JChemPhy_2014

A molecular dynamics study of ambient and high pressure phases of silica: Structure
and enthalpy variation with molar volume
Chitra Rajappa, S. Bhuvaneshwari Sringeri, Yashonath Subramanian, and J. Gopalakrishnan
Citation: The Journal of Chemical Physics 140, 244512 (2014); doi: 10.1063/1.4885141
View online: http://dx.doi.org/10.1063/1.4885141
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THE JOURNAL OF CHEMICAL PHYSICS 140, 244512 (2014)
A molecular dynamics study of ambient and high pressure phases of silica:
Structure and enthalpy variation with molar volume
Chitra Rajappa, S. Bhuvaneshwari Sringeri, Yashonath Subramanian,
and J. Gopalakrishnan
Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India
(Received 28 January 2014; accepted 13 June 2014; published online 30 June 2014)
Extensive molecular dynamics studies of 13 different silica polymorphs are reported in the
isothermal-isobaric ensemble with the Parrinello-Rahman variable shape simulation cell. The van
Beest-Kramer-van Santen (BKS) potential is shown to predict lattice parameters for most phases
within 2%–3% accuracy, as well as the relative stabilities of different polymorphs in agreement with
experiment. Enthalpies of high-density polymorphs – CaCl2-type, α-PbO2-type, and pyrite-type –
for which no experimental data are available as yet, are predicted here. Further, the calculated en-
thalpies exhibit two distinct regimes as a function of molar volume—for low and medium-density
polymorphs, it is almost independent of volume, while for high-pressure phases a steep dependence
is seen. A detailed analysis indicates that the increased short-range contributions to enthalpy in
the high-density phases arise not only from an increased coordination number of silicon but also
shorter Si–O bond lengths. Our results indicate that amorphous phases of silica exhibit better op-
timization of short-range interactions than crystalline phases at the same density while the magni-
tude of Coulombic contributions is lower in the amorphous phase. © 2014 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4885141]
I. INTRODUCTION
Silica (SiO2) is the most abundant oxide on the earth’s
crust. It exists in many different crystalline polymorphs with
SiO4 tetrahedra as the principal structural unit, except for
high-pressure forms in which SiO6 octahedra are observed.
The polymorphs differ only in the connectivity of the basic
structural units and hence are quite close in energy. The struc-
tural complexity and subtle energy differences between dif-
ferent silica polymorphs make the modelling of these systems
non-trivial. Many accurate forcefields have been developed to
model silica polymorphs.1–15
Several workers have investigated different silica phases
using a variety of computational techniques. Tsuneyuki
et al.16
carried out molecular dynamics (MD) simulations to
show that α-cristobalite transforms to an orthorhombic phase
at around 16.5 GPa, which then transforms to stishovite at
23 GPa. Tse and Klug17
performed molecular dynamics sim-
ulations using the van Beest-Kramer-van Santen (BKS) po-
tential to calculate static and dynamical properties of several
silica polymorphs. They find that the equilibrium structures,
elastic constants as well as the calculated vibrational spectra
agree well with experimental values.
Keskar and Chelikowsky18
calculated the thermody-
namic properties of quartz, coesite, and stishovite from the
phonon density of states using the interatomic potentials de-
rived by Tsuneyuki et al.1
Bourova and coworkers19
have
investigated the high-temperature structure and dynamics of
coesite using lattice dynamics and molecular dynamics sim-
ulations. The high-pressure behaviour of α-cristobalite has
been investigated by Garg and Sharma20
by molecular dy-
namics simulations using the BKS and Tsuneyuki-Tsukada-
Aoki-Matsui (TTAM) potentials.
Poole and coworkers21
and Herzbach et al.22
have car-
ried out simulation studies to understand the phase diagram of
silica polymorphs, structural phase transformations, and ther-
mochemical properties of silica using three different poten-
tials. Huang and coworkers23,24
as well as Liang et al.25
have
both investigated the mechanism of phase transformations be-
tween different phases of silica.
In recent times there is increased interest in ab initio
studies of silica.26–29
Durandurdu has investigated the effect
of anisotropic stress on α-cristobalite by means of constant
pressure ab initio simulations. He observed the formation
of several structures: anatase-like, stishovite, and CaCl2-type
structures.26
In order to understand the energy dependence on
TOT angle ab initio simulations were carried out on silica and
germania crystalline systems. They found that SiOSi angle is
nearly flat over a wide range of angles whereas this was not
the case with GeOGe angle.27
Shi et al.28
explored the effect
of temperature on ring diameters in zeolites sodalite, ferrierite
and silicalite to understand negative thermal expansion coef-
ficients as well as variation in ring diameter with temperature
using both ab initio and classical simulations. Continuous ran-
dom network model of an amorphous glass has been studied
using ab initio approaches and the short-range order has been
compared with that in crystalline α-quartz.29
There are several ab initio studies investigating band
structure, and the phase boundary of α-quartz and other high
pressure phases up to stishovite with the help of Hartree-
Fock and density functional theory (DFT) calculations. The
effect of Hamiltonian and basis set on the vibrational fre-
quency of α-quartz has also been studied. Ab initio studies
have also been used to calculate elastic properties as well as
investigate the surface structure of α-quartz and high pressure
silica.12–15,31
0021-9606/2014/140(24)/244512/11/$30.00 © 2014 AIP Publishing LLC140, 244512-1
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244512-2 Rajappa et al. J. Chem. Phys. 140, 244512 (2014)
coesite CaCl2−typestishovit e
alpha−PbO2−type pyrite−type
Cristobalite Tridymitealpha−quartz
ZSM−5 ZSM−11 Zeolite Beta
Faujasite EMT
(a)
(b)
(c)
FIG. 1. Various polymorphs of silica investigated in this study are shown: (a) high pressure phases; (b) medium density phases at ambient temperature and
pressure; and (c) porous phases of silica.
