entropy
Article
Ion Hopping and Constrained Li Diffusion Pathways
in the Superionic State of Antifluorite Li2O
Ajay Annamareddy and Jacob Eapen *
Department of Nuclear Engineering, North Carolina State University, Raleigh, NC 27695, USA;
[email protected]
* Correspondence: [email protected]; Tel.: +1-919-515-5952
Academic Editors: Giovanni Ciccotti, Mauro Ferrario and Christof Schuette
Received: 21 March 2017; Accepted: 15 May 2017; Published: 18 May 2017
Abstract: Li2 O belongs to the family of antifluorites that show superionic behavior at high
temperatures. While some of the superionic characteristics of Li2 O are well-known, the mechanistic
details of ionic conduction processes are somewhat nebulous. In this work, we first establish an
onset of superionic conduction that is emblematic of a gradual disordering process among the Li ions
at a characteristic temperature Tα (~1000 K) using reported neutron diffraction data and atomistic
simulations. In the superionic state, the Li ions are observed to portray dynamic disorder by hopping
between the tetrahedral lattice sites. We then show that string-like ionic diffusion pathways are
established among the Li ions in the superionic state. The diffusivity of these dynamical string-like
structures, which have a finite lifetime, shows a remarkable correlation to the bulk diffusivity of
the system.
Keywords: Li2 O; atomistic simulations; antifluorites; superionics; string-like; ionic conduction; diffusion
1. Introduction
Lithium oxide (Li2 O) has long been of interest because of its potential application as a
tritium-breeding material in fusion reactors [1]. Li2 O is also a superionic (or fast-ion) conductor,
attaining liquid-like ionic conductivities within the solid state [2]. It is one of the simplest Li-based
superionic conductors having just two species [3–7]. The structural and dynamic characteristics of
Li2 O thus have implications for important technologies ranging from future fusion reactors to solid
state batteries [8]. The focus of this paper is to study the collective dynamics of Li ions in the highly
conducting superionic state of Li2 O using atomistic simulations and statistical mechanics.
Li2 O has an antifluorite structure (space group: Fm3m) with oxygen ions positioned on a
face-centered cubic (FCC) lattice and the lithium ions occupying all the eight tetrahedral sites of
the FCC lattice [9,10]. The crystal structure can be viewed alternatively as cations (Li) arranged on a
simple cubic lattice, with anions occupying alternate cube centers. Both depictions are illustrated in
Figure 1, with cations and anions represented by smaller and larger spheres, respectively. The empty
cube centers are the octahedral sites of the FCC lattice and are possible locations for interstitials.
The lithium ions diffuse at temperatures lower than the melting point of the material; this leads to the
superionic character of Li2 O with the lithium ions contributing to the bulk of the ionic conductivity,
while the less mobile oxygen ions play the important role of maintaining the crystalline structure.
A transposition of cations to the FCC lattice and anions to the tetragonal sites gives rise to a popular
variant—the fluorite structure. In Figure 1, the anion and cation positions in antifluorites can simply
be interchanged to obtain this structure. Fluorite materials such as PbF2 , SrCl2 , and CaF2 are also
superionic conductors with anions making a dominant contribution to the ionic conductivity [9,11,12].
Entropy 2017, 19, 227; doi:10.3390/e19050227
www.mdpi.com/journal/entropy
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Figure
1. (Left)
Lattice
structureofofananantifluorite
antifluoritecrystal.
crystal. The
The larger
Figure
1. (Left)
Lattice
structure
larger spheres
spheres(shown
(shownininred)
red)depict
depict
anions
that
form
an
FCC
structure
while
the
smaller
spheres
(in
green)
represent
cations
occupying
anions that form an FCC structure while the smaller spheres (in green) represent cations occupying
available
tetrahedralsites
sitesofofthe
theFCC
FCClattice;
lattice. (Right)
(Right) In
In an
an alternate
alternate illustration,
all all
thethe
available
tetrahedral
illustration,the
thecations
cations
form
a
simple
cubic
lattice
while
the
anions
occupy
alternative
cube
centers.
A
fluorite
structure
is is
form a simple cubic lattice while the anions occupy alternative cube centers. A fluorite structure
realized when the cations and anions are interchanged. Neutron scattering/diffraction experiments
realized when the cations and anions are interchanged. Neutron scattering/diffraction experiments
and atomistic simulations indicate that the empty cube centers or the octahedral sites of the FCC
and atomistic simulations indicate that the empty cube centers or the octahedral sites of the FCC lattice,
lattice, which are the interstitial locations, are increasingly avoided in the superionic state for fluorites
which are the interstitial locations, are increasingly avoided in the superionic state for fluorites and
and antifluorites [9,11,13,14].
antifluorites [9,11,13,14].
Superionic conductors can be categorized into two types depending on the nature of their
Superionic
conductors
beType
categorized
into
two types
depending
on the nature
of their
transition to the
conductingcan
state:
I materials
typically
show an
abrupt transition,
at a particular
transition
to the to
conducting
state:state
Typethrough
I materials
typicallyphase
showtransition,
an abruptwhile
transition,
a particular
temperature,
the conducting
a solid-solid
Type IIatsuperionics
portray a more
gradual
transition
a widea range
of temperatures
[9]. Thewhile
superionic
of Type
temperature,
to the
conducting
stateover
through
solid-solid
phase transition,
Typestate
II superionics
II materials
is also
characterized
by over
a quasi-second-order
thermodynamic
a characteristic
portray
a more
gradual
transition
a wide range of
temperaturestransition
[9]. Theatsuperionic
state
temperature
T
λ
,
below
the
melting
point,
where
the
specific
heat
(c
p
)
shows
an
anomalous
peak
[15].
of Type II materials is also characterized by a quasi-second-order thermodynamic transition at
Several fluorites
that fall inTλthe
Type the
II category
this anomalous
behavior
Otheran
a characteristic
temperature
, below
melting portray
point, where
the specific
heat (c[16].
