
Double transition observed (see text). Pressure applied directly
to the C60 sample itself.
number of such studies are collected in table 1, subdivided into
three groups. The rst group contains studies carried out in the
intercalating noble gases neon and helium, the second the results
of studies using non-intercalating hydrostatic media and methods
that give well de ned transition pressures, and the third group
contains studies carried out in solid media or over small ranges in
T (such as RT only). While the results given in the third group
show a very large scatter, as might be expected, those given in the
rst two groups are relatively well de ned but di er between the
groups. This is well illustrated in gure 4, reprinted from Samara
et al. [71] and showing the di erence between phase lines obtained
in helium and N2 gas respectively. Inspection of table 1 shows that
ve of the values in the second group are in almost perfect
agreement and thus that the `best value for the phase line slope of
pure C60 can be taken as 162 6 2K GPa- 1 , while in helium and neon
a signi cantly smaller average slope dTo /dp = 113KGPa- 1 is found.
The accepted explanation for the lower slope under these conditions
[71, 75]is that the presence of intercalated gas atoms in the
lattice interstitials increases the bulk modulus of the lattice.
Since the transition is observed to occur at the same
intermolecular distance in all noble gases [72] this means that at
any given temperature the transition occurs , at a higher pressure
in the intercalated materials. The slope dTo /dP can be calculated
from simple thermodynamic considerations. Samara et al. [71] used
the known volume change D V at the transition [33, 66]
Fullerenes under high pressures
13
Figure 4. Pressure-temperature phase diagram of C60, showing the
fcc-to-sc phase boundary as measured in helium and in N2.
(Reprinted with permission from Samara et al. [71] ) (1GPa= 10kbar.
) .
together with the Clausius Clapeyron relation dTo /dp = D V /D S
to calculate a value for D S at the transition which turned out to
be in very good agreement with experimental data [67] The volume
change at the transition is close to 1% at zero . pressure [33,
66]but there are reports that indicate that D V decreases with
increasing pressure. Lundin and Sundqvist [80] found that, for pure
C60 , D V decreased to about 0.7% near 0.5 GPa, while Grube [73]
found a similar rate of decrease in an experiment in helium gas to
0.4 GPa. A similar rapid decrease in D V under pressure was also
observed at the fcc-to-rh transition in C70 [88], where D V was
even found to approach zero near 1.5GPa. Since the slope is
independent of pressure, this indicates that D S must also
decrease, and Lundin and Sundqvist [80] speculated that in C60 the
transition might change character under pressure from rst order to
continuous in analogy with the orientational transition in NH4 Cl
[89] However, very recent . accurate neutron studies of C60 by
Pintschovius et al. [74]showed no such decrease in D V with
increasing pressure. Surprisingly, some studies of V against p in
C60 fail to observe any transition anomaly at all [90 92] For the
C60 C70 mixture studied by Lundin et al. [90]this is . not
unexpected, since in such mixtures the transition is often quite
smeared in T (or p), but both the `mechanical study of Bao et al.
[91] and the neutron work by Blaschko et al. [92] were carried out
on pure C60 which should have a clearly detectable volume anomaly.
The slope of the phase line has also been calculated theoretically
using other methods. Several groups have calculated this slope as a
test of di erent intermolecular potentials, basically van der
Waals-type potentials with various added electrostatic interactions
(to be brie y discussed in section 4.2.1). Although Burgos et al.
[93] calculate a very low dTo /dp = 40 KGPa- 1 , better agreement
with experiments is found in other calculations with slightly di
erent potentials; Lu et al. [94]reported a value of 115KGPa- 1 and
Cheng et al. [37]120KGPa- 1 . Lamoen
14
B. Sundqvist
and Michel [95] instead calculated the slope in a model which
assumed that the transition originates in a coupling between
molecular orientations and acoustic lattice displacements and found
dTo /dp = 182KGPa- 1 , which like the two former values is in
reasonable agreement with experiments. Finally, it should be
pointed out that many researchers have observed anomalies more
complicated than would be expected for a simple rst-order
transformation. Already in their rst study Samara et al. [69]found
that the DTA peaks developed from being slightly anisotropic at
zero pressure to having a de nite shoulder, perhaps indicating the
presence of more than one peak, at the highest pressures used.
Although they could not rule out a direct impurity e ect, Samara et
al. suggested that the double-peak structure may result from
partial inhibition of the molecular rotation by interstitial
impurities, such that the molecules can have two rotational states,
hindered and unhindered rotation, about 5 K apart in energy. Rasolt
[96]instead discussed the precursor e ects observed by Samara et
al. in terms of uctuations in short-range orientational order and
deduced that the transition was a uctuation-driven transition. More
clearly de ned double transitions were observed in the resistance
studies by Ramasesha and Singh [85] who found that the , di erence
in transition temperature depended on pressure and approached zero
at zero pressure, and very recently a double transition with a
splitting of 0.1 0.3 K has also been observed in highly pure C60 at
zero pressure in a thermal study using modulated di erential
scanning calorimetry [97]. It is thus possible that an intermediate
structural state exists in a very narrow range close to To. (The
anomaly could in principle be connected with surface `melting, but
the surface layers have been observed to be rotationally disordered
already at temperatures signi cantly below To [98].) 4.1.3. The
orientational structure in the sc phase As discussed in section 2
there are two possible orientational states, the P (pentagon) and H
(hexagon) orientations, in the sc phase at zero pressure. Far below
260K the relative fractions of P- and H-oriented molecules are
given by the usual thermal distribution 1 (4.1) f (T) = , 1 + exp
(- /kT ) where is the energy di erence between the two states.