Zwijnenburg et al.32
have compared experimental ther-
mochemical data for silica polymorphs with simulation
studies using different force ﬁelds and with DFT calculations.
More recently, Liang and coworkers33
have reported molec-
ular dynamics studies of the structural transitions and phase
transformations in melanophlogite, a low-pressure silica
polymorph.
Experimental investigations of the thermodynamic prop-
erties of silica polymorphs are well-known. Johnson and
coworkers have studied the thermodynamic properties of
silicalite.34
Petrovic et al. have measured the enthalpy of for-
mation of a number of high-silica zeolites using high temper-
ature solution calorimetry and drop calorimetry.35
They ﬁnd
that the enthalpy of the high-silica zeolitic forms are about
7–14 kJ/mol higher than the enthalpy of quartz.35,36
Spec-
troscopic (IR, Raman) studies of silica polymorphs are also
available.37,38
Navrotsky et al. have measured the heat of for-
mation of various phases of high-silica zeolites as well as
some dense forms of silica as a function of molar volume.39
They have also reported the experimentally measured entropy
of formation for some silica polymorphs.39
Figure 1 shows the different silica polymorphs inves-
tigated here. Coesite, stishovite, CaCl2-type, α-PbO2-type,
and pyrite-type silica are high-pressure polymorphs. α-quartz,
cristobalite, and tridymite are medium-density polymorphs,
while ZSM-5, ZSM-11, zeolite beta, faujasite, and EMT
(hexagonal faujasite) are low-density or porous silica poly-
morphs.
In spite of innumerable studies, there is lack of a com-
prehensive and comparative study of the low, intermediate
and high density polymorphs of silica. Is there a single in-
termolecular potential that can predict the complexity of
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the phase diagram and the innumerable structures of silica
correctly? How well is the thermodynamics of silica phases
at various densities reproduced?
In the present study we have chosen to use the BKS po-
tential to model 13 different phases of silica polymorphs over
a range of densities since (i) it has a simple form, (ii) we focus
on structural properties in this study, and the BKS potential is
known to model structural and thermodynamic properties of
even high-pressure phases quite well.17,22
BKS potential is known to have certain limitations—
many-body terms are not accounted for, polarization effects
are not factored in, and there is an implicit assumption that
there are no significant changes in electronic structure un-
der different temperatures and pressures.7
Also, vibrational
properties estimated using BKS are known to be only qual-
itatively in agreement with experiment.17
A major drawback
of the BKS model is that the dispersive term coefficients are
unphysically large and the partial charges used are too low.40
By far the most serious shortcoming of BKS is that it leads to
both coesite and stishovite being more stable than α-quartz—
this can probably be attributed to the large dispersive term
in the potential.22,40
This is contrary to experiment as well
as DFT studies (in the generalized gradient approximation)
which predict α-quartz to be the most stable polymorph.12
BKS is also known to overestimate the transition pressure for
the stishovite I → II transition.22
We focus on the structure and thermodynamic properties
of SiO2 polymorphs and the results are compared with the
experimental data. Simulation of silica glass is also reported
and provides interesting insights into its metastability and its
origin. Finally, metastability of silica crystalline polymorphs
is briefly discussed.
II. METHODS
A. Molecular dynamics
Molecular dynamics simulations were performed in the
NPT ensemble using a variable shape simulation cell. The
simulation cell is represented by three cell vectors, a, b, and c,
following Parrinello-Rahman.41
Unlike in the original formu-
lation, we have restricted the degrees of freedom associated
with the cell matrix, h = (a,b,c) to six by choosing a to be
along the x-axis, b to be in the xy-plane, and c in any direc-
tion. This prevents the rotation of the simulation cell observed
when the cell matrix has nine degrees of freedom.42
B. Intermolecular potential
The BKS potential was used to model all the silica
polymorphs:3
φαβ(r) = Aαβexp[−Bαβr] − Cαβ/r6
+ qαqβe2
/r, (1)
with α, β = Si, O. r is the distance between the two ions of
types α and β. Charges of +2.4 and −1.2 are assigned to the
Si and O atoms, respectively. The coefficients, Aαβ, Bαβ, and
Cαβ of the Buckingham potential are given in Table I.
The BKS potential is known to exhibit an unphysical di-
vergence to −∞ at small distances. This could be a problem
TABLE I. Intermolecular interaction potential for molecular dynamics sim-
ulation of silica polymorphs.
Interaction Aαβ B−1
αβ Cαβ
type (kJ/mol) Å (kJ/mol)Å6
O–O 1.339962d+05 0.362319 1.688493d+04
Si–O 1.737098d+06 0.205205 1.288446d+04
especially while simulating high pressure phases (pyrite-type
and α-PbO2-type silica) or while simulating an amorphous
phase at high temperature before quenching it. To circumvent
this, MD simulations were carried out on such phases using
the modified BKS potential (i.e., the original BKS potential
with an additional 30-6 Lennard-Jones-type term):43
φαβ(r) = φBKS
αβ (r) + 4 αβ
σαβ
rαβ
30
−
σαβ
rαβ
6
, (2)
where φBKS
αβ (r) is the original BKS potential and rαβ is the
separation between atoms α and β, which may be either sili-
con or oxygen atoms. Si–Si and σSi–Si are both set to zero. Si–O
and O–O are taken to be 0.2988995 and 0.10140858 kJ/mol,
respectively. σSi–O and σ−O–O are given the values 1.3136 and
1.7792 Å, respectively.43
The additional term does not alter the form of the original
BKS potential at large separations, but effectively prevents the
negative divergence at short distances. This form of the mod-
ified BKS potential has been used by several groups, specifi-
cally while simulating silica at high temperatures and/or high
pressures.44–46
Several other modifications have also been
proposed to correct for the unphysical divergence at short
distances.47–52
In the classical molecular dynamics simulations whose
results we report here, the total interaction energy has contri-
butions from atom-atom (Si–O, O–O, Si–Si) repulsive terms
which are always positive. The contribution for this comes
from the first term on the right hand side of Eq. (1). The sec-
ond contribution is the attractive dispersion interaction term
which is always negative. The sum of these two terms is what
is termed by us here as the van der Waals interaction energy,
Evdw. This is also referred to as short-range interaction en-
ergy. The electrostatic interaction due to charges on the atoms
gives rise to ECoul. This is also termed as long-range interac-
tion energy. The total interaction energy is
Epot = Evdw + ECoul, (3)
where
Evdw = Erepul + Edisp
=
α β
(Aαβexp(−Bαβr) − Cαβ/r6
). (4)
There is an additional contribution arising from the PV
term:
H = Epot + PV. (5)
At high pressures, the PV term is significant. Ab initio
calculations report energies which are typically of the order12
of around −23 eV. Ab initio energies include electronic
contributions, zero-point energy of optical modes, thermal
192.38.33.18 On: Fri, 04 Sep 2015 07:33:02

corrections, etc., while empirical potentials include disper-
sion (fluctuating dipole-fluctuating dipole moment, and other
higher order terms), repulsion term (due to Pauli exclusion
principle arising from electron cloud overlap) and long-range
electrostatic contributions. It is therefore difficult to com-
pare the ab initio energies with the energies from empiri-
cal calculations. However, the differences in energies, for ex-
ample, between silica polymorphs should be comparable in
magnitude.