p ) shows
thermodynamic
and
mechanical
properties
also
exhibit
a
change
in
behavior
near
T
λ
:
for
example,
a
anomalous peak [15]. Several fluorites that fall in the Type II category portray this anomalous
significant
decrease
in
the
value
of
elastic
constant
C
11 is observed near Tλ for fluorites [17,18].
behavior [16]. Other thermodynamic and mechanical properties also exhibit a change in behavior
Although
2O is known to be a Type II conductor with a gradual increase in the ionic conductivity
near
Tλ : forLiexample,
a significant decrease in the value of elastic constant C11 is observed
[2], the divergent-like behavior of cp has not yet been observed experimentally. Instead, Tλ for Li2O
near Tλ for fluorites [17,18]. Although Li2 O is known to be a Type II conductor with a gradual
has been estimated approximately from neutron diffraction [10], diffusivity measurements [19], and
increase in the ionic conductivity [2], the divergent-like behavior of cp has not yet been observed
variations in properties such as lattice parameter [10] or elastic constants [20]. While these
experimentally. Instead, Tλ for Li2 O has been estimated approximately from neutron diffraction [10],
measurements do not agree on the numerical value, it is commonly accepted that λ transition takes
diffusivity measurements [19], and variations in properties such as lattice parameter [10] or elastic
place in the vicinity of 1200 K [10,19].
constants [20]. While these measurements do not agree on the numerical value, it is commonly accepted
Recently, we have shown that many fluorites exhibit an onset of disorder or an onset of
thatsuperionicity
λ transition at
takes
place in the vicinity
of 1200
K [10,19].
a characteristic
temperature
(denoted
as Tα) that can be obtained from neutron
Recently,
we
have
shown
that
many
fluorites
an onset
of disorder
an onset
diffraction, ionic conductivity, and specific heat dataexhibit
[16], which
is distinct
from Tλ.orAtomistic
of simulations
superionicity
at
a
characteristic
temperature
(denoted
as
T
)
that
can
be
obtained
from
α
have further helped in elucidating the structural and dynamical
changes observed across
neutron
diffraction,
ionic
conductivity,
and specific
data
[16],II ionic
which
is distinct
Tα [13,21–23],
as well
as across
the λ transition.
We thusheat
regard
a Type
conductor
to befrom
in theTλ .
Atomistic
simulations
have
further
helped
in
elucidating
the
structural
and
dynamical
changes
superionic state at temperatures above Tα. In this work, we first apply our theoretical and
observed
acrossdata
Tα [13,21–23],
as well as across
the λ transition.
We thus
regard
II ionic
experimental
analysis methodologies
to the antifluorite
Li2O to show
that the
onsetaofType
superionic
conductor
to be
in the
at temperatures
Tα . string-like
In this work,
we firststructures
apply our
conduction
occurs
at superionic
Tα ≈ 1000 K.state
We then
show that Liabove
ions form
dynamical
with a finite
lifetime, and data
the string
diffusivity,
which is
on peak participation
Li the
ionsonset
in
theoretical
and experimental
analysis
methodologies
to based
the antifluorite
Li2 O to showof
that
strings,
depicts
a
remarkable
correlation
to
the
bulk
diffusivity
of
the
system.
of superionic conduction occurs at Tα ≈ 1000 K. We then show that Li ions form string-like dynamical
structures with a finite lifetime, and the string diffusivity, which is based on peak participation of Li
Simulation
Methods
ions2.in
strings, depicts
a remarkable correlation to the bulk diffusivity of the system.
Atomistic simulations are widely used today to predict the properties of materials. They are also
2. Simulation
immensely Methods
useful to uncover atomistic mechanisms, which are difficult to establish solely with
experimental
techniques. are
In this
work,
wetoday
use classical
atomistic
simulations
to investigate
Atomistic simulations
widely
used
to predict
the properties
of materials.
Theythe
are
dynamics
of Li ions
in the superionic
state
of Li2are
O. We
select to
a rigid-ion,
alsospatiotemporal
immensely useful
to uncover
atomistic
mechanisms,
which
difficult
establish two-body
solely with
Entropy 2017, 19, 227
3 of 11
experimental techniques. In this work, we use classical atomistic simulations to investigate the
spatiotemporal dynamics of Li ions in the superionic state of Li2 O. We select a rigid-ion, two-body
Buckingham-type potential developed by Asahi et al. [24] from first principles electronic structure
simulations. The potential takes the following form:
Vαβ rαβ =
1
4πε 0
zα z β
Cαβ
+ Aαβ e(−rαβ /ραβ ) − 6
rαβ
rαβ
(1)
The indices α and β label the ionic species, r denotes the distance between the ions, and z is the effective
charge. The first term in Equation (1) represents the Coulombic interaction between the charged ions,
while the second and third terms represent the repulsive interaction due to the overlap of the electron
clouds and the attractive van der Waals or dispersion interaction, respectively.
In the parametrization used by Asahi et al. [24], the dispersion forces are excluded for all ion-pairs,
while the short-range repulsive interaction is attributed to Li-O pair only. It is interesting to note
that the training set for developing the potential is mainly driven by the melting point for Li2 O
(Tm = 1711 K). While other potentials have been developed in the past [20,25–29] for studying the
superionic behavior of Li2 O, the potential developed more recently by Asahi et al. [24] reproduces the
experimentally measured density of Li2 O in the entire solid phase and more importantly forecasts the
λ transition, based on thermal expansivity, to occur at ~1200 K [24], which is close to what is deduced
from neutron diffraction experiments. Remarkably, the Asahi et al. potential also predicts the onset of
disorder at a lower temperature (Tα in our terminology), which is nearly identical to the temperature
at which a change of slope is detected in the ionic conductivity [2]. Thus with a minimal training set
and parameters (see Table A1 in Appendix A), the potential developed by Asahi et al. [24] predicts the
transitions that are most critical to explaining the atomistic mechanisms that underpin the superionic
state of Li2 O. A comparison of different transition temperatures in Li2 O is shown in Table 1.
Table 1. Comparison of transition temperatures in Li2 O. Tλ and Tm refer to the λ transition temperature
and the melting point, respectively, and Tα represents the disorder-onset temperature.