(Close to To uctuations or other e ects tend to bring the actual
measured f closer to 50%.) The P-oriented state has a slightly
lower energy and is the preferred orientation at zero pressure, but
the H state has a slightly smaller molecular volume and should thus
be energetically preferable at high pressures. Because is very
small, pressure has a large e ect on the details of the
intermolecular interaction in C60 . Using the observed di erence in
molecular volume and simple thermodynamic arguments, David and
Ibberson [99] showed that the relative energies of the two states
should cross already at quite low pressures. They also veri ed this
experimentally by neutron di raction studies under pressure between
150 and 200K. At 150K, their data showed that the ratio [P]/[H]of
P- to H-oriented molecules decreased from about 70/30 at zero
pressure to 50/50 at an `equilibrium pressure peq = 191MPa (where
obviously = 0). The equilibrium pressure is probably weakly
temperature dependent, since the orientational potential should
primarily depend on the intermolecular distance and thus the volume
rather than the pressure. If we assume that the equilibrium occurs
at a certain (temperature-
Fullerenes under high pressures
15
Figure 5. Low-temperature phase diagram of C60 as suggested by
Sundqvist et al. [103] .
independent) molecular volume, we can use data for the thermal
expansion [29, 35] and the compressibility (section 4.2.1) to nd
that peq should decrease to near 165MPa at 100K and to increase to
about 217MPa and 242MPa at 200K and 250K respectively. This is, of
course, a rst approximation only, but later highpressure studies of
the thermal expansion [73] the compressibility [74] and the ,
thermal conductivity [100] as well as high-pressure Raman studies
[101]all indicate , that the equilibrium pressure near 90 100K
indeed is closer to 150MPa than to 200MPa. Above peq , the stable
orientational state should be the H orientation. At zero pressure
the energy di erence between the two states is too small for any
completely P-oriented state to be observed at any temperature but,
since peq is quite low, it should be possible to apply pressures
high enough for to be much greater than kT, and thus an almost
completely H-oriented structure. This was rst suggested by
Sundqvist et al. [102, 103]to explain anomalies observed in their
experimental data for the compression properties [79]and the
thermal conductivity [104]of C60 under pressure. By making a simple
linear extrapolation of the fraction f of P-oriented molecules
against pressure at 150K they deduced an approximate `phase line
for the formation of such an H-oriented phase and obtained the
low-temperature phase diagram shown in gure 5, as well as an
approximate value for f ( T ) at any pressure [14] As discussed in
detail elsewhere [14] the phase diagram obtained agreed with . ,
their observations of anomalies in the measured bulk modulus under
high pressures ( gure 19 below) and the thermal conductivity. Soon
after, Wolk et al. [101] made Raman scattering studies as a
function of pressure and temperature and observed the appearance of
new lines due to libron modes. From these they also deduced the
existence of a preferably H-oriented phase above peq < 150MPa,
as shown in gure 6. Blaschko et al. [92]later carried out
high-pressure neutron scattering studies which showed much better
agreement with an assumed predominantly H-oriented phase [P]/[H]=
70/30 than with a predominantly P-oriented phase at 1.6 GPa and RT.
Although the existence of a predominantly H-oriented phase has thus
been veri ed, the form of equation (4.1) in principle forbids a
completely P- or H-oriented
16
B. Sundqvist
Figure 6. Low-temperature phase diagram of C60 as suggested by
Wolk et al. [101] (reprinted with permission from the authors).
Figure 7. The function f ( T ) from equation (4.1), showing the
quasilinear low-T behaviour (---) and the observed constant f below
Tg for C60 (- -). ( Reprinted from Lundin and Sundqvist [80]) .
phase to exist at any temperature above 0K, and no well de ned
`phase boundary should exist for an H-oriented phase. It is
therefore surprising that anomalies in bulk physical properties
have indeed been observed [79, 80]near pressures corresponding to
extrapolated `phase transition pressures. Figure 7 shows the
theoretical fraction of P-oriented molecules as a universal
function of kT / [80] Because of the form of . the equation there
is always a quasilinear increase or decrease in f ( T ) over
[P]/[H] ratios from about 80/20 to about 98/2, after which there is
a nal more gradual exponential decrease in the minority
orientation. Any property sensitive to the orientational structure
should thus experience a fairly sharp change in its temperature
dependence near kT < 0.25. From equation (4.1) together with
data for
Fullerenes under high pressures
17
Figure 8. The function f ( T , p) calculated from equation (4.1)
and data for D (p) given by David and Ibberson [99] at pressures
from 0 to 1 GPa in steps of 100MPa.
against p given by David and Ibberson [99] it is possible to
calculate approximate values for the fraction f of P-oriented
molecules at any pressure or temperature. The result is shown in
gure 8, which veri es that an almost perfectly (H) orientationally
ordered phase must exist above some critical pressure. (The
corresponding gure in [14]was calculated in a simpli ed model
assuming a linear dependence of f on p and is thus less accurate.)
Although in principle no perfectly ordered phase exists, in
practice any structure with, say, 1% or 0.1% misoriented molecules
must be considered very well ordered. Note also that this phase is
the only truly orientationally ordered phase existing in the phase
diagram of C60 . The pressure dependence of f at constant
temperature is shown in gure 9, which has been calculated for pure
C60 at a temperature of 150K, that is with 70% Poriented molecules
at zero pressure. Near 5 10% there is a fairly sharp bend in the
curve, but again it is di cult to see how this rather smooth change
in slope can give rise to the sharp anomalies in B(p) reported by
Lundin et al. [79, 80] (see above). However, as discussed by Burgos
et al. [93] the crystal potential for molecular reorientation is
more complicated than usually assumed. These workers calculated the
intermolecular energy not only for the rotation of one molecule
surrounded by neighbours all in the P orientation, but also for
several other cases. As an example, gure 10 shows the
intermolecular energy for one molecule as a function of the
rotation angle about the [111] crystal axis for these cases. The
dotted curve is the standard result assuming all 12 nearest
neighbours are in the P-oriented state, while the full curve
assumes coherent collective rotation of all molecules and the chain
curve assumes all neighbours in the H orientation. In the last
curve there is only one minimum; that is, if all (or possibly even
a majority?) of the neighbours are H oriented, the molecule studied
will also end up in the same state. Burgos et al. concluded that sc
C60 will probably contain a large number of well oriented (both P
and H) microdomains rather than consist of a mixture of randomly
oriented molecules. Since the energy of the H-oriented state also
decreases with increasing pressure, it is not unreasonable that a
nal `lock-in to a completely H-oriented phase might also occur when
the fraction of H-oriented molecules exceeds some critical value
which we may guess is not very far from 11 /12 < 0.92 (i.e. on
average all
18
B. Sundqvist
Figure 9. The function f calculated from equation (4.1) as a
function of pressure p at a nominal temperature of 150K.