III. COMPUTATIONAL DETAILS
The DLPOLY set of programs53
was used to carry out
molecular dynamics simulations at room temperature in the
NPT ensemble using a variable shape simulation cell for each
of the silica polymorphs. The simulation cell size was cho-
sen sufficiently large so that a cut-off radius of 12 Å could be
used in the calculation of short-range interactions. In order to
get good statistics, larger system sizes were used for the high-
pressure phases. For these, a larger cut-off of 14 Å has been
employed. The simulation cell consisted of 192 SiO2 units
(for low density faujasite) to 2916 SiO2 units (for high den-
sity pyrite-type silica). Table II lists the system size used for
each of the polymorphs studied here. Note that simulations
have been carried out on large systems and for reasonably
long durations. For high pressure phases simulations have
been carried out on even larger systems. All simulations were
performed so that the conservation of energy is better than 1
in 104
.
Two different glassy phases of silica were also simulated,
each one starting from the crystalline structure of α-quartz
and tridymite. We label these Glass-A and Glass-T, respec-
tively. The modified BKS potential was used in these simu-
lations. The simulation cell consisted of 882 SiO2 units and
2016 SiO2 units for simulating Glass-A and Glass-T, respec-
tively (see the supplementary material54
for details of prepa-
ration of the amorphous silica phases).
Periodic boundary conditions were employed. Electro-
static interactions were evaluated using the Ewald summation
technique. A time step of 1 fs was used for simulating most
of the crystalline polymorphs of silica. For the high-pressure
phases – stishovite, CaCl2-type, α-PbO2-type, and pyrite-type
– the timestep had to be reduced to 0.5 fs in order to obtain
good energy conservation. The durations of the equilibration
and production runs are also listed in Table II.
IV. RESULTS AND DISCUSSION
A. Structure
Although calculations on silica polymorphs have been re-
ported previously, they have been restricted to low density
polymorphs.9
This is the first study which reports results of
both low, intermediate, and high density polymorphs using
a single potential. In general many potentials include terms
such as bonded term, three-body angle dependent, and finally
four-body dihedral angle terms. However, it is well-known
that the existence of many different phases of silica arises
from the possibility to explore a large range of angles in the
three-body term. Hence, it would be difficult for a potential
with three-body terms to account for such a large number of
phases. BKS potential, on the other hand, prescribes a simple
atom-atom “non-bonded” interaction which, as we shall see,
can account for the extraordinarily wide range of structures
that silica can exist in.
Table III shows the average values of the lattice param-
eters obtained from our variable shape NPT ensemble simu-
lations for each silica polymorph. The corresponding exper-
imental values are also indicated (see caption to Table III).
Clearly, the BKS forcefield (which we have used in this study)
is able to correctly model lattice parameters across the whole
spectrum of densities and molar volumes. It is encouraging
to see that in most cases, the deviation from the experimental
value is less than 5%. Although the BKS potential was primar-
ily parametrized for tetra-coordinated silica,3
we find that this
TABLE II. The simulation cell consists of na, nb, and nc unit cells along each of a, b, and c directions. In
addition, the number of SiO2 units in the simulation cell is tabulated for each polymorph.
Silica No. of Structure
Run length
polymorph na nb nc SiO2 units reference Equilibration Production
Pyrite-type 9 9 9 2916 55 1 ns 750 ps
α-PbO2-type 9 7 8 2016 56 1 ns 1 ns
CaCl2-type 8 8 15 1920 56 1 ns 1 ns
Stishovite 8 8 15 1920 57 1 ns 1 ns
Coesite 5 3 9 2160 58 1 ns 1 ns
Glass-A 7 7 6 882 250 ps 250 ps
α-quartz 7 7 6 882 59 1 ns 1 ns
Cristobalite 6 6 4 576 60 250 ps 250 ps
Glass-T 2 7 3 2016 250 ps 250 ps
Tridymite 2 7 3 2016 61 250 ps 250 ps
ZSM-5 2 2 2 768 62 1 ns 1 ns
ZSM-11 2 2 2 768 63 250 ps 250 ps
Zeolite beta 2 2 1 256 64 250 ps 250 ps
Faujasite 1 1 1 192 65 250 ps 250 ps
EMT 2 2 2 768 66 250 ps 250 ps
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TABLE III. The average values of simulation cell volume, VMD , as well as the average unit cell parameters— a , b , c , α, β, and γ as obtained from the
variable shape MD simulations are tabulated for each silica polymorph. For the unit cell edge lengths and molar volume, the percentage deviation with respect
to experimental values for the crystalline form is indicated in parentheses. For the angles between unit cell axes, the experimental value for the crystalline form
is indicated in parentheses, when it is different from the experimental value.