Tα (K)
Tλ (K)
Tm (K)
†
Experiments
Atomistic (MD) Simulations
1000 [2,10]
1200 [10]
1711 [24]
1000 †
1200 [24]
1715 [24]
Estimated from the long-range order parameter (LRO)—see Figure 4.
An 8 × 8 × 8 antifluorite system containing 6144 ions is used for performing the atomistic
simulations. The Newton equations of motion are integrated using Velocity-Verlet algorithm under
periodic boundary conditions. A time step of 1 fs is observed to maintain long-time energy
conservation. The system is first equilibrated in a NPT ensemble (constant number of ions, pressure and
temperature) under zero pressure. The dynamical quantities of interest are then evaluated in
the energy conserving micro-canonical ensemble. The Coulomb interactions are evaluated using
the Wolf-summation method [30], which we have benchmarked to the standard Ewald method
(see Appendix B). In the Wolf method, both the Coulomb energies and forces are evaluated by applying
a spherical truncation at a finite cut-off radius with charge compensation on the surface of the truncation
sphere [30]. It is computationally more efficient than the widely used Ewald or fast-multipole methods,
with comparable accuracy for bulk and nanostructures. A cut-off radius of 10 Å is applied in the
evaluation of both the short-range and electrostatic interactions. All the simulations are performed
using the atomistic code—qBox that has been developed in-house.
Entropy 2017, 19, 227
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3. Results
3.1. Onset of Disorder from Reported Experimental Data
0.100
2
(a) Neutron Diffraction Data
(b) Ionic Conductvity
1
log(σ/Sm )
0.075
-1
Fraction of displaced Li ions (nf)
We begin by estimating the crossover temperature (Tα ) in Li2 O that corresponds to the onset
Entropy 2017,
19, 227
4 of 11
of disorder
from
experimental data. For this purpose, we utilize the reported data on neutron
diffraction [10] and ionic conductivity of Li ions using the nuclear magnetic resonance (NMR)
[10] and ionic conductivity of Li ions using the nuclear magnetic resonance (NMR) method [2]. The
method [2]. The neutron diffraction measurements give quantitative information on disorder
neutron diffraction measurements give quantitative information on disorder by measuring the
by measuring the fraction of Li ions that are displaced from their native lattice sites at different
fraction of Li ions that are displaced from their native lattice sites at different temperatures, as shown
temperatures, as shown in Figure 2a. Using the best fit to the diffraction data (shown by the dotted
in Figure 2a. Using the best fit to the diffraction data (shown by the dotted line), the onset of disorder
line), the onset of disorder can be estimated to occur at Tα ≈ 1000 K. In our earlier study, we had
can be estimated to occur at Tα ≈ 1000 K. In our earlier study, we had established that fluorite
established
that
fluorite
such as PbF2 , SrCl
a rapidatrise
in the with
ionic
2 , and
2 show
conductors
such
as PbFconductors
2, SrCl2, and CaF2 show a rapid
rise
in theCaF
ionic
conductivity
Tα though
conductivity
at Tα though
with
a higher
activation
energy
[16,21].
shown inofFigure
2b, the falls
ionic
a higher activation
energy
[16,21].
As shown
in Figure
2b, the
ionic As
conductivity
Li2O broadly
conductivity
of
Li
O
broadly
falls
into
two
Arrhenius
segments
with
a
noticeable
change
in
slope
at Tα
2
into two Arrhenius segments with a noticeable change in slope at Tα ≈ 1000 K. Both neutron scattering
≈ 1000
K.
Both
neutron
scattering
and
ionic
conductivity
measurements,
therefore,
establish
that
and ionic conductivity measurements, therefore, establish that the onset of Li disorder takes place the
at
onset
Li disorder takes
place atTthe
characteristic
temperature Tα ≈ 1000 K.
theofcharacteristic
temperature
α ≈ 1000
K.
0.050
0
-1
-2
-3
0.025
-4
0.000
800
1000
1200
Temperature (K)
1400
1600
-5
1.0
1.5
2.0
1000/T (K-1)
Figure
2. Fraction
(a) Fraction
of lithium
ions
f) that are displaced from regular sites, obtained from neutron
Figure
2. (a)
of lithium
ions
(nf(n
) that
are displaced from regular sites, obtained from neutron
diffraction measurements in Li2O [10]. The onset of disorder estimated from the best-fit to the
diffraction measurements in Li2 O [10]. The onset of disorder estimated from the best-fit to the diffraction
diffraction data occurs at Tα ≈ 1000 K. (b) Variation of ionic conductivity with temperature in Li2O
data occurs at Tα ≈ 1000 K; (b) Variation of ionic conductivity with temperature in Li2 O measured
measured using nuclear magnetic resonance (NMR) [2]. As evident, the ionic conductivity also shows
using nuclear magnetic resonance (NMR) [2]. As evident, the ionic conductivity also shows a noticeable
a noticeable change in slope at Tα ≈ 1000 K. A quantitative correlation between the onset of disorder
change in slope at Tα ≈ 1000 K. A quantitative correlation between the onset of disorder and the change
and the change in the slope of ionic conductivity is observed in several fluorite conductors such as
in the slope of ionic conductivity is observed in several fluorite conductors such as SrCl2 , PbF2 and
SrCl2, PbF2 and CaF2 [16].
CaF2 [16].
We will now use atomistic simulations to determine the structural disorder among the Li cations
We
now use
atomistic
determine
structural
disorder among
the Li
O. Similar
to our
earliersimulations
investigationtoon
UO2, we the
calculate
the long-range
order (LRO)
in Li2will
parameter
[13,31]
for to
theour
Li sub-lattice.
The LRO, measured
the Fourier
transformorder
of the (LRO)
local
cations
in Li2 O.
Similar
earlier investigation
on UO2 , through
we calculate
the long-range
N
iq⋅rj
parameter
[13,31]
for
the
Li
sub-lattice.
