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On packing these layers in three dimensions, one can obtain a
number of polytypes. Figure 9 shows four of these; others may be
found in our detailed study.
41
We
first found these structures
(except for graphane I) by using an evolutionary algorithm
structure search (USPEX) at one pressure. The structures so
obtained were then taken over a wide pressure range.
Figure 10 shows the static lattice enthalpies of various CH
systems, including benzene phase III, benzene phase V, the two
polymeric structures mentioned earlier, and the graphane-type
phases obtained by us using the evolutionary algorithm search.
The corresponding most stable carbon (graphite below 10 GPa
and cubic diamond above 10 GPa) and H
2
phases
50
are taken as
references. The graphite-to-diamond phase transition around 10
GPa causes the apparent kink in the graphane relative enthalpy
curves at that pressure; there is no discontinuity in their intrinsic
enthalpy curves.
For the CH system, the most stable phase we found in the
range 0
À10 GPa is the all-chair graphane -AA- stacking or -AB-
stacking, both dynamically stable (see SI). These phases are close
in energy over a wide pressure range. Graphane III becomes the
most stable phase in the range of 10
À200 GPa. The calculated
phonon dispersion for both graphanes III and IV shows no
imaginary frequencies at 200 and 300 GPa, indicating that both
are dynamically stable. The reasons for the increasing stabiliza-
tion of graphanes III and IV with pressure (relative to I and II) are
discussed elsewhere; the di
fferential can be traced to the balance
of inter- and intrasheet H 3 3 3 H interactions.
41
Graphanes I and II should interconvert easily, for what is
involved is merely the sliding of layers. But the interconversion
between this pair and graphane III or IV should be di
fficult—
many CC bonds would have to break in the process. These are
geometrical isomers, not conformers.
So why do the benzene phases (III and V) not convert
spontaneously to these four-connected polymers? Because there
are likely signi
ficant barriers to polymerization. Organic chem-
istry as a whole is an exercise in kinetic persistence. The isomers
of benzene itself (Figure 11) are but one example of this. So
prismane, benzvalene, and Dewar benzene are respectively no
less than 114, 81, and 80 kcal/mol less stable than benzene.
However, they were made in the 1960s and 1970s;
51
À53
despite
slow decomposition to benzene, they have a half-life long enough
Figure 8.
Four isomeric single-sheet graphanes. Side views are at left,
top views at right.
Figure 9.
Graphane I (at 10 GPa), II (at 10 GPa), III (at 200 GPa), and
IV (at 300 GPa) structures.
Figure 10.
Enthalpy of various CH systems as a function of pressure.
The reference is the most stable phase of carbon and H
2
: graphite below
10 GPa and cubic diamond above 10 GPa; for H
2
, P6
3
/m (<100 GPa),
C2/c (100
À250 GPa), and Cmca-12 (>300 GPa). Note that zero-point
energies are not included in these calculations.
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to allow isolation. The reversions of these to benzene are forbidden
reactions.
54
To summarize: at every pressure studied, saturated, four-coordi-
nate CH phases, and graphanes in particular, are more stable than
any phase retaining discrete benzene molecules. Furthermore, the
transformation from benzene to saturated hydrocarbon may well
be local or nucleated. The outcome would be an amorphous,
approximately four-coordinated polymer. We will return to the
first stages of nucleated polymerization later in this paper. It
seems to us likely that such polymers, di
fficult if not impossible to
calculate, are in fact the experimentally observed polymeric
products of the pressurization of benzene.
DFT computations apply to the T = 0 K and static situation.
To analyze the e
ffect of temperature on benzene phase III, we
performed an annealing simulation using the ab initio molecular
dynamics (MD) method.
55
For the annealing process, the initial
temperature of the system is increased to 300 K, and a
final
temperature of 0 K is requested. The annealing analysis indicates
that the chemical transformation (to a saturated polymer) occurs
in phase III when the pressure is above 150 GPa at elevated
temperature. If the temperature is higher still, the transformation
may occur at a still lower pressure.
Ciabini et al.