Silica VMD a b c α β γ
polymorph (Å3) (Å) (Å) (Å) (deg.) (deg.) (deg.)
Pyrite-type 44 758.0 (1.2) 4.048 (3.0) 3.881 (−1.3) 3.909 (−0.5) 90.0 90.0 90.0
α-PbO2-type 37 835.4 (4.6) 3.845 (3.6) 4.644 (−0.2) 4.204 (1.1) 90.0 90.0 90.0
CaCl2-type 36 350.1 (4.7) 3.861 (1.4) 3.861 (1.2) 2.540 (2.0) 90.0 90.0 90.0
Stishovite 43 026.3 (−3.6) 4.112 (−1.6) 4.112 (−1.6) 2.651 (−0.5) 90.0 90.0 90.0
Coesite 73 790.9 (−1.7) 7.001 (−1.9) 12.391 (0.2) 7.229 (0.8) 90.0 121.0 (120.0) 90.0
α-quartz 33 224.2 (1.7) 4.938 (0.5) 4.938 (0.5) 5.443 (0.7) 90.0 90.0 120.0
Cristobalite 24 639.0 (−7.6) 4.911 (−1.2) 4.911 (−1.2) 6.557 (−5.3) 90.0 90.0 90.0
Tridymite 89 170.6 (−10.5) 18.087 (−2.4) 4.688(−6.3) 23.268(−2.3) 90.0 105.6 (105.82) 90.0
ZSM-5 42 621.1 (5.0) 20.263 (1.1) 20.256 (1.9) 13.629 (1.9) 90.0 90.0 90.0
ZSM-11 43 203.2 (4.6) 20.332 (1.3) 20.333 (1.3) 13.660 (1.9) 90.0 90.0 90.0
Zeolite beta 16 931.5 (1.7) 12.679 (0.2) 12.679 (0.2) 26.769 (1.4) 90.0 90.0 90.0
Faujasite 14 273.9 (6.1) 24.739 (2.0) 24.739 (2.0) 24.739 (2.0) 90.0 90.0 90.0
EMT 57 658.4 (5.1) 17.496 (1.6) 17.496 (1.6) 28.568 (1.7) 90.0 90.0 120.0
forceﬁeld yields reasonable results for six-coordinated high-
pressure phases as well. This is in agreement with the work of
Tse and others.17,67
We believe the atom-atom potential form
is responsible for the wonderful ability of BKS potential to
account for polymorphism in silica.
B. Thermodynamics
1. Metastability of silica polymorphs
Table IV lists the average temperature, pressure, and mo-
lar volumes obtained from the variable shape NPT ensem-
ble simulations for all the silica polymorphs. Also indicated
are the values of the enthalpies and the total energies of in-
teraction, along with the short-range and long-range energy
components. We note that the Evdw is the van der Waals con-
tribution which is the sum of the Edisp and Erepul. The former
(dispersion term) is always negative and favorable while the
latter (repulsive) is always positive and destabilizes. As we
can see, the sum of these two terms is negative and favor-
able up to stishovite but for still higher pressure polymorphs,
the energy is unfavorable. The error bars are also reported in
Table IV. The errors are small suggesting that the obtained
values are reliable.
Experimentally, α-quartz is known to be the most sta-
ble silica polymorph at ambient conditions. We ﬁnd that at
TABLE IV. The total energy of interaction, as well as the short-range and long-range contributions are tabulated for all the silica polymorphs simulated in
this study (variable shape MD simulation), as a function of molar volume. The average volume, temperature, and pressure obtained from the simulation are
indicated along with the set values in parentheses. Values reported are averaged over 250 ps, except for those polymorphs marked with an asterisk for which the
averages are over a period of 1 ns.
Silica V T P Evdw ECoul Epot H
polymorph (cm3/mol) (K) (katm) (kJ/mol) (kJ/mol) (kJ/mol) (kJ/mol)
Pyrite-type∗ 9.24(9.14) 299.7(300) 2674.8(2674.8) +827.30(1.73) − 5941.30(0.82) − 5114.00(1.09) − 2597.64(4.96)
α-PbO2-type∗ 11.30(10.81) 299.0(300) 1188.1(1188.1) +258.06(1.54) − 5761.90(0.93) − 5503.97(0.71) − 4132.19(4.74)
CaCl2-type∗ 11.40(10.89) 298.3(300) 1188.2(1188.1) +256.22(11.7) − 5781.77(6.85) − 5525.52(4.90) − 4142.03(36.2)
Stishovite∗ 13.49(14.02) 299.2(300) 99.1(99.0) − 167.54(5.5) − 5475.52(5.1) − 5643.23(0.48) − 5496.35(19.9)
Coesite∗ 20.57(20.58) 297.5(300) 44.4(44.4) − 79.60(1.93) − 5555.09(1.78) − 5634.72(0.25) − 5532.87(6.44)
Glass-A 22.68(22.68) 299.7(300) 0.0(0.0) − 93.45(1.54) − 5481.07(1.63) − 5574.49(6.07) − 5563.38(4.99)
α-quartz∗ 22.68(22.71) 298.1(300) 0.0(0.0) − 90.25(1.38) − 5533.79(1.38) − 5624.04(0.17) − 5612.93(3.75)
Cristobalite 25.76(25.49) 288.6(298) 0.1(0.0) − 87.08(2.26) − 5520.49(2.32) − 5607.64(0.93) − 5596.70(12.2)
Glass-T 26.64(23.87) 297.6(300) 0.0(0.0) − 86.50(1.45) − 5478.67(1.98) − 5564.98(1.63) − 5554.07(6.46)
Tridymite 26.64(27.52) 290.5(298) 0.2(0.0) − 85.56(1.82) − 5524.80(2.05) − 5610.12(1.31) − 5599.21(16.3)
Glass (Expt.) 27.27
ZSM-5∗ 33.42(33.65) 298.1(298) 0.0(0.0) − 55.46(4.32) − 5529.30(1.51) − 5584.77(0.27) − 5573.57(4.32)
ZSM-11 33.88(34.02) 373.2(373) 0.0(0.0) − 55.27(1.78) − 5525.13(1.79) − 5580.34(0.22) − 5566.41(5.15)
Zeolite beta 39.83(38.56) 298.3(298) 0.0(0.0) − 53.99(2.53) − 5518.36(2.54) − 5572.66(0.23) − 5561.33(7.19)
Faujasite 44.77(44.29) 298.8(298) 0.0(0.0) − 51.37(4.11) − 5504.69(4.12) − 5556.25(0.32) − 5544.79(12.0)
EMT 45.21(46.51) 293.0(293) 0.0(0.0) − 51.32(2.72) − 5504.82(2.72) − 5556.25(0.19) − 5545.18(8.02)