The
LRO,
measured
through
the
Fourier
transform
of
the
density given by ρ ( q ) =  j =1 e
where q is the wave vector, is a simple metric to assess quasi-static
iq·r j
N
local density given by ρ(q) = ∑ j=1 e
where q is the wave vector, is a simple metric to assess
disorder at different scales. For crystalline solids, the magnitude of LRO remains steady with time
quasi-static disorder at different scales. For crystalline solids, the magnitude of LRO remains steady
and tends to approach unity with decreasing temperature while for liquids it quickly decays to zero
withindicating
time andthe
approaches
with decreasing
temperature
while for
liquids
it quickly
decays
absence ofunity
any long-range
order. As
shown in Figure
3, the
magnitude
of LRO
for Lito
zeroions
indicating
absence
anyfor
long-range
order. As
Figure 3, the
magnitude crystalline
of LRO for
remainsthe
constant
inof
time
all temperatures
tillshown
meltinginindicating
an underlying
Li ions
remains
constant
in
time
for
all
temperatures
till
melting
indicating
an
underlying
structure. However, the decrease in magnitude with temperature suggests perceptiblecrystalline
disorder
structure.
the decrease
in magnitude
temperature
perceptible
disorder among
amongHowever,
Li ions while
preserving
a crystallinewith
sub-lattice,
whichsuggests
is somewhat
counter-intuitive.
One
Li ions
whileway
preserving
a crystalline
sub-lattice,
somewhat
Onedynamic
possible
possible
to accommodate
disorder
among which
Li ionsiswithin
the Licounter-intuitive.
sub-lattice is through
wayhopping
to accommodate
Li ions
within the Li
sub-lattice
through in
dynamic
hopping
of Li ionsdisorder
betweenamong
lattice sites,
a mechanism
that
has beenisobserved
prior atomistic
of Lisimulations
ions between
sites,
a mechanism
that has
observed
insuperionic
prior atomistic
simulations
of Lilattice
2O [26,27],
and
ubiquitous among
thebeen
anions
of fluorite
conductors
in the
We
will demonstrate
later
that such
cation hopping
between
sites
of Lisuperionic
and[13,21,23].
ubiquitous
among
the anions of
fluorite
superionic
conductors
in thenative
superionic
2 O [26,27],state
takesWe
place
Li ions diffuse
through
dynamical
structures.
In contrast,
the
stateindeed
[13,21,23].
willand
demonstrate
later that
such string-like
cation hopping
between
native sites
indeed takes
oxygen
anions
show
limited
variation
in
the
LRO
magnitude
with
temperature
indicating
a
rigid
place and Li ions diffuse through string-like dynamical structures. In contrast, the oxygen anions show
anion
sub-lattice.
limited
variation
in the LRO magnitude with temperature indicating a rigid anion sub-lattice.
Figure 4a shows the time-averaged LRO magnitude against temperature (scaled to Tλ = 1200 K).
The growth of disorder on the Li sub-lattice accelerates at ~1000 K, which is the characteristic
temperature at which the neutron diffraction data shows a discernible increase in the disorder.
Interestingly, at temperatures below Tα (1000 K), and above Tλ (1200 K), the LRO shows a near-linear
variation for the range of temperatures considered in this study. As expected, the LRO magnitude
hovers near unity and does not change significantly for the oxygen anions.
Entropy 2017, 19, 227
5 of 11
Figure 4a shows the time-averaged LRO magnitude against temperature (scaled to Tλ = 1200 K).
The growth of disorder on the Li sub-lattice accelerates at ~1000 K, which is the characteristic
temperature at which the neutron diffraction data shows a discernible increase in the disorder.
Interestingly, at temperatures below Tα (1000 K), and above Tλ (1200 K), the LRO shows a near-linear
variation for the range of temperatures considered in this study. As expected, the LRO magnitude
hovers Entropy
near unity
and
2017, 19,
227 does not change significantly for the oxygen anions.
5 of 11
Entropy 2017, 19, 227
1.0
0.9
(a) Lithium
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
800 K
1100 K
800 KK
1400
1100 K
201400 K 40
0
0
900 K
1200 K
900 K
1200 K
60
Time
40 (ps)
60
20
5 of 11
1.0
(a) Lithium
Long-range
order
(LRO)
Long-range
order
(LRO)
Long-range
order
(LRO)
Long-range
order
(LRO)
1.0
1000 K
1300 K
1000 K
1300 K
80
100
80
1.0
0.9
0.9
0.8
(b) Oxygen
0.7
0.6
800 K
1100 K
800 KK
1400
1100 K
1400 K
0.6
0.5
0.5
0.4
0.4
100
(b) Oxygen
0.8
0.7
0
20
0
20
40
900 K
1200 K
900 K
1200 K
60
1000 K
1300 K
1000 K
1300 K
80
Time
40 (ps)
60
100
80
100
Time (ps)
Time(b)
(ps)
Figure 3. Magnitude of long-range order (LRO) parameter for (a) lithium and
oxygen ions across
Figure 3. Magnitude of long-range order (LRO) parameter for (a) lithium and (b) oxygen ions across
reciprocal
lattice
vectorfor
q (a)
is chosen
[1 oxygen
0 0] and
[1 across
−1 1]
several
temperatures
in long-range
Li2O. The order
Figure 3.
Magnitude of
(LRO)
parameter
lithiumalong
and (b)
ions
several directions
temperatures
Li2 O.and
The reciprocal
latticerespectively,
vector q is and
chosen
along [1 0of 0]
and [1 −1 1]
for thein
lithium
sub-lattices,
the magnitude
q corresponds
The reciprocal
lattice vector q is
chosen
along [1 0 0]
and [1 −1 1]
several temperatures
in Li2O.oxygen
directions
forlongest
the for
lithium
and oxygen
sub-lattices,
respectively,
andand
thethe
magnitude
to
the
wavelength
in
the
system.
directions
the lithium
and
oxygen
sub-lattices,
respectively,
magnitudeofofq qcorresponds
corresponds to
5
(a)
0.9
5
4
(a)
0.9
0.8
0.8
Lithium
Oxygen
Lithium
Oxygen
0.7
0.7
0.6
0.6
Tλ=1200 K
0.9
0.9
Tα≈ 1000 K
1000 K
1.0 Tλ=1200
1.1 K1.2 Tα≈1.3
1.4
1.0
1.1 T /T
1.2
λ
T /T
1.3
1.4
Scaled
disorder
Scaled
disorder
Long
range
order
(LRO)
Long
range
order
(LRO)
the longest
wavelength
in the system.
to the
longest wavelength
in the system.