7
used MD simulations to study several amorphi-
zation events in benzene from con
figurations equilibrated (phase
III) at 23 GPa and 540 K. All the reactive events are irreversibly
initiated when the nearest-neighbor (nonbonded) C 3 3 3 C dis-
tance approaches 2.5
À2.6 Å. In our case, the nearest-neighbor
C 3 3 3 C distance in phase III at 190 GPa and 0 K is 2.4 Å, which is
similar to the distance at 23 GPa and 540 K obtained by Ciabini
et al. In fact, Ciabini et al. did not observe any reaction in a pressure
range up to 150 GPa
—only an increase of the atomic kinetic energy
(up to 1500 K at 23 GPa) was able to induce the amorphization in a
few hundred femtoseconds. This indicates, as suspected, that
temperature is also an important factor in the phase transitions of
benzene. Note that there is a di
fference in the structures obtained by
Ciabini et al. and us: our polymer II is an ordered crystal structure
and not amorphous. The length of the simulation and the tempera-
ture may be responsible for the di
fference.
Does this mean that we should desist and not study the e
ffect
of pressure on benzene? Not at all. As our calculations also show,
the various benzene phases are local minima, dynamically stable
over a wide pressure range. They will face substantial barriers to
transforming into graphanes. Phase III in particular is a benzene
structure that in our calculations is persistent to
∼200 GPa at 0 K.
Thus, it makes sense to inquire into the properties of the most
persistent benzene phases, such as phase III, focusing on their
eventual metallization.
Approaches to Metalization while Maintaining a Benzene
Structure.
As we saw earlier in our calculations, benzene remains
in phase III until
∼200 GPa at T = 0 K, despite saturated phases
being more stable. At
∼200 GPa it transforms spontaneously to a
polymeric saturated CH phase. As phase III is compressed, the
band gap becomes smaller, and at
∼180 GPa it vanishes, within
the tolerances of the GGA computational approach we use. This
region, 180
À200 GPa, in which one has a metallic benzene phase
that has a barrier to rearrangement to a saturated polymer, is of
substantial interest to us. We next describe in some detail the
approach to metallization.
Figure 12 shows the computed relative compressions of
benzene phase III as a function of pressure. The Wigner
ÀSeitz
radius r
s
is de
fined by 4πr
s
3
/3 = 1/
F, where F is the average
valence-electron density. Since
F = N/V, where N is the number
of valence electrons in the unit cell and V is the volume of the unit
cell,
F scales as V
À1/3
. The volume reduction for all the benzene
phases, not only phase III, is rapid between 0 and 10 GPa and is
attributable to the squeezing out of the van der Waals space
between benzene units. Such behavior has been observed in
many molecular crystals under pressure.
56,57
The remarkably useful Goldhammer
ÀHerzfeld criterion for
metallization
58
holds that an insulator or semiconductor is likely
to become a metal when the conditions on the density are such
that the bulk polarizability diverges; that is, electrons can be
stripped o
ff the atoms or molecules with a diverging local field
associated with an in
finitesimal perturbation. To be specific, the
Goldhammer
ÀHerzfeld criterion says that a material becomes
metallic when the quantity (1
À fRV
m
)
À1
diverges, where
R is
the molecular polarizability, V
m
the volume per molecule in the
solid, and f a dimensionless factor determined only by the
packing of the molecules in the crystal. The divergence occurs
when f
R/V
m
= 1. For cubic systems f = 4
π/3, and this gives V
m
=
4
πR/3. Using the definition of r
s
, 4
πr
s
3
/3 = V
m
/N
ve
(where N
ve
is the number of electrons), we then have the following condition
for cubic systems, namely r
s
= (
R/N
ve
)
1/3
. The average polariz-
ability of benzene,
R, is 67.55 bohr;
3
N
ve
= 30 in one benzene
molecule. One then gets r
s
= 1.31 for the metallization of
benzene.
The real benzene structure (phase III is under discussion) is, of
course, not cubic. Still the arguments made above carry over well
—as Figure 13 shows, the band gap goes to zero at P ≈ 180 GPa,
where r
s
= 1.31 and the relative compression is just around 3.0.
Figure 11.
Three known isomers of benzene.
Figure 12.
Computed relative compression as a function of pressure in
benzene phase III. The relative compression is de
fined by [r
s
(0)/r
s
(p)]
3
,
where r
s
(0) and r
s
(p) are the Wigner
ÀSeitz radius at ambient pressure
and calculated high pressure, respectively.