Stishovite∗ 13.81(14.02) 299.7(300) 0.0(0.0) − 207.27(3.6) − 5436.98(3.6) − 5644.27(0.09) − 5632.81(12.5)
192.38.33.18 On: Fri, 04 Sep 2015 07:33:02

ambient conditions (300 K and 1 atm pressure), the BKS po-
tential indicates that stishovite is the most stable phase and
is marginally lower in energy (by about 20 kJ/mol) than α-
quartz. Poole and coworkers also found that stishovite is more
stable than α-quartz at 300 K and 1 atm in their simulation
study of the phase diagram using BKS potential.21
Thus, it is
clear that BKS does not predict the order of stability correctly
at room temperature and pressure. Our NVE simulations of
stishovite and α-quartz (data not shown here) also indicate
that the BKS potential favours the stishovite phase at ambient
conditions.
However, under the experimental conditions at which
each of these phases is known to exist, the order of sta-
bility could be different. Hence, we carried out simula-
tions at the corresponding pressure and temperature to see
what order of stability is indicated. The results from the
variable shape Parrinello-Rahman isothermal-isobaric sim-
ulations performed at different pressures and temperatures
showed significant changes in the order of stability. Under
these conditions, we find that α-quartz is the most stable
phase. The enthalpies of various phases at their respective
pressures and temperatures are listed in Table IV. Note that
α-quartz is the most stable phase by about 117 kJ/mol as com-
pared to stishovite.
We also attempted to study a few phases which share a
co-existence line in the phase diagram as a function of pres-
sure. We have shown in Figure 2(a) the variation in enthalpy
as a function of pressure for coesite and α-quartz. In the case
of coesite, the pressure was decreased (the arrow indicates the
direction of pressure change) from 44 katm to 1 atm, while in
the case of α-quartz the pressure was increased from 1 atm to
100 katm. Note that the lines do not intersect in the range of
pressures investigated. However, this is not surprising since
the hysteresis associated with the first-order transitions will
ensure that the transitions actually take place at much higher
or lower pressures. In the case of stishovite-α-quartz (shown
in Figure 2(b)), the two come close to intersection but this
appears to be at some negative pressure. More detailed stud-
ies are required for understanding phase transitions in silica
polymorphs and these are under progress.
0 20 40 60 80 100
-5700
-5650
-5600
-5550
-5500
-5450
-5400
H(kJ/mol)
α-quartz
coesite
0 20 40 60 80 100
P (katm)
-5700
-5650
-5600
-5550
-5500
-5450
-5400
H(kJ/mol)
α-quartz
stishovite
(a)
(b)
>
<
>
<
FIG. 2. Variation of enthalpy with pressure for α-quartz, coesite, and
stishovite.
0 10 20 30 40 50
-300
0
300
600
900
Evdw
(kJ/mol)
0 10 20 30 40 50
-6000
-5800
-5600
-5400
ECoul
(kJ/mol)
0 10 20 30 40 50
V (cm
3
/mol)
-5800
-5600
-5400
-5200
-5000
Epot
(kJ/mol)
1
2, 3
4
5 6 7, 8 9, 10 11 12, 13
FIG. 3. Plot of total intermolecular interaction energy, Epot, short-range
van der Waals interaction energy, Evdw, and Coulombic energy contri-
bution, ECoul obtained from molecular dynamics simulations in the vari-
able shape NPT ensemble at the temperatures and pressures indicated in
Table IV. The following polymorphs are shown in the figure: (1) pyrite-type
silica, (2) α-PbO2-type silica, (3) CaCl2-type silica, (4) stishovite, (5) coesite,
(6) α-quartz, (7) cristobalite, (8) tridymite, (9) ZSM-5, (10) ZSM-11, (11) ze-
olite beta, (12) faujasite, and (13) EMT. The dotted lines are high-order fitted
polynomials and serve to guide the eye. Filled symbols represent the corre-
sponding values for Glass-A and Glass-T. Error bars are also indicated for all
quantities. In all cases, the error bars are smaller than the size of the symbol
used.
2. Energy contributions as a function of molar volume
Previous calculations suggest that the differences in the
enthalpies between large pore zeolites are small.9
They fur-
ther suggest that the short-range contributions are also not
very different for these zeolites. In order to understand the
metastability of the various polymorphs that we have simu-
lated in the present study, we have listed the short-range and
the long-range contributions in Table IV. Further, Figure 3
shows the variation of the total interaction energy and the as-
sociated short-range and long-range components as a function
of molar volume for all the polymorphs studied here.
The values obtained in the present study and previ-
ous calculations show good agreement with each other.