(b)
Neutron diffraction
LRO
Neutron diffraction
LRO
(b)
4
3
3
2
2
1
1
0
1.5
0
1000
1100
1300
1400
1.5
1000
1100Temperature
1200
1300
(K)
1400
1200
Temperature (K)
λ
Figure 4. (a) Variation of the
time-averaged long-range order (LRO) for lithium and oxygen ions from
800
to 1400
(b) Comparison
of the (scaled)
neutron
diffraction
withand
theoxygen
(shifted/scaled)
Figure
4. (a) K.
Variation
of the time-averaged
long-range
order
(LRO) fordata
lithium
ions from
Figure 4. (a) Variation of the time-averaged long-range order (LRO) for lithium and oxygen ions from
disorder
metric
from theofLRO
Li ions.neutron
The statistical
errordata
for LRO
less (shifted/scaled)
than 1%.
800 to 1400
K. evaluated
(b) Comparison
the of
(scaled)
diffraction
withis the
800 to 1400 K; (b) Comparison of the (scaled) neutron diffraction data with the (shifted/scaled) disorder
disorder metric evaluated from the LRO of Li ions. The statistical error for LRO is less than 1%.
metric evaluated
the LRO
of Li ions.
statistical
error
is less than
1%.
We have from
calculated
a metric
for The
Li-ion
disorder
as for
theLRO
difference
(modulus)
between the
magnitude
of LRO
and its ideal
valuefor
(unity)
anddisorder
shifted itastothe
givedifference
a null value
at Tα. In Figure
4b, we
We have
calculated
a metric
Li-ion
(modulus)
between
the
compared
thisaand
disorder
metric
from
simulations
the
fraction
Li ions
the
native
sites
magnitude
of LRO
its ideal
value
(unity)
and
it to
give(modulus)
aof
null
valueleaving
at Tα. In
Figure
4b,
we
Wehave
have
calculated
metric
for
Li-ion
disorder
asshifted
thetodifference
between
the
magnitude
evaluated
from
neutron
scattering
data,
both
scaled
to
their
respective
magnitudes
at
T
λ.
have
compared
this
disorder
metric
from
simulations
to
the
fraction
of
Li
ions
leaving
the
native
sites
of LRO and its ideal value (unity) and shifted it to give a null value at Tα . In Figure 4b, we have
Interestingly,
the
disorder
metric
from
atomistic
simulation
is
able
to
capture
the
variation
of
the
evaluated
from
neutron
scattering
data,
both
scaled
to
their
respective
magnitudes
at
T
λ.
compared this disorder metric from simulations to the fraction of Li ions leaving the native sites
neutron
diffraction
data at temperatures
1000 to ~1300
(shown
by the shaded
region);
Interestingly,
the disorder
metric from ranging
atomisticfrom
simulation
is ableK to
capture
variation
of the
evaluated from neutron scattering data, both scaled to their respective magnitudes at Tλ . Interestingly,
at
higher diffraction
temperatures,
predicts a larger
disorder
leading
to melting
1715 K,region);
which
neutron
dataLRO
at temperatures
ranging
fromeventually
1000 to ~1300
K (shown
by theatshaded
the disorder metric from atomistic simulation is able to capture the variation of the neutron diffraction
isatclose
to temperatures,
the experimental
melting
point
(1711
K).
higher
LRO
predicts
a larger
disorder
eventually leading to melting at 1715 K, which
data at istemperatures
ranging from
1000
to(1711
~1300
close to the experimental
melting
point
K). K (shown by the shaded region); at higher
temperatures,
LRO
predicts
a
larger
disorder
eventually
leading to melting at 1715 K, which is
3.2. Hopping and Cooperativity among LI Ions
close to3.2.
theHopping
experimental
melting point
and Cooperativity
among(1711
LI IonsK).
A number of atomistic simulations on fluorites and antifluorites have shown that the mobile
anions
from of
oneatomistic
native lattice
site to another
at temperatures
close tohave
Tλ [13,26,27,32,33].
Wemobile
have
Ahop
number
simulations
on fluorites
and antifluorites
shown that the
3.2. Hopping and Cooperativity among LI Ions
observed
similar
hopping
Li ions,
shown in
5. The close
Li ions
vibrate about
anions hop
from one
nativeoflattice
siteas
to another
at Figure
temperatures
to Tmostly
λ [13,26,27,32,33].
We their
have
sites,
punctuated
byofsporadic
toina Figure
neighboring
Occasionally,
concerted
observed
similar
hopping
Li ions, hopping
as shown
5. Thesite.
Li ions
mostly
vibrate
their
A lattice
number
of
atomistic
simulations
on
fluorites
and
antifluorites
have
shown
thatabout
the ionic
mobile
hopping
can
also
be
noticed,
in
addition
to
a
few
large
amplitude
displacements.
The
characteristic
lattice
sites,
punctuated
by
sporadic
hopping
to
a
neighboring
site.
Occasionally,
concerted
ionic
anions hop from one native lattice site to another at temperatures close to Tλ [13,26,27,32,33]. We have
motion
ofcan
all Li
ions,
on average,
can be extracted
usingamplitude
Gs(r,t), the displacements.
self-part of the The
van characteristic
Hove spacehopping
also
be noticed,
in addition
to a few large
time
correlation
Gs(r,t) gives
conditional
probability
of finding
anofion
an infinitesimal
motion
of all Li function.
ions, on average,
canthe
be extracted
using
Gs(r,t), the
self-part
theinvan
Hove spacevolume
dr around
r at timeGt,s(r,t)
given
thatthe
it has
occupiedprobability
the origin at
t = 0.anFigure
Gs(r,t)
time correlation
function.
gives
conditional
oftime
finding
ion in5b
anshows
infinitesimal
for
the
Li
ions
for
a
correlation
time
of
30
ps
at
1200
K.