Figure 10. Intermolecular energy as a function of rotation angle
about the [111]crystal axis: ( ) curve obtained assuming all
molecules rotate coherently; ( ) curve assuming neighbouring
molecules are all in the 98 (P) orientation; (- -) curve assuming
neighbours are in the 38 (H) orientation. (Reprinted with
permission from Burgos et al. [93]) .
nearest neighbours except one are in the H orientation). One
interesting and testable consequence of this model is that, once
formed, the H-oriented state should be very stable, and a
`completely H-oriented phase might remain stable down to peq or the
fcc transition pressure, whichever is higher, and at low T possibly
even at zero pressure. Very recent calculations by L. Pintschovius
(1997, private communication), assuming to be a linear function of
f , also showed that the combination of such an e ect and the di
erent compressibilities of the two states (section 4.2.1) will
result in an abrupt nal transition. As hinted at above, the P H
reorientation under pressure has been observed to have signi cant e
ects on thermophysical properties such as
Fullerenes under high pressures
19
the bulk modulus and the thermal conductivity. This will be
discussed further in the relevant parts of section 4.2 below.
Before closing this section we note that a thermodynamic model
which describes the relative fractions of frozen-in, rotating and
ratcheting molecules in the lattice has been developed by Saito et
al. [105]but, since this model has not yet been applied to the
material under high pressures, it will not be discussed further
here. Also, the activation enthalpy D Ha for molecular rotation at
constant pressure and other thermodynamic quantities have been
measured by NMR in the sc phase under pressure and will be
discussed further in section 4.2.6. Finally, it should be noted
that anomalies are observed in many physical properties, mechanical
[106, 107] , structural [108] acoustic [109] dielectric [110] and
others [3], in the sc phase near , , 150K at zero pressure. The
cause of these anomalies is still not undisputed, but in some cases
it can be shown that they are connected with the glass transition
which at high frequencies falls in this temperature range. The only
study showing anomalies in this range under pressure is the thermal
conductivity study by Andersson et al. [84] . The anomaly shifts to
higher temperatures with increasing pressure, indicating either a
correlation with the sti ening of the lattice under pressure or
with the increase in glass transition temperature Tg because of the
decrease in molecular reorientation frequency. 4.1.4. The glassy
crystal transition and the properties of the glassy phase The glass
transition near Tg = 90K has also been studied in a few experiments
under high pressures. In the glassy crystal phase (or orientational
glass) below Tg , molecular motion has become slow enough that no
reorientational motion can usually be detected during the course of
a normal measurement. This means, for example, that, once a sample
has been cooled through the transition, the fraction of P-oriented
molecules can be considered constant at the limiting value obtained
on approaching Tg from above. The pressure dependence of Tg has
been found from thermal conductivity studies by Andersson et al.
[84, 100, 104] shows de nite anomalies at Tg because of its .
sensitivity to orientational disorder. To a rst approximation, the
thermal resistivity Wdis due to orientational disorder should be
given by Wdis ~ f (1 - f ) and thus have a maximum at f = 0.5 in
analogy with the electrical resistivity of concentrated alloys
[111] and on cooling towards Tg there must always be a decrease in
W dis . When Wdis , suddenly becomes constant at Tg, there will be
a sharp anomaly in d /dT. From data such as those shown in gure 11
the slope dTg /dp of the glass transition line has been deduced in
the range up to 0.7GPa, with the result dTg /dp = +62 KGPa- 1 [84,
100] (Similar slopes have later been reported by Grube [73]and by
Pintschovius et . al. [74] ) Above 0.7 GPa the glass transition
anomalies become small because of the . very low fraction of
P-oriented molecules (see above). In the range 0.15 0.2 GPa the
anomalies are also very small, since the [P]/[H] ratio is close to
50/50 at all temperatures. Again, the molecular reorientation in sc
C60 has interesting e ects on the thermodynamic properties. For
most glasses, the slope dTg /dp can be determined from the formula
[112] D a dTg (4.2) = Tg V g , D cp dp where D a and D cp are the
changes in the thermal expansivity and the speci c heat capacity
respectively at the transition. For C60, direct use of this formula
gives a
20
B. Sundqvist
Figure 11. Thermal conductivity of C60 against temperature at
the pressures (in gigapascals) indicated, showing typical glass
transition anomalies (arrows). ( Reprinted from Andersson et al.
[100] ) .
large negative value dTg /dp < - 130KGPa- 1 [104] because the
P-to-H molecular reorientation above Tg leads to a negative D a at
the transition (see gure 2 above). Correcting for this e ect a
positive slope in better agreement with theory can be found [104] .
If the pressure is changed while the sample is kept at a
temperature below Tg , no change in the orientational structure can
occur. On heating such a sample towards Tg, interesting relaxation
anomalies are observed [100] as shown in gure 12. A , comparison
with gure 11 shows that the character of the anomalies changes, and
both minima and maxima occur in when the molecules relax towards
the normal orientational structure at the new pressure. If the
sample is cooled at low pressures, a majority of the molecules will
be in the P orientation. If the pressure is then increased to a
pressure where H orientation should normally dominate, the fraction
of Poriented molecules will decrease from its initial high value,
through 50% where will have a minimum because of the form of Wdis,
towards a low value. This will be observed as an initial decrease
in , followed by a nal increase. Close to Tg , such a molecular
relaxation can also be observed by measuring as a function of time
[100] . Since the H orientation is connected with a smaller
molecular volume, the reorientation of each extra P oriented
molecule to the H state increases the volume available for other
molecules to reorient, and as a result the e ective glass
transition temperature becomes lower if the low-temperature phase
has a surplus of P-oriented molecules, and vice versa. Wolk et al.
[101] tried to determine the glass transition slope by cooling at
zero pressure, then applying pressure at low temperature and
measuring the glass transition temperature under pressure during
heating. As pointed out by Sundqvist et al. [113] the resulting
slope, dTg /dp = 35 KGPa- 1 , , shown in gure 6 above, is signi
cantly lower than the value of 62 KGPa- 1 obtained by Andersson et
al. [84, 100] by measurements at constant pressure ( gure 5) but
agrees well with the slope obtained from the position of the minima
in observed at
Fullerenes under high pressures
21
Figure 12. Thermal conductivity of C60 against temperature at
the pressures (in gigapascals) indicated, showing modi ed glass
transition anomalies. In all cases, the samples had been cooled
near 0.1GPa and pressurized to the nal pressure below 90K before
the measurements were carried out during heating. (Reprinted from
Andersson et al. [100]) .