For crystalline cristobalite, the Coulombic contribution esti-
mated from this MD study is −5520 kJ/mol. This compares
well with values of −5525 kJ/mol and −5530 kJ/mol, ob-
tained from simulation studies by Yamahara et al.68
and Li
et al.,69
respectively, for BKS potential. The values reported
by Li and coworkers69
for the Coulombic components for α-
quartz, tridymite, ZSM-11, and zeolite beta are −5540.53,
−5547.72, −5537.16, and −5526.54 kJ/mol which are com-
parable to our values of −5533.79, −5524.80, −5525.13, and
−5518.36 kJ/mol, respectively. The maximum difference is
for tridymite which is around 0.4%.
Before we discuss the metastability of the various crys-
talline polymorphs, it is interesting to understand the differ-
ence between the crystalline and the amorphous forms of
silica. There have been a few ab initio studies which have
investigated pressure-induced amorphization of silica.10,30,70
The short-range contributions to the total energy is slightly
192.38.33.18 On: Fri, 04 Sep 2015 07:33:02

0 30 60 90 120 150 180
0
0.003
0.006
0.009
0.012
0.015
f(θ)
crystal
glass
0 30 60 90 120 150 180
0
0.002
0.004
0.006
0.008
0 30 60 90 120 150 180
θ (degrees)
0
0.002
0.004
0.006
0.008
f(θ)
0 30 60 90 120 150 180
θ (degrees)
0
0.001
0.002
0.003
0.004
0.005
α-quartz tridymite
O-Si-O
Si-O-Si
O-Si-O
Si-O-Si
FIG. 4. The distribution of different angles is shown for the glass as well as
well as the crystal of comparable density.
larger in magnitude for the glassy phase as compared to the
crystalline polymorph with similar molar volume. This means
that the amorphous phases have a lower short-range energy
as compared to the crystalline form at the same density!
In other words the glasses optimize the short-range interac-
tions better than their crystalline counterparts. Thus, Glass-A
and Glass-T are found to have short-range interaction ener-
gies of −93.45 kJ/mol and −86.50 kJ/mol, while the corre-
sponding crystalline phases – α-quartz and tridymite – are
associated with short-range energies of −90.25 kJ/mol and
−85.56 kJ/mol, respectively.
The reason for better optimization of the short-range en-
ergy in the case of glasses can be understood if we look
at the structure of the glass. In the glassy phase, it is seen
that although the tetrahedra remain intact, the interconnec-
tion between two tetrahedra are altered significantly. As is
evident from Figure 4 the distribution of Si–O–Si angles is
altered significantly. The distribution changes from bimodal
to monomodal and also extends over a much larger range of
angles. This is consistent with what is known of the “soft”
bending potential at the oxygen atom; i.e., the energetic cost
associated with distorting the Si–O–Si angle is very small.71
This suggests that the link between tetrahedral units is much
more flexible and provides for much larger choice. This is
not the case for O–Si–O angle which is broader for the glass
as compared to the crystal. From the figure it is also evident
that in tridymite much smaller angles are seen as compared to
α-quartz. We see that better optimization and larger flexibil-
ity and existence of short-range order leads to lower energies
(higher in magnitude) for the glassy phase.
In contrast, the Coulombic contribution is higher (lower
in magnitude) for the glassy phase. The crystalline polymorph
has a significantly lower Coulombic energy as compared to
the glassy phase of similar density. This is understandable
since the absence of long range order in the glassy phase leads
to lower contributions from the Coulombic component of the
interaction energy (see Table IV for the actual values).
This trend in the relative values of the Coulombic and
short-range contributions between the amorphous and the
crystalline forms is similar to that observed by Yamahara
et al.68
in their MD studies of crystalline and amorphous
cristobalite. They find that the Coulombic component of in-
teraction energy for the crystal is about 25 kJ/mol more
favourable than that for the amorphous cristobalite. In the
present study, we find that the Coulombic contributions
for crystalline α-quartz and crystalline tridymite are about
52.7 kJ/mol and 46.1 kJ/mol lower (i.e., more favourable) than
the respective amorphous phases. Yamahara et al. have used a
different silica potential and so only a qualitative comparison
is possible.
Two distinct regimes can be seen in the variation of the
enthalpy and other quantities with molar volume—a sharp de-
pendence at high-pressures (for stishovite and beyond) and
a very slight variation for lower density polymorphs. Note
that stishovite and still higher pressure phases all are six-
coordinated. Thus, it appears that the change from weak de-
pendence to strong dependence on molar volume is associated
with the change from 4-coordinated to 6-coordinated phases.
This is an important link between the thermodynamics of sil-
ica phases and the structure. The present finding suggests that
there is a thermodynamic signature of the change in coordina-
tion of silicon. This has not been observed previously, to the
best of our knowledge.
The underlying reason for this marked change in the
dependence of Epot, the total potential energy of interaction
with molar volume becomes clear if we look at Table IV and
Figure 3. From Figure 3 it is evident that the variations with
molar volume of Epot and Evdw are similar. From Table IV,
we see that at high pressures, the contribution of the short-
range interaction energy, Evdw, to the total energy becomes
significant. Thus, the steep variation of Epot with molar vol-
ume, V , seen at high pressures, is due to significant increase
in the short-range energy with decrease in molar volume. The
Coulombic contribution, on the other hand, does not exhibit
any such noticeable change. This is to be expected since there
is no significant change in long-range order. Thus, at higher
densities, the short-range interaction energy has a stronger de-
pendence on molar volume than at smaller pressures.
To understand the reasons for the sudden change in the
short-range interaction energy with molar volume, we have
further analysed the nearest neighbour structure in detail.
Figure 5 shows the dependence of the BKS potential for the
Si–O interaction on the Si–O distance. The sharp increase
in the contribution from the short-range interaction can be
understood in terms of the nearest neighbour structure. In
α-quartz, we see that there are just four oxygens at a dis-
tance of 1.61 Å. We see from Figure 5 that energy becomes
unfavourable below 1.52 Å. Further, the contribution to the
short-range energy rapidly increases with decrease in Si–O
distance. In coesite, the Si–O distance and the coordination
number of Si are very similar to α-quartz. But as the pressure
increases, first there is an increase in the coordination num-
ber of silicon. Stishovite exhibits 6 coordination with most of
the Si–O bonds longer than in α-quartz (4 at 1.76 Å and 2 at
1.81 Å as obtained from MD simulation). The increased co-
ordination leads to larger short-range contribution. With fur-
ther increase in pressure, the coordination number remains
essentially unchanged but the Si–O bond distance gradually
192.38.33.18 On: Fri, 04 Sep 2015 07:33:02

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4
r (A)
-250
0
250
500
750
1000
1250
U(Si-O)(kJ/mol)
BKS potential
Modified BKS potential
pyrite−type alpha−PbO2−type CaCl2−type
alpha−quartz coesite stishovite
FIG. 5. The BKS potential for the Si–O interaction is plotted as a function
of the Si–O distance. The modified form of the potential is also indicated in
the figure.
decreases: 4 at 1.64 Å and 2 at 1.62 Å for both CaCl2- and α-
PbO2-type. In the highest pressure polymorph – pyrite-type
silica – all the six Si–O bonds are of length 1.61 Å. These
changes can be more clearly seen in Figure 6 for various high-
pressure phases of silica and α-quartz. The changes in coor-
dination numbers with distance for various phases are also
shown in this figure. Thus, the shorter Si–O bond lengths from
stishovite onwards lead to a sharp increase in the contribution
from short-range interactions. Thus, the present analysis sug-
gests that the increase in the enthalpies of the various high
pressure phases arises principally from changes in the first
0 10 20 30 40 50
V (cm
3
/mol)
-6000
-5000
-4000
-3000
-2000
H
MD
(kJ/mol)
0 10 20 30 40 50
-920
-910
-900
-890
-880
-870
-860
-850
ΔH
expt
(kJ/mol)
1
2,3
4
5
6 7
7 8
9 10
11
12
13
(a)
(b)
4 5 6 9 10 11 12 13
FIG. 7. Variation of enthalpy with molar volume for different silica poly-
morphs (a) from experimental data, and (b) from variable shape NPT ensem-
ble molecular dynamics simulations. The following polymorphs are shown
in the figure: (1) pyrite-type silica, (2) α-PbO2-type silica, (3) CaCl2-type
silica, (4) stishovite, (5) coesite, (6) α-quartz, (7) cristobalite, (8) tridymite,
(9) ZSM-5, (10) ZSM-11, (11) zeolite beta, (12) faujasite, and (13) EMT. The
dotted lines are high-order polynomials and are meant to guide the eye. Filled
symbols represent the corresponding values for Glass-A and Glass-T. Error
bars are indicated for values from MD; in all cases, the error bars are within
the size of the symbol used.
coordination shell: increase in coordination number as well
as shorter Si–O bond lengths.
Figure 7 shows (a) the experimental heats of forma-
tion from available literature, and (b) the enthalpies obtained
from our MD simulations plotted as a function of molar vol-
ume for all the polymorphs investigated in this study. There
can be no quantitative comparison between the enthalpies
of formation obtained from experiment and the enthalpies
obtained from simulation. We show the variation of H =
H(polymorph) − H(α-quartz) as a function of molar volume
from experiment and MD in Figure 8. We can see that while
the trends are similar, there are quantitative differences be-
tween them. We have not shown the H for polymorphs
beyond stishovite as there is no experimental data for high
pressure phases beyond stishovite. Experimental data show
1.4 1.5 1.6 1.7 1.8 1.9 2
r (A)
0
5
10
15
20
25
gSi-O
(r)
α-quartz
coesite
stishovite
CaCl2
-type
α-PbO2
-type
pyrite-type
1.4 1.5 1.6 1.7 1.8 1.9 2
r (A)
0
1
2
3
4
5
6
7
nSi-O
(r)
α-quartz
coesite
stishovite
CaCl2
-type
α-PbO2
-type
pyrite-type
FIG. 6. Si–O radial distribution functions and associated coordination numbers are shown for the high-pressure polymorphs studied here. The corresponding
quantities for α-quartz have also been plotted for comparison.
192.38.33.18 On: Fri, 04 Sep 2015 07:33:02

10 20 30 40 50
V (cm
3
/mol)
0
50
100
150ΔH(kJ/mol)
ΔH from exptl data
ΔH from MD
4
5
6
7
9
11
8
12,13
10
FIG. 8. Dependence of H (=H(polymorph) − H(α-quartz)) on molar vol-
ume from experimental data and from the present variable shape NPT MD
study. The following polymorphs are shown in the figure: (4) stishovite, (5)
coesite, (6) α-quartz, (7) cristobalite, (8) tridymite, (9) ZSM-5, (10) ZSM-11,
(11) zeolite beta, (12) faujasite, and (13) EMT.
that the enthalpy of stishovite is considerably higher than
that of coesite, suggesting a steeper variation of enthalpy
with molar volume at high pressures. Our own MD simu-
lations suggest that this is indeed the case. Note, however,
that there is a qualitative agreement between the BKS derived
curve and the experimental curve suggesting that the order
of the enthalpies of the various phases is in agreement with
experiment.
Further, the additional high pressure phases (CaCl2-type,
α-PbO2-type, pyrite-type) studied here show that this steep
dependence becomes even steeper with reduction in molar
volume. From Table IV one can get an approximate esti-
mate of the magnitude of enthalpy difference between these
phases and α-quartz. Thermodynamic data from experiments
on these high pressure phases are not yet available. The
present study thus reports the enthalpies of the high pressure
phases of silica for the first time. It will be interesting to com-
pare these values with experimental data as and when they
become available.
The value of enthalpy, H, obtained from MD simulation
for α-quartz (−5613 kJ/mol) is in good agreement with those
obtained by Murashov and Svishchev who report a value of
about −5620 kJ/mol for MD simulations carried out with the
BKS potential.72
Note that the lowest enthalpy is obtained for
α-quartz which is the most stable phase as indicated by our
variable shape NPT ensemble simulations.