The
self-correlation
decays
rapidly
with
r, sbut
volume dr around r at time t, given that it has occupied the origin at time t = 0. Figure 5b shows G
(r,t)
for the Li ions for a correlation time of 30 ps at 1200 K. The self-correlation decays rapidly with r, but
Entropy 2017, 19, 227
6 of 11
Entropy 2017, 19, 227
6 of 11
observed similar hopping of Li ions, as shown in Figure 5. The Li ions mostly vibrate about their lattice
thepunctuated
multiple peaks
the proclivity
Li ions to settle
these positions
simultaneously.
To
sites,
by show
sporadic
hopping of
to the
a neighboring
site.atOccasionally,
concerted
ionic hopping
identify
the
peak
locations,
the
radial
distribution
function
(RDF)
of
the
Li
ions
at
1200
K
is
also
shown
can also be noticed, in addition to a few large amplitude displacements. The characteristic motion
in Figure 5b. Interestingly, the peak positions of Gs(r,t) show excellent agreement with the nearest
of all
Li ions, on average, can be extracted using Gs (r,t), the self-part of the van Hove space-time
neighbor distances obtained from RDF, indicating that Li ions engage in discrete hops from one
correlation function. Gs (r,t) gives the conditional probability of finding an ion in an infinitesimal
native lattice site to another.
volume dr around r at time t, given that it has occupied the origin at time t = 0. Figure 5b shows Gs (r,t)
The ionic jumps are thermally driven and do not depend on having permanent vacancies/defects
forinthe
ions for
a correlation
of 30 psweathave
1200established
K. The self-correlation
decays
with r,
theLi
system.
In our
recent worktime
on fluorites,
that such concerted
ionicrapidly
hops can
butpersist
the multiple
peaksofshow
the proclivity
the Li ionsintoforming
settle atquasi-one-dimensional
these positions simultaneously.
for hundreds
picoseconds
and are of
instrumental
stringTo like
identify
the peak
locations,
the radial
distribution
function
(RDF)
of the Li among
ions atthe
1200
K is also
structures
[23]. Not
surprisingly,
a similar
mechanism
is seen to
be operational
Li ions
shown
in
Figure
5b.
Interestingly,
the
peak
positions
of
G
(r,t)
show
excellent
agreement
above the characteristic order-disorder transition temperature
Tα. In a departure from with
the the
s
conventional
interpretation
of defects,
weRDF,
willindicating
now showthat
thatLidisorder
among
the Li ions
is from
nearest
neighbor
distances obtained
from
ions engage
in discrete
hops
through
dynamic
string-like structures, which have a finite lifetime.
onemanifested
native lattice
site to
another.
Lithium ions
(a)
10-2
1.5
10-3
1.0
10-4
0.5
1
Gs(r,t)
Δx(t), a/2
2
2.0
Lithium ions
(b)
0
-1
-2
T = 1200 K
T = 1200 K
-5
0
20
40
60
Time (ps)
80
100
10
1
2
3
4
5
6
7
8
Radial Distribution Function
3
0.0
r(Å)
Figure 5. (a) Normalized x-displacement of four randomly chosen lithium ions at 1200 K. Recall that
Figure 5. (a) Normalized x-displacement of four randomly chosen lithium ions at 1200 K. Recall that
the closest cation sites are separated by a/2, along the [1 0 0] direction. (b) The van Hove selfthe closest cation sites are separated by a/2, along the [1 0 0] direction; (b) The van Hove self-correlation
correlation function Gs(r,t) of lithium ions for a correlation time of 30 ps at a temperature of 1200 K.
function
Gs (r,t)
of lithium ions for a correlation time of 30 ps at a temperature of 1200 K. The G (r,t)
peaks are nearly coincident with the nearest neighbor peaks of the partial Li radial s
The Gs(r,t)
peaks
are
nearly
coincident
distribution function
(RDF). with the nearest neighbor peaks of the partial Li radial distribution
function (RDF).
3.3. String-Like Dynamic Structures
The
ionic
jumpsofare
thermally
and do not depend
ontwo
having
permanent
vacancies/defects
The
hopping
Li ions
mostly driven
occurs cooperatively
between
or more
neighbors.
We observe
in the
system.
In our recent
work
on fluorites,
we have
established
that
concerted ionic hops
from
our simulations
that the
mobile
Li ions tend
to follow
each other
insuch
quasi-one-dimensional
or can
persist
for hundreds
of picoseconds
areas
instrumental
in of
forming
string-like
trajectories.
We identify aand
string
an ordered set
ions in quasi-one-dimensional
which an ion replaces thestring-like
next
one over[23].
a certain
interval; a string
thusmechanism
contains at least
two ions.
criterion isamong
similarthe
to the
structures
Nottime
surprisingly,
a similar
is seen
to beThis
operational
Li ions
one
used
in
the
study
of
supercooled
liquids
to
analyze
string-like
cooperative
motion
in
the
arrested
above the characteristic order-disorder transition temperature Tα . In a departure from the conventional
state [34,35]. If
is thewe
probability
of show
detecting
of length
L over
δt, the mean
interpretation
ofP(L)
defects,
will now
thata string
disorder
among
the aLitime
ionsinterval
is manifested
through
string
length
(Δ)
is
computed
as
Σ
(
)/Σ
(
).