[P]/[H]= 50/50 in the experiments with a frozen-in orientational
distribution [100] . An extrapolation of the `50/50 line to lower
pressures shows that it crosses the `normal glass transition line
at 0.15 6 0.05GPa, indicating that this is the equilibrium pressure
peq near 100K. 4.1.5. L ow-pressure phase diagram of C6 0 The
information given in the preceding sections can be collected and
condensed in the form of the pressure temperature phase diagram for
C60 shown in gure 13. The various phases have already been
discussed above. The slope of the fcc-to-sc phase line is taken as
162KGPa- 1 as discussed in section 4.1.2 and that of the glass
transition line is taken from Andersson et al. [84, 100] To
describe the orientational . states in the sc phase, two
`boundaries are given. The almost vertical broken line near 0.2 GPa
is a weakly temperature-dependent boundary line corresponding to
peq ( = 0) , calculated for a constant transition volume as
discussed in section 4.1.3 above. For p < peq the P-oriented
state is more stable while above this the H orientation dominates.
The shaded area at higher pressures corresponds to the region of
the transition into the H-oriented state discussed above. The
transition region boundaries shown have been calculated in the
following way. The lowerpressure side corresponds to a fractional P
state occupancy f = 15%, calculated using a constant peq to obtain
a lower limit. The high-pressure side corresponds to f = 5% but
calculated using the temperature-dependent peq to ensure that we nd
an upper limit. These limits on f were chosen because inspection of
the theoretical curve for f (p) in gure 9 indicates that the
macroscopic `transition should occur near f = 10%, very close to
the value f = 0. 92 for which I speculated (from the theoretical
potential given by Burgos et al. [93] and shown in gure 10) that a
`lock-in transition to an oriented phase might occur. The observed
anomalies in the
22
B. Sundqvist
Figure 13. Phase diagram of monomeric C60 as discussed in the
text. The full line [71] denotes the fcc-to-sc transformation and
the dotted line the glass transition [100] The . almost vertical
broken line delineates the equilibrium line between regions with P
(low p) and H (high p) orientations and the shaded area is the
region where a transition into an H-oriented phase should occur
(see text). Symbols denote anomalies observed in various properties
by Lundin and Sundqvist [80] ( j ), Jeon et al. [117](n ), Huang et
al. [118](, ), Meletov et al. [115]( r ), Meletov et al. [82](h ),
Bao et al. [91] ( d ), Blank et al. [114] ( s ) and Jephcoat et al.
[116] ( m ).
macroscopic properties should thus fall between these limits if
the assumptions made above are correct. Above the high-pressure
side of the shaded area less than 5% of the molecules should be P
oriented even in a pessimistic calculation and, if an orientational
`lock-in transition occurs as suggested by the potential of Burgos
et al., the crystal should to a very good approximation be
`completely H oriented. In both cases the dominant type of disorder
is probably no longer orientational but probably stacking faults,
or vacancies and interstitials. If domains of well ordered material
form, as suggested by Burgos et al., spectroscopic anomalies may
occur even at pressures below this region. The fcc-to-sc phase line
is well known and has been well studied, and the same is true to a
smaller extent for the low-pressure part of the glass transition
line. However, although the existence of the P H orientational
ordering transition is proved beyond doubt [92, 101] the actual
transition `line or area is not well , investigated. The symbols in
gure 13 denote compression anomalies observed by Lundin and
Sundqvist [80] at low temperatures as well as spectroscopic and
structural anomalies observed at room temperature. Several groups
have reported [81 83, 91, 114, 115]the existence of anomalies in
spectroscopic and other properties between 2 and 2.5 GPa at room
temperature, and Jephcoat et al. [116]observed the formation of a
structurally ordered phase near 2.5 GPa. Surprisingly, this
highpressure phase is reported to be completely P oriented,
although Jephcoat et al. stated that data at 1.3 GPa are in better
agreement with an H-oriented state.
Fullerenes under high pressures
23
Whether the unexpected orientation reported is correct, a
mistake or owing to the use of He as pressure medium (section 7.4)
is not clear. Anomalies have also been reported in the Raman
[117]and infrared (IR) [118]spectra of C60 in the range above 1
GPa. It has sometimes been assumed [81 83] that the anomalies
observed near 2 GPa correspond to the glass transition, but the
slope of the glass transition line is only 62KGPa- 1 at low
pressures and a linear extrapolation thus gives a glass transition
pressure of 3.4 GPa at 293K. Since the normal curvature of glass
transition lines is in such a direction as to increase this
pressure value, it is more likely that the anomalies observed are
connected with the formation of an (H) orientationally ordered
phase. However, this still remains to be proved by lowtemperature
high-pressure experiments. 4.1.6. Does liquid C6 0 exist? Most
neutral materials exist in three states: gas, liquid and solid. As
already discussed in section 4.1.1 above, C60 sublimes near 700K
without forming a liquid phase. As shown in gure 14 [119] the
standard graphite diamond phase diagram of , C contains a liquid
phase at very high temperatures above 5000K, but no such phase has
yet been observed for C60 . It might be argued that it should be
possible to observe such a state if sublimation was suppressed by
the application of a high pressure, and a number of theoretical
studies have been carried out to try to predict the properties
[120]and range of stability of such a phase. Interestingly, there
is still no consensus even on the question of whether a liquid
phase can exist at all. Most
Figure 14. Phase diagram of carbon. Solid lines are equilibrium
phase boundaries and the broken curve BFG the threshold for rapid
graphite diamond transformation. Commercial diamond synthesis is
carried out at A, C denotes rapid diamond-tographite conversion, DE
is the area where graphite is converted into hexagonal diamond and,
along the dotted line, HIJ graphite reversibly transforms into
diamond-like structures. ( Reprinted with permission from Bundy et
al. [119]) .