The enthalpy values for glass (molar volume = 27.27
cm3
/mol) are from experimental data by Piccione et al.73
In the same paper, they also report the enthalpy for crys-
talline tridymite (molar volume = 27.52 cm3
/mol) to be
−910.00 kJ/mol. This is about 8.2 kJ/mol lower than the
glassy silica with similar density. We also find that the en-
thalpies of our simulated glasses are higher than that of the
corresponding crystalline polymorphs at the same density.
Glass-A and Glass-T have enthalpies of 49.55 kJ/mol and
45.14 kJ/mol higher than that of α-quartz and tridymite, re-
spectively.
0
10
20
30
40
50
60
ΔH
q
298.15
,ΔG
q
298.15
(kJ/mol)
Enthalpy
Free energy
10 20 30 40 50
V (cm
3
/mol)
-3
-2
-1
0
1
2
3
TΔS
q
298.15
(kJ/mol)
Entropy
1
2
3
4 5
6
8
12
13
1
2
3
4
5
6
7 8 9
10
11
12
13
1
2
3. 4 5
6
8. 11
12 13
g
g
g
7 9 10
11
7
9
FIG. 9. Plot of Hq
298.15, Gq
298.15, and T Sq
298.15 vs. molar volume for
different silica polymorphs using experimental data available in the literature.
Here the Hq
298.15, Gq
298.15, and T Sq
298.15 are the formation enthalpy,
free energy, and entropy of the given polymorph minus the same quantity
for α-quartz. Data for the following polymorphs are shown in the figure: (1)
stishovite, (2) coesite, (3) α-quartz, (4) cristobalite, (5) tridymite, (6) ZSM-
23, (7) ZSM-12, (8) ZSM-5, (9) ZSM-11, (10) SSZ-24, (11) zeolite beta, (12)
faujasite, and (13) EMT. The data corresponding to glass are marked with g.
Metastability is associated with many interesting phe-
nomena such as polymorphism and superconductivity.74
The
metastability of most polymorphic forms of silica arises from
rather small differences in free energy. We have compiled
the entropic contribution as well as the free energies and en-
thalpies of formation from experimental data available in the
literature.34,35,73,75
The relative values of these quantities with
respect to α-quartz are plotted as a function of molar volume
in Figure 9 for a number of polymorphs for which data are
available. For the polymorphs shown in the figure, we note
that the entropic contribution to free energy is less than 10%.
While T S◦
298.15 varies only over a range of about 6 kJ/mol,
H◦
298.15 varies over a range of almost 60 kJ/mol (see
Figure 9). That is, the entropic contribution to metastability
is very small. Thus, although we have not computed the free
energies of the various silica polymorphs, it is likely that the
trend in free energy will be dictated largely by the trend in
the enthalpy. Hence, enthalpy as well as the total interaction
energy we have obtained here from MD are indicative of the
mestability of the various silica polymorphs. It is satisfying
to note that the results from simulation predict the order of
metastability of the various polymorphs of silica correctly, in
agreement with experiment. Further, these studies show that
there are two different regimes of enthalpy dependence on
molar volume, giving rise to two distinct types of metastabil-
ity: the one at high pressures is dictated by the short-range in-
teractions and the one at lower pressures by both short-range
and long-range interactions.
V. CONCLUSIONS
This simulation study provides a comprehensive inves-
tigation of 13 different silica polymorphs, with a focus on
192.38.33.18 On: Fri, 04 Sep 2015 07:33:02

high-density phases. There are several interesting findings.
First, the present study indicates that the BKS potential can
predict the lattice parameters of all the 13 phases of silica in
agreement with reported X-ray diffraction values. The modi-
fied BKS was crucial in reproducing the high density phases.
Further, the predicted values of the thermodynamic enthalpies
of low density phases are in good agreement with experimen-
tal data. This study predicts the enthalpies of the high-density
polymorphs for the first time—these can be verified as and
when experimental thermodynamic data become available.
The predicted enthalpies of CaCl2-type, α-PbO2-type, and
pyrite-like phase of silica are, respectively, 4142.03, 4132.19,
and 2597.64 kJ/mol. Note that the enthalpies of CaCl2-type,
α-PbO2-type are rather close to each other. This is consistent
with the fact their molar volumes are not very different. Re-
cently, there has been growing interest in prediction of high-
pressure crystalline phases.76,77
Our study shows that amorphous phases of silica are as-
sociated with lower short-range interaction energies than their
crystalline counterparts at the same density. This interesting
behaviour suggests a possible reason why silica glasses are
so commonly seen. The crystalline phase, however, exhibits
long-range order (unlike the glassy phase) and is, therefore,
associated with a predominantly favourable long-range ener-
getic contribution.
For high-density polymorphs, the enthalpy is seen to ex-
hibit a marked change from a weak to a steep dependence on
molar volume—experimental data indicate that the enthalpy
of stishovite is significantly higher than that of coesite. The
present study reports enthalpy for polymorphs denser than
stishovite and we show that the reason for the sharp change in
dependence of the enthalpy on molar volume at higher densi-
ties has its origins in the larger contributions from short-range
interactions. We further demonstrate that the increased short-
range contribution to enthalpy at high densities arises from
(i) an increase in coordination number of silicon atom from
four to six, as well as (ii) reduced Si–O bond lengths. Since
most of these are unfavourable interactions, this leads to
the observed steep increase in energy with decreasing molar
volume.
ACKNOWLEDGMENTS
J.G. wishes to thank INSA, New Delhi, for the award of
a Senior Scientist position. C.R. thanks DST, New Delhi, for
the award of a grant (DSTO1225) under the Women Scientist
Scheme. S.Y. thanks DST, New Delhi, for a Raja Ramanna
fellowship (DST1065) and grant under nano-mission for 100
teraflop computing facility (DST1169).
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