We
have
varied
the
time
interval
of
observation
(δt)
dynamic string-like structures, which have a finite lifetime.
from 0.1 ps to several hundred ps; these time intervals are generally sufficient to establish the kinetics
string formation
andStructures
annihilation. For computational efficiency, we have limited our analysis to
3.3.ofString-Like
Dynamic
the top 5% mobile Li ions. As observed in our previous work on fluorites [23], the string characteristics
hopping of
Li ions
mostly
occursmobile
cooperatively
areThe
dynamically
similar
among
different
groups. between two or more neighbors. We observe
from our
simulations
the mobile
Li mean
ions tend
follow
Figure
6a showsthat
the variation
of the
stringtolength
(Δ)each
withother
time, in
forquasi-one-dimensional
temperatures ranging or
string-like
trajectories.
a string
as an ordered
ofneeded
ions infor
which
an ion replaces
the next
from 1000
to 1400 K. We
Theidentify
strings have
a lifetime
with the set
time
establishing
the longest
as τ*interval;
(T), decreasing
ForThis
anycriterion
temperature,
withto the
onestring,
over adenoted
certain time
a stringwith
thus increasing
contains attemperature.
least two ions.
is similar
time,
the strings
latch on toliquids
new ions
grow in
length until
the peak time
τ*. Clearly,
oneincreasing
used in the
study
of supercooled
to to
analyze
string-like
cooperative
motion
in thethis
arrested
dynamic
process
is
favorable
at
lower
temperatures
rather
than
at
higher
temperatures.
Thus,
strings
state [34,35]. If P(L) is the probability of detecting a string of length L over a time interval δt,
the mean
of different
lengths
can be formed
at(all
temperatures,
but longer
are more
probable
at lower (δt)
string
length (∆)
is computed
as ΣLP
L)/ΣP
variedones
the time
interval
of observation
( L). We have
temperatures. It is interesting to note that the peak length of the strings approaches two at the highest
from 0.1 ps to several hundred ps; these time intervals are generally sufficient to establish the kinetics
temperatures, which indicates that the string motion at these temperatures mostly involves a set of
of string formation and annihilation. For computational efficiency, we have limited our analysis to the
two Li ions where one replaces the other. Since the formation of strings is a random process, coherent
top 5% mobile Li ions. As observed in our previous work on fluorites [23], the string characteristics are
dynamically similar among different mobile groups.
Entropy 2017, 19, 227
7 of 11
1
1000 K
1050 K
1150 K
1250 K
1350 K
14
12
10
8
(a)
1100 K
1200 K
1300 K
1400 K
6
(b)
900 K
1000 K
1200 K
0.1
P(L)
Mean string length (Δ)
Figure 6a shows the variation of the mean string length (∆) with time, for temperatures ranging
from 1000 to 1400 K. The strings have a lifetime with the time needed for establishing the longest
string, denoted as τ* (T), decreasing with increasing temperature. For any temperature, with increasing
time, the strings latch on to new ions to grow in length until the peak time τ* . Clearly, this dynamic
process is favorable at lower temperatures rather than at higher temperatures. Thus, strings of different
lengths can be formed at all temperatures, but longer ones are more probable at lower temperatures.
It is interesting to note that the peak length of the strings approaches two at the highest temperatures,
Entropy
2017, 19,that
227 the string motion at these temperatures mostly involves a set of two Li ions
7 ofwhere
11
which
indicates
one replaces the other. Since the formation of strings is a random process, coherent and well-defined
and
well-defined
stringan
formation
follows
an exponential distribution—a
feature
that is also
string
formation
follows
exponential
distribution—a
feature that is also
observed
in observed
supercooled
in supercooled liquids and jammed systems [34,36]. As noted in Figure 6b, the probability of string
liquids and jammed systems [34,36]. As noted in Figure 6b, the probability of string formation
formation follows an exponential distribution above the order-disorder transition temperature Tα
follows an exponential distribution above the order-disorder transition temperature Tα (~1000 K).
(~1000 K). At lower temperatures, the distribution is uneven, which suggests a lack of spatio-temporal
At lower
temperatures,
distribution
is auneven,
which
suggests
a lackbehavior
of spatio-temporal
coherence
coherence
among thethe
Li ions
in forming
string-like
structure.
A similar
is also observed
in
among
the Liwherein
ions in well-characterized
forming a string-like
structure.
A similar
behavior
is also observed
in fluorites,
fluorites,
strings
are identified
only above
the characteristic
temperature
wherein
well-characterized strings are identified only above the characteristic temperature Tα [23].
Tα [23].
0.01
4
0.001
2
0.1
1
10
Time-interval, δt (ps)
100
1E-4
0
20
40
60
String length, L
Figure 6. (a) The evolution of string-like dynamical structures in time among the most mobile (top
Figure
6. (a) The evolution of string-like dynamical structures in time among the most mobile (top
5%) Li ions. The mean string length (Δ) is measured by the number of ions constituting the string.
5%) Li ions. The mean string length (∆) is measured by the number of ions constituting the string;
(b) The probability of finding a string of length L at different temperatures during the string lifetime
(b) The
probability of finding a string of length L at different temperatures during the string lifetime
*(T), which is defined as the time needed for establishing the longest string.
τ
*
τ (T), which is defined as the time needed for establishing the longest string.
The population of lithium ions that are involved in strings throws more light on the string
kinetics.
Figure 7aofshows
theions
mean
of ions
that are
participating
in the
process.
The population
lithium
thatpopulation
are involved
in strings
throws
more light
on string
the string
kinetics.
Strikingly,
near
Tα mean
(1000 K),
close to 80%
the that
Li ions
the top 5%)
participate
in the formation
Figure
7a shows
the
population
ofof
ions
are(among
participating
in the
string process.
Strikingly,
Atclose
temperatures
λ (1200 K), as the peak string length decreases, the peak
nearofTαstrings.
(1000 K),
to 80% ofabove
the Li Tions
(among the top 5%) participate in the formation of strings.
participation
also
drops
significantly.