24
B. Sundqvist
investigations have used the same intermolecular potential [121]
but even so the , results di er between di erent groups. Hagen et
al. [122]found that no liquid phase can exist, since in their
calculation the liquid vapour coexistence line always fell at lower
temperatures than the solid uid coexistence curve, while almost
simultaneously Cheng et al. [123] found a narrow range above 1800K
where a liquid phase could possibly exist under a low pressure of a
few megapascals (tens of bars). Molecular dynamics simulations by
Abramo and Caccamo [124] also show an anomaly near 2250K at 2.2 MPa
which might be associated with melting. Although theoretically the
C60 molecule should be stable to about 4000K [125] experiments ,
have shown it to break down at very much lower temperatures
(section 4.1.1) and it is not clear whether liquid C60 at 1800
2300K would be stable on a molecular scale long enough to be
observed even in a rapid transient heating experiment. Later
theoretical studies [126 129]have been even less conclusive and the
question is still open. Rascon et al. [130]suggested that the
reason for the di erent results obtained by di erent groups might
be the form of the intermolecular potential used [121] , which they
showed is close to the critical potential for which no liquid phase
can occur. (Although Shchelkacheva [128]used a di erent potential
due to Yakub [131] and found results very similar to those obtained
in previous studies, it should be noted that the Yakub potential is
only an approximation to that of Girifalco [121]) . Finally,
Vorobev and Eletskii [61, 62] argued from thermodynamic scaling
arguments that the similarity between C60 and the noble gases shows
that a liquid phase should indeed exist. The question of whether a
liquid phase does exist or not has thus not yet been answered but,
since the physical properties of such a phase should be quite
unusual [120] the search will probably continue until either such a
phase is , found or its existence has been conclusively disproved.
4.2. Physical properties of C6 0 under high pressures 4.2.1.
Compressibility and other elastic properties Having discussed how
changes in temperature and pressure modify the structure of C60, I
now turn to the interesting question of how the physical properties
of C60 are modi ed by high pressures. As long as the molecules
remain stable it is obvious from the enormous strength of the
intramolecular bonds that the molecular properties will change
little under an applied pressure, but it is equally obvious that
the weakness of the intermolecular bonds means that the lattice
properties will be quite sensitive to the applied pressure. The
pressure in itself is not, of course, important; what is important
is the change in volume V (and thus intermolecular distance)
brought about by the applied pressure. In this section the
important relation between volume and pressure will therefore be
discussed in some detail. If the pressure is hydrostatic and the
material is isotropic, the compression will also be isotropic,
there will be no change in sample shape and the resulting volume
decrease can be described completely by the isothermal bulk modulus
B B=
-V
dp dV
T
,
(4.3)
or, equivalently, the isothermal compressibility = B- 1. As
discussed in section 3, this ideal situation rarely occurs in real
experiments. Although C60 itself is isotropic at low pressures
because of its cubic structure, the pressure in real experiments is
not often perfectly hydrostatic but contains uniaxial or shear
stress components. In
Fullerenes under high pressures
25
particular, as mentioned in section 3, all pressure media
solidify below 12 GPa [51 53], and soft solid noble gases such as
helium cannot be used to extend the quasihydrostatic pressure range
[52] because they intercalate into the C60 lattice under high
pressures. High-pressure studies of C60 are thus often carried out
using liquid media such as pentane, Freons or oils, or by
compressing pure C60 without a medium, relying on its small shear
strength [48] to produce quasihydrostatic conditions. The
discussion below will concentrate on data for the isothermal bulk
modulus B obtained by direct measurements of the volume change
under pressures, but available data for the adiabatic bulk modulus
Bs obtained from acoustic or optical studies will also be discussed
brie y. The relation between B and Bs is most clearly written in
terms of the associated compressibilities and s through the
thermodynamic relation [132, 133] = s +
and Bs = - 1 should thus be expected to be slightly larger than
B. s The compressibility or bulk modulus of a material not only is
of technical and practical interest but is also important from a
basic scienti c point of view. The compression behaviour is
determined by the intermolecular potential, and measured data for B
are therefore very important in forming a benchmark against which
theoretical models for the intermolecular interactions can be
tested. A large number of calculations have therefore been carried
out for the bulk modulus of C60 and will be brie y discussed below.
Many experimental methods have been devised for studies of B, and
the most common method today is to measure simultaneously the
lattice constant and the structure using X-ray or neutron di
raction methods. Unfortunately, this is a relatively slow method
which gives only a limited number of experimental points. The most
common method is to use X-ray di raction and to pressurize the
sample in a DAC but the small sample size (less than 0.001mm3)
implies very long exposure times and most studies are therefore
carried out using synchrotron sources. With the small aperture
available and the low intensity of the scattered radiation, only a
small number of di raction lines can usually be seen and it can be
di cult to identify small changes in the lattice structure. Neutron
scattering studies require larger sample volumes and are limited to
the signi cantly lower pressures produced by piston-and-cylinder or
large anvil devices. In very-highpressure studies the sample is
usually compressed without the use of an additional medium, even if
a quasihydrostatic material such as NaCl can be added in the cell
to function as a combined pressure-transmitting medium and in-situ
pressure calibrant, but at lower pressures a uid pressure medium is
often used. An alternative method is to measure the compression
properties mechanically in the piston-and-cylinder geometry, where
the piston movement can be measured very accurately but the maximum
pressure is limited to 3 4 GPa. With large samples the
piston-and-cylinder methods give a very high resolution and
accuracy provided that the device is carefully calibrated and that
accurate corrections for the compression of pistons and gaskets as
well as for the radial expansion of the cylinder are applied [134]
In . most cases, no pressure transmitting medium is used since it
is then necessary to carry out additional corrections for the
usually not very well de ned compression of the medium. This method
gives a semicontinuous record of V against pressure and can thus be
used to detect phase transitions and to map the pressure
temperature phase diagram, but the accuracy is sometimes limited
because of hysteresis e ects
9TV a 2 B2 , cV
(4.4)
26
B. Sundqvist
arising from friction both in the gaskets and inside the sample
itself, which su ers plastic ow in the experiment. After the
suggestion by Ruo and Ruo [135, 136] that compressed C60 might have
a bulk modulus signi cantly higher than the value of 441GPa for
diamond [137] the bulk modulus of C60 has been investigated
experimentally by a large , number of groups using all the above
methods and a variety of pressure-transmitting media. Surprisingly,
although the methods used should all have a similar accuracy, the
results from di erent groups often di er by signi cant amounts. The
compressibility study by Fischer et al. [68]was the rst
high-pressure study of C60 . Using X-ray di raction in a DAC under
hydrostatic pressure they obtained one single experimental point at
1.2 GPa, with a pressure calibration based on an empirical pressure
against load relation. This method is rarely used with DACs because
of the large inherent inaccuracy. The phase diagram in section
4.1.5 shows that 1.2GPa is well into the stability range of the fcc
phase and the measured volume change up to 1.2GPa thus included
both the continuous volume changes within the two phases involved
and the volume drop (about 1%) at the fcc-to-sc transition. The
average linear compressibility up to 1.2 GPa was d(ln a) /dp = -
2.3 10- 12 cm2 dyn- 1 = - 2.3 10- 2 GPa- 1 , corresponding to a
volume compressibility = 6.74 10- 2 GPa- 1 or a bulk modulus B =
14. 8 GPa. These values con rmed the basic assumption that the
intermolecular interaction in C60 must be very similar to the
interplanar interaction in graphite, since the c-axis
compressibility of graphite is d(ln c) /dp = - 2.8 10- 2 GPa- 1
[138] Because of the very small in-plane compress. ibility of
graphite the bulk modulus is close to the inverse of this value, at
B = 33.8 GPa [138] . Very soon afterwards Duclos et al.
[49]presented the results of a more complete investigation of the
compressive properties up to 20GPa, under both hydrostatic and
non-hydrostatic conditions as deduced from the deformation of the
ruby chip used for pressure calibration. As shown in gure 15, the
relative volume V / V o is a smooth function of p at all pressures
investigated under hydrostatic conditions (experiment 1) but not
under non-hydrostatic conditions (experiments 2 and 3). The data
from experiment 1 were tted to both the Vinet et al. [139] and the
Murnaghan [140] equations of state with identical results, yielding
an initial zero pressure bulk modulus B(0) = 18.1 GPa with a
pressure derivative B = dB /dp = 5. 7. Because of the experimental
procedure used, no data were obtained below about 0.5 GPa and
contrary to statements in the original paper the measured values
should thus apply to the sc phase only. Again, no account was taken
of the volume change at the fcc-tosc transition which makes the
low-pressure value B(0) rather dubious. At very high pressures,
Duclos et al. observed repeatable changes in the X-ray di raction
diagrams above 20 and 16.5 GPa in experiments 2 and 3 respectively.
The compression data also indicate a larger V / V o at very high
pressures in these experiments than in the hydrostatic experiment.
Since a volume increase at a structural transition is highly
unlikely for stability reasons, such a change in molecular volume
should be connected with a transition at lower pressures. If the
high-pressure compression data are extrapolated to lower pressures,
the extrapolations cross the data from experiment 1 between 5 and
7GPa, in good agreement with the observations of the formation of
an orthorhombic polymeric phase to be discussed in section 5. From
the slopes of the curves it is also possible to deduce very
uncertain values B > 200GPa for the average bulk modulus between
5 and 20GPa in this high-pressure phase. These values agree to
within an order of
Fullerenes under high pressures
27
Figure 15. RT compression behaviour of C60. Hydrostatic
conditions are believed to extend to 20GPa in experiment 1 and to
10GPa in experiment 2. Experiment 3 was carried out without a
pressure-transmitting medium. The curve has been tted to the data
using the Vinet et al. [139]equation of state. (Reprinted with
permission from Duclos et al. [49]) .
magnitude with the predictions of Ruo and Ruo [135, 136]and
others (see below) for the compressibility of the C60 molecule
itself. Although more accurate numerical data have been obtained in
later experiments, these two early studies are still widely cited
because they illustrate both the fact that C60 is basically
`three-dimensional (3D) graphite in its intermolecular properties
and the fact that measurements of the high-pressure properties are
complicated by the formation of high-pressure phases. Later
experiments have concentrated on either increasing the accuracy of
the measurements or on measuring the compression properties over
larger ranges in temperature and in other structural phases, or all
of these. To give some examples, David and Ibberson [99] carried
out a highly accurate neutron di raction study of the
compressibility up to an Ar pressure of 0.4GPa at 150 200K, with
the primary objective of studying the molecular orientation as a
function of pressure (see section 4.1.3). Lundin and Sundqvist
[79]were the rst to measure the bulk modulus in the fcc phase at
and above RT and also carried out careful studies of the
compression properties of sc C60 up to 1 GPa over a wide range in
T, from 150 to 430K [80] The . pressure range was extended signi
cantly by Ludwig et al. [141] who measured V , against p in both sc
and fcc C60 up to above 8 GPa at temperatures down to 70 K. Very
careful studies of the compressibility in both the sc and the fcc
phases up to 0.6 GPa at RT were carried out using neutron
scattering in a gas environment by Schirber et al. [72]. This study
also included the e ects of gas intercalation on the structure,
bulk modulus and phase diagram on C60 (see section 7.4). Finally,
Pintschovius et al. [74] have very recently carried out a neutron
scattering study up to a pressure of 0.5GPa over an extended
temperature range and answered several questions that remained
unanswered in the rst version of this review. To date, over 20
publications have been devoted to measurements of the bulk modulus
of C60, some by direct compression as those presented above, some
by other methods. Numerical data from most of these are presented
in table 2, and digitalized
28
B. Sundqvist
Table 2. Experimental data for the bulk modulus B and for B = dB
/dp for C60 under various conditions. The values given are usually
the initial bulk modulus B(0) at atmospheric (zero) pressure. RT
denotes room temperature; X indicates X-ray diffraction in a DAC, M
mechanical measurements and N neutron scattering. T (K) RT RT RT RT
RT RT 298 317 336 RT RT RT RT RT RT RT RT 298 RT 290 RT RT RT RT RT
RT RT RT RT 262 236 180 170 152 150 Pressure range (GPa) B(0) 0 1.2
0 0 0 0 0.38 0.05 0.2 0 0.13 0 0.25 0 0.3 0 0.35 0 0.3 0 0.25 0
0.25 0 0.2 0 0.2 0 20 14.8a 6.4b 8.4b 11.3b 8.4c 8.5 6.8 6.7 6.8
6.7a a
Phase fcc+ sc fcc fcc fcc fcc fcc fcc fcc fcc fcc fcc fcc fcc
fcc fcc sc sc sc sc? sc sc sc sc sc sc sc sc sc sc sc sc sc sc sc
sc
B
Reference Fischer et al. [68] Coufal et al. [142] Kobelev et al.
[143, 144] Fioretto et al. [145]
Comments and techniques X Surface acoustic wave technique, thin
lm Ultrasound Brillouin scattering, thin lm M M, in Ar M M M X X N,
in He N, in Ne N, in Ar N, in Ar X; hydrostatic pressure M M M M X
X X X N N, in He N, in Ne N, in Ar N, in Ar N, in Ar M M, in Ar X M
N continued
13.4 12.5 12 11.8 9.6 18.1
6.7c Lundin et al. [90] 32 Grube [73] 29 Lundin and Sundqvist
[79, 80] 24 Lundin and Sundqvist [79, 80] 21 Lundin and Sundqvist
[79, 80] Komori and Miyamoto [146] Ludwig et al. [141] 15 Schirber
et al. [72] 12 Schirber et al. [72] 5 Schirber et al. [72] 20
Pintschovius et al. [74] 5.7 Duclos et al. [49] 4.2c Lundin et al.
[90] 11.1 Lundin and Sundqvist [79, 80] 6.9 Bao et al. [91] Bashkin
et al. [147] 5.7 Wang et al. [148] 10.6 Haines et al. [149] 2.53d
Ludwig et al. [141] Komori and Miyamoto [146] Blaschko et al. [92]
Schirber et al. [72] Schirber et al. [72] Schirber et al. [72] 14
Pintschovius et al [74] 13 Pintschovius et al. [74] 13.1 Lundin and
Sundqvist [79, 80] Grube [73] Ludwig et al. [141] 16.2 Lundin and
Sundqvist [79, 80] David and Ibberson [99]
0.38 1.1 11.6c 0.5 1.1 9.5 0 2 0 2.7 0 13.7 0 10 0.3 12 0.35 1.6
0 1.8 0.4 0.6 0.4 0.6 0.3 0.55 0.23 0.5 0 0.5 0 0.5 0 0.1 0 9 0 0.6
15.4 16.4 20.9 14.4 13.4d 28a 18.5a 16.5a 15.3a 13.2a 8.3 9.5 9.6
11.5 14.2d 10.4
0 0.28 12.75a
Fullerenes under high pressuresTable 2 (concluded ) T (K) 150
110 110 70 60 RT 290 RT Pressure range (GPa) B(0) 0 0.5 0 0.1 0 0.5
0 9 0 0.1 0 3 0 2.7 5 20 0 12 12.5 13 14.7d 13.75 9.1 10.5 32.9
> 200a 540b
29
Phase sc sc sc sc sc orhg orhh orh?h
B e e
Reference Pintschovius et al. [74] Grube [73] Pintschovius et
al. [74] Ludwig et al. [141] Grube [73] Shimomura et al. [150]
Bashkin et al. [147] Duclos et al. [49] Blank et al. [151]
Comments and techniques N, in Hef M, in Hef N, in Hef X M, in
He; in glass phase X M X; for polymer phase? (see text)
Ultrasound
RT Amorphous 2ha b c d e f g h
Average value over pressure range indicated (linear
approximation). Adiabatic bulk modulus BS . Mixture of C60 and C70;
no sharp fcc-to-sc transition observed. Non-standard modi ed Birch
equation of state. B nonlinear in p; negative initial dB /dp. He
reported not to intercalate in C60 below 180 K [73]. Orthorhombic
C60 crystal obtained by crystallization from solution. Polymeric
material; see section 5.5.
data from most of those carried out at RT have been plotted
together to bring out the di erences and similarities. Some of
these curves are shown in gures 16 18, plotted on di erent scales
to enable comparisons in di erent ranges of pressure. Of the values
shown in table 2, some are given in the original papers, while some
have been extracted with reduced accuracy from graphs. It is
obvious from table 2 that the scatter in the data is surprisingly
large, considering the accuracy expected in such experiments.
Because of the large number of investigations all these data cannot
be discussed in detail, but a number of comments should be made on
the results. First, Lundin and Sundqvist [80] noted that an
internally consistent set of data now exists for the compression
properties of the sc phase up to above 1 GPa. This is clearly seen
in gure 16, showing the similarity of the RT data of Lundin and
Sundqvist [79, 80] Ludwig et al. [141]and Schirber et al. [72],
which all agree very , closely above 0.25 GPa. To these data sets
we can add the very recent data of Pintschovius et al. [74]; in the
sc phase, the data from [72] [74]and [80]are, in fact, , almost
indistinguishable at all points in their common pressure range. The
lowtemperature data of Ludwig et al., Lundin and Sundqvist and
Pintschovius et al. also agree very well with those of David and
Ibberson [99] To give some numerical . examples, the relative
volumes V / V o observed agree extremely well between the groups.
At 150 152K and 280MPa, Lundin and Sundqvist obtained V / V o =
0.9780, Pintschovius et al. V / V o = 0. 9778, and David and
Ibberson V / V o = 0.9776 while at 1 GPa Lundin and Sundqvist
obtained V / V o = 0.9460 at 152K and Ludwig et al. V /V o = 0.9464
at 170K. The RT data for the bulk modulus B given in table 2 appear
to di er greatly, but this seems to result mainly from di erences
in the extrapolations of the data since the data for the volume
compression in gure 16 actually agree very well between di erent
groups. A
30
B. Sundqvist
Figure 16. Measured volume change against pressure for C60 in
the low-pressure range at RT. The `reference set for the sc phase
discussed in the text contains data from Schirber et al. [72] (d ),
Pintschovius et al. [74] (---), Lundin and Sundqvist [79, 80] ( )
and Ludwig et al. ( j ). Other data are from Duclos et al. [49](n
), Blaschko et al. [92] ( s ) and Bashkin et al. [147] ( h ).
Figure 17. Measured volume change against pressure for C60 up to
3 GPa at RT. Data are from Duclos et al. [49] ( n ), Fischer et al.
[68] (, ), Schirber et al. [72] (d ), Lundin and Sundqvist [79, 80]
( ), Bao et al. (---), Blaschko et al. [92] (s ), Ludwig et al.
[141] ( j ) and Bashkin et al. [147] (h ).
calculation of the average bulk modulus B over the common
pressure range 0.3 0.6 GPa gives values greater than 13.2 GPa from
Schirber et al., 14.5 GPa from Ludwig et al., 14.7 GPa from
Pintschovius et al. and 14.5GPa from Lundin and Sundqvist, all in
very good agreement. Below 260K, on the other hand, the sc phase is
stable at zero pressure and no extrapolations are involved in the
calculation of B(0) . Data for the zero-pressure bulk modulus B(0)
from the latter three data sets, plus the low-temperature data of
Grube [73] and David and Ibberson [99] have , therefore been
plotted against temperature in gure 19. All data points in the
range
Fullerenes under high pressures
31
Figure 18. Measured volume change against pressure for C60 up to
28GPa at RT. Data are from Duclos et al. [49] ( n ), Ludwig et al.
[141] ( j ), Wang et al. [148] (---), Haines and Leger [149] (s )
and Nguyen et al. [152] (d ).
Figure 19. Low-temperature zero-pressure bulk modulus B(0) for
sc C60. Data are from Grube [73] (n ), Lundin and Sundqvist [79,
80] (, ), Pintschovius et al. [74] ( d ), David and Ibberson [99](h
) and Ludwig et al. [141]( j ). The full line has been tted to the
data up to 262K only.
below 260K can be well described by a straight line B(0) = (15.6
- 0.023T ) GPa,
(4.5)
with T in kelvins. An extrapolation of this line to RT gives a
value for B(0) of 8.9 GPa which agrees very well with the data of
Lundin et al. and Pintschovius et al. (also shown on the plot).
This shows again that we have a very good agreement between six
recent sets of data which thus seem to form a reliable reference
set for sc C60 in the low-pressure range. In particular, it should
be noted both in gures 16 and 19 and in table 2 that all recent
accurate values for B are signi cantly (25 55%) lower than the
value of 18.1 GPa given by Duclos et al. [49], which unfortunately
is
32
B. Sundqvist
Figure 20. Measured bulk modulus B as a function of pressure at
150 and 236K. The broken line has been corrected to show B against
p for a hypothetical state with constant f . (Reprinted from Lundin
and Sundqvist [80]) .
still widely used as a reference standard in calculations (see
below). The temperature dependence of the bulk modulus also seems
to be unusually strong in C60 . The P H reorientation with pressure
and temperature has a large e ect on the measured compression data
in the sc phase. First, the two states have recently been shown to
have di erent compressibilities [74] with the P-oriented state
being about , 10% `harder. Second, every increase in pressure is
accompanied by a reorientation of a number of molecules from the P-
to the H-oriented state. Because the H orientation has a lower
molecular volume, this gives an extra volume decrease which adds to
the volume change that would have been observed at a constant
[P]/[H]ratio. The sc phase thus has an anomalously low B in the
pressure and temperature range where this reorientation occurs, and
theoretical calculations assuming a constant [P]/ [H] ratio should
be expected to give higher values for B(0) than those observed in
experiments. Figure 20 shows the measured B as a function of
pressure at 152 and 236K [80] (Because of the high experimental
resolution in this experiment, B could . be calculated directly
from consecutive measured pairs of V and p.) At both temperatures,
anomalies are observed in the pressure range where the
reorientational transformation is expected to be approaching
completion, as discussed in section 4.1.3. The reorientation in
fact mainly a ects the slope B = dB /dp, as shown by the broken
line which is the result of a simple approximate calculation of how
B would depend on the pressure if the [P]/[H] ratio (see section
4.1.3) had stayed constant. Even more pronounced anomalies,
including changes in the sign of B , have recently been observed
below 150K by Pintschovius et al. [74] Below the glass transition
no . further molecular reorientation occurs, and both Pinschovius
et al. and Grube [73] showed that this results in a strong increase
in B on cooling through Tg , as would be expected. At pressures
well above 1 GPa the di erence between the results of di erent
studies become larger but data from all groups tend to fall on or
in between the extreme curves found by Duclos et al. [49] for
hydrostatic and non-hydrostatic conditions respectively ( gures 17
and 18; see also gure 15). Ludwig et al. [141] found a very strong
curvature in V against pressure and noted that no reasonable
standard equation of state is able to describe their data over the
full range of
Fullerenes under high pressures
33
pressures studied. Considering the C60 molecules as
incompressible spheres they suggested a modi ed Birch Murnaghan
equation of state in which the constant volume of the spheres is
subtracted from the crystal lattice. This modi ed equation is then
able to describe the compression of the `compressible fraction of
the lattice very well at all temperatures, but the structure of the
equation makes it di cult to compare their numerical data with data
from other sources. (The actual physical e ect behind the observed
rapid increase in B with increasing pressure is probably a
continuous polymerization with increasing pressure, as will be
discussed further in section 5.) The individual C60 molecules
indeed seem to be quite incompressible. This has been shown both
theoretically, as will be discussed brie y below, and
experimentally. Duclos et al. [49]showed from the disappearance of
the (200) di raction peak that the molecular form factor [32] must
be almost unchanged near 20 GPa, indicating a decrease in molecular
radius of the order of 1% and a `molecular B of at least 670GPa.
Although Singh [153] tried to explain the di erences between
hydrostatic and non-hydrostatic studies from elasticity theory, the
e ects discussed above and in sections 4.1.3 and 5 suggested that
the large di erences observed simply re ect the fact that the
translational, the orientational and even the molecular structure
of C60 may di er between experiments depending on the rate of
compression and the pressure medium used. These factors lead to di
erent structures, depending on the degree of polymerization, the
degree of lattice disorder created by inhomogeneous pressure and
possibly pressure quenching of the orientational state, all factors
which will a ect the measured compression properties. Returning to
the low-pressur