City, University of London Institutional Repository
Citation: Khoo, K. L., Dissado, L. A., Fothergill, J. and Youngs, I. J. (2004). Molecular
dynamics simulation of high frequency (1010 to 10 12 Hz) dielectric absorption in the
Hollandite Nax(Ti 8-xCrx)O16. Proceedings of the 2004 IEEE International Conference on
Solid Dielectrics ICSD 2004, 2, pp. 550-553. doi: 10.1109/ICSD.2004.1350490
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Permanent repository link: https://openaccess.city.ac.uk/id/eprint/1388/
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City Research Online
Molecular dynamics simulation of high frequency (10
10
to 10
12
Hz) dielectric absorption in the Hollandite
Na
x
(Ti
8-x
Cr
x
)O
16
K L Khoo
1*
, L A Dissado
1
, J C Fothergill
1
& I J Youngs
2
1
Engineering Department, University of Leicester, Leicester, University Road,
LE1 7RH, United Kingdom
2
QINETIQ, Room 1146, Building A7, Cody, Technology Park, Ively Road,
Farnborough, Hampshire, GU14 0LX, United Kingdom
* E-mail : klk4@le.ac.uk
Abstract: The charge-compensating sodium ions that
reside interstitially in the one-dimensional tunnels of the
hollandite Na
x
(Ti
8-x
Cr
x
)O
16
are used as a simple model
for a fluid. Molecular dynamics are used to calculate the
motions of the ions at a range of temperatures between
200 K and 373 K. The polarization response of the system
to a step-up electric field is calculated for field strengths
between 7.43 MV/m and 74.3 GV/m, and converted to an
ac susceptibility. A resonance absorption is found,
peaking at frequencies between4.5x10
10
and 8.8x10
10
Hz
at 297K. The origin of the response is shown to be the
anharmonically coupled ion vibrations damped by ion
hopping to neighbouring sites. The relationship of the
result to the experimentally observed Poley absorption is
explored, and a brief comparison of the calculated
dynamics to previous theoretical models is made.
INTRODUCTION
Hollandite is a non-stoichiometric material that is based
on a family of compounds of general formula
A
x
M
4-x
N
y
O
8
. In these materials a fraction of the cations
M are replaced by a cation in a different oxidation state N,
with charge neutrality maintained by the presence of
interstitial monovalent cations A. A special feature of
their structure is the existence of one-dimensional (1-D)
tunnels within which the interstitial ions are located at
specific binding sites [1,2]. The material investigated
here is a sodium priderite (titania-based hollandite,
Na
x
(Ti
8-x
Cr
x
)O
16
, (x = 1.7) [2] in which Na
+
is the tunnel
ion, see Fig. 1. The ions in the lattice cage produce a
potential surface with minima at the binding sites about
which the sodium ions vibrate. However because of the
non-stoichiometric nature not all of the possible binding
sites are occupied [2,3], and the sodium ions have some
freedom to move from site to site. The sodium ions
therefore behave rather like a “simple fluid” with each ion
moving in a potential that fluctuates because of the
changing interactions with the other sodium ions.
Each Na
+
ion forms a dipole with the counter ion in the
lattice cage. Na
+
ion vibrations under the action of the
co-operative forces of the cage and the other Na
+
ions
correspond to dipole librations if restricted to the same
binding site, while displacement of the ion from site to
site corresponds to rotational hopping. Dissado and Hill
[4] have presented a theory for dielectric relaxation that
takes into account such cooperative many-body motions
and it has been shown [5] that the theory predicts an
additional absorption peak at frequencies of 10
10
to 10
12
Hz. Peaks such as this have been observed by Poley [6],
Davies [7] and others [8,9] in both liquids and solids and
have been named after Poley. The theories [5,6] suggest
that this peak is caused by the cooperative librations of
the dipoles. Here we use Molecular Dynamic (MD)
simulations to see if such motions do indeed produce the
predicted Poley absorption.
Fig. 1: Hollandite model projected slightly off the c-axis
to give a clearer view of the 3-dimensional structure.
METHOD
The MD simulation has been carried out using programs
that we have written specifically for the purpose. The
potential between two ions is taken to have the form:
ij
ji
p
ij
ijij
r
ezz
r
rV
0
2
4
)(
(1)
where z
i
and z
j
are the formal charge of ith and jth ions
respectively, e is the unit charge, r
ij
is the distance
between the ith and jth ions. We have neglected the Van
der Waals‟ contribution as negligible compared to the
Ti
4+
O
2-
Na
+
b
a
coulombic potential. The parameters λ of the repulsive
term for each ion-pair have been chosen to give as good
an agreement as possible with the published
6-exponential potentials [2]. We have used a rigid-lattice
approximation in which only the sodium ions are free to
move.
These ions are not restricted to the unit cell in
which Cr
3+
ion replaces a Ti
4+
ion, they may also move to
neighbouring cells if the site is unoccupied. Electrostatic
interactions between the sodium ions along the tunnel
lead to a fluctuating potential environment for the motion
of the sodium ions.
Our hollandite model consists of 30 unit cells extended
along the c-axis to form a single tunnel. Twenty four
titanium ions are selected at random and replaced by
chromium ions. Charge neutrality is maintained by an
equal number of sodium ions. These preferentially reside
at interstitial sites within the same unit cell that contains a
chromium ion, whose location depends upon that of the
Cr
3+
ion within the cell [2,3]. Initial positions for the ions
were taken from the results of the x-ray analysis structure
refinement [3]. In our MD simulations the ion dynamics
were calculated using a time step of 10
-15
s, and reflective
boundaries at the ends of the tunnel. The temperature of
the system is defined via the kinetic energy of the sodium
ions.
The system was found to equilibrate in 5000 time steps.
Ten simulations were carried out initiating with different
sets of Na
+
velocities, but with the same average kinetic
energy, i.e. temperature. Mean locations for the Na
+
locations were obtained by averaging over the
simulations. The time development of a polarization
response was obtained by performing a simulation in
which a step dc electric field along the c-axis was
switched on at the 5001
th
time step. The resulting c-axis
displacement of the Na
+
from their mean locations gives
the polarization of the system, whose time development
was followed for 95000 time steps (= 9.5 10
-11
s). The
starting conditions here were a mean of those in the ten
simulations used to obtain the Na
+
locations. The
polarization current dP/dt divided by the field strength
gives the dielectric response function whose one-sided
Fourier transform yields the linearized frequency
dependent susceptibility = ‟ - i” [10]. Simulations
were performed for temperatures in the range 200 K to
373 K, and step fields of 7.43MV/m to 74.3 GV/m.
RESULTS
During the simulation the Na
+
ions may hop back and
forth between neighbouring binding sites when they are
unoccupied and vibrate in the binding site in which they
reside, as shown in Fig. 2. The frequency dependence of
‟, see Figs 3(a) and 3(b), show clearly that the response
has the form of a broadened resonance in the frequency
region around 5x10
10
Hz. An absorption peak in ” is
associated with this feature. This is the case for all
temperatures and fields investigated.
Fig 2: Trajectories of movement in three-dimensions of
the 6
th
sodium ion in the hollandite model with a field of
743MV/m at 273K.
Resonance absorption in vibrating systems can be
broadly classified as Gaussian, where there is a
distribution of independent vibrations, and Lorentzian
where a single fast vibration mode is damped by coupling
to a range of slower vibrations. The many body nature of
the forces in the Na
+
system imply that the Lorenztian
should be the better description. Figures 3(b) and 3(c)
show the best fits for ‟ and ” respectively to the
Lorentzian function, given in equations (2) and (3).
22
)(4
2
)("
wxx
wA
x
c
(2)
22
0
)(4
4
)('
wxx
xxA
yx
c
c
(3)
where x
c
, w, A and y
0
are the parameters used for the
resonant frequency, full width at ½ maximum, amplitude
factor and constant respectively.
DISCUSSION
The hopping of charges between alternative sites is
usually assumed to yield a relaxation peak at the hopping
frequency [10]. Ionic vibrations are expected to take
place at a much higher frequency and give a resonance
behaviour [11]. Their influence on the hopping charges is
regarded as an average series of impulses corresponding
to thermal noise, giving a Debye peak. In contrast the
theory of Dissado and Hill [4] envisages the hopping
dipoles as coupled to displacements in the centres of
motion of the vibrating system so that there is no
separation between the hopping and vibration timescales.
The hopping motion will then damp the extended
vibrations giving a 10
10
to 10
12
Hz absorption peak in
addition to a broadening of the relaxation peak.
(a)
(b)
(c)
Fig. 3: (a) The real and imaginary parts of the
susceptibility as a function of frequency with an electric
field of 743MV/m at 273K. (b) The real part of the
susceptibility with the best curve fitting to equation (3).
(c) The imaginary part of the susceptibility with the best
curve fitting to equation (2).
Simulation of the dynamics of a single Na
+
ion with the
other ions held rigid as well as the lattice cage gives a
harmonic vibration frequency of 4x10
12
Hz. The
simulated absorption peaks in ” (resonance frequency)
obtained for the complete ionic system lie between
4.5x10
10
and 8.8x10
10
Hz at 297K. Therefore the
anharmonic coupling of the Na
+
motions has
progressively connected larger groups of ions and
extended the vibrations to lower frequencies as suggested
by Dissado and Hill [4]. Ion hopping will break the
connection of an ion with one cluster of coupled
vibrations and link it to another, thereby providing the
damped resonance-like behaviour observed in Fig. 3. The
ion hopping can be taken to be equivalent to the friction
between annulus and disc in the Itinerant Oscillator
model [12], however this model treats the coupling
between the oscillator and the vibrations as random
impulses and does not include the anharmonic coupling
we have shown to be important, as implied in [8].
(a)
(b)
Fig. 4: (a) The resonance frequency as a function of
temperature (b) The peak of ” as a function of
temperature for a range of electric field (7.43MV/m –
7.43GV/m).
The absorption peaks obtained lie at the lower end of
Poley‟s predicted range, which is typically observed in
polar liquids in the 1.2 - 70 cm
-1
(3.6x10
10
– 2.1x10
12
Hz)
region at room temperature [6]. It would therefore be
reasonable to assume that our calculated absorption
corresponds to what would be the Poley absorption for
this material. However our calculated resonance
frequency and resonance peak height are essentially
independent of temperature, see Fig. 4 (a) & (b). In
contrast Johari [8] deduced that the Poley absorption
frequency in ice clathrate crystals decreased with an