Interestingly,
thelength
peak participation
lifetime
(τP), which wealso
At temperatures above Tλ (1200 K), as the
peak string
decreases, the
peak participation
define as the time when most ions participate, is somewhat larger thanPthe peak string length lifetime
drops* significantly. Interestingly, the peak participation lifetime (τ ), which we define as the time
τ as shown in Figure 7b for temperatures below Tλ. While in typical supercooled liquids, these
when most ions participate, is somewhat larger than the peak string length lifetime τ* as Pshown in
lifetimes are nearly identical [23,34], our earlier work on fluorites (CaF2 and UO2) also shows τ to be
Figure 7b for temperatures
below Tλ . While in typical supercooled liquids, these lifetimes are nearly
larger than τ* at temperatures below
Tλ [23]. However, at a temperature of Tλ and above, both
identical
[23,34],
earlier
work on
fluorites
(CaF
) also
shows τP to be larger than τ* at
2 and UO
timescales
are our
almost
coincident,
a behavior
that
is identical
in2CaF
2 and UO2. It is instructive to note
temperatures
below T
and above,
timescales
are almost
λ [23]. However,
that the lifetime
timescales
are O(1)atpsa temperature
at Tλ, which of
is Taλ signature
forboth
liquid-like
diffusivity.
coincident,
a
behavior
that
is
identical
in
CaF
and
UO
.
It
is
instructive
to
note
that
the
lifetime
Summarizing, two distinct timescales can be recognized
in
2
2 antifluorites (and fluorites [23]): the time
*
P).
timescales
are O(1)topsthe
at peak
Tλ , which
a signature
Summarizing,
two (τ
distinct
corresponding
string is
length
(τ ), andfor
theliquid-like
time whendiffusivity.
most ions participate
in strings
*
P
Whereas
τ isbeassociated
withinindividual
strings,
τ fluorites
dictates the
collective
relaxation
of the strings.
Wepeak
timescales
can
recognized
antifluorites
(and
[23]):
the time
corresponding
to the
P (and not τ*) will control the diffusive relaxation P
*
*
therefore
anticipate
that
τ
of
the
system:
in
fluorites,
string length (τ ), and the time when most ions participate in strings (τ ). Whereas τ is associated
have shown
that this
is indeedthe
thecollective
case [23]. relaxation of the strings. We therefore anticipate that
withwe
individual
strings,
τP dictates
τP (and not τ* ) will control the diffusive relaxation of the system: in fluorites, we have shown that this
is indeed the case [23].
Entropy 2017, 19, 227
Entropy 2017, 19, 227
8 of 11
8 of 11
(a)
0.8
*
τ* and τPτ(ps)
and τP (ps)
(a)
0.8
0.4
Fraction of ions
Fraction of ions
0.6
0.6
0.2
0.4
0.0
0.2
1000 K
1150 K
1300 K
1050 K
1200 K
1350 K
0.1
0.0
1 1000 K
1100 K
1250 K
1400 K
10
1050 K
1200 K
1350 K
1150 K
1300 K
0.1
1
10
(b)
τ*
τP
1
1
0.8
1100 K
1250 K
1400 K
10
8 of 11
10
100
Time interval, δt (ps)
(b)
τ*
τP
Entropy 2017, 19, 227
0.9
1.0
1.1
1.2
1.3
Tλ/T
100
0.8
0.9
1.0
1.1
1.2
1.3
Figure 7. (a) The fraction of ions participating in string-like dynamical structures among the most
Figure 7. (a) The fraction
of ionsδtparticipating
in string-like dynamical structures
among the most
interval,
(ps) corresponding
mobile (top 5%) Time
Li ions.
(b) Lifetimes
to peak string length (τT*λ)/Tand maximal Li ion
mobile (top 5%) Li ions; (b) Lifetimes corresponding to peak string length (τ* ) and maximal Li ion
P). While these timescales are nearly identical above Tλ (1200 K), they diverge below Tλ.
participation
(τThe
P
Figure 7. (τ
(a)
fraction
ions participating
in string-like
dynamical
structures
among
the most
participation
). While
theseoftimescales
are nearly
identical above
Tλ (1200
K), they
diverge
below Tλ .
mobile (top 5%) Li ions. (b) Lifetimes corresponding to peak string length (τ*) and maximal Li ion
We
will now(τP).attempt
to timescales
evaluateare
annearly
effective
string
Asdiverge
strings
areTλ.transient
While these
identical
above diffusivity.
Tλ (1200 K), they
below
participation
We
will
now
attempt
to
evaluate
an
effective
string
diffusivity.
As
strings
are
transient
structures,
structures, diffusivity cannot be computed as the longtime (t → ∞ ) limit of the slope of the
mean
2
diffusivity
cannot
be (Δr
computed
as thewe
longtime
(t →string
∞ ) the
limit
of diffusivity
theAs
slope
ofcontrolled
the
will now
attempt
evaluate
an
effective
diffusivity.
strings
are mean
transient
) [31].toInstead,
will
assume
that
string
is
bysquare
the
square We
displacement
2 ) [31]. cannot
P [23].
structures,
diffusivity
be τcomputed
as the
longtime
(t →
∞a) characteristic
limit
of the slope
of the mean
displacement
(∆r
Instead,
we
will
assume
that
the
string
diffusivity
is
controlled
byδPthe
peak
string participation
lifetime
Since
strings
experience
displacement
of
2) [31]. Instead,
we
will strings
assume
that the string
diffusivity
is controlled
by the
(Δr
in square
time
τPdisplacement
,participation
we will regard
the string
diffusivity
to be proportional
(δP)2/τP, analogous
to the
slope
peak
string
lifetime
τP [23].
Since
experience
atocharacteristic
displacement
of δP
P
P
peak
participation
lifetime
τ diffusivity
[23]. Since
strings
experience
a characteristic
displacement
of
δ slope
Pstring
P )2 /τ
P , analogous
theτmean
displacement.
Figure
8, we
the (scaled)
diffusivity
of the
in of
time
, we square
will
regard
the
stringIn
to delineate
be proportional
to (δbulk
tolithium
the
in time τP, we will regard2 the string diffusivity to be proportional to (δP)2/τP, analogous to the slope
evaluated
as displacement.
compare
it against
thethe
(scaled)
string
function
of ions
the mean
square
In Figure
8, we
delineate
(scaled)
bulkdiffusivity,
diffusivityasofa the
lithium
lim Δr 6t and
t →∞
of the mean square
displacement.
In Figure 8, we delineate the (scaled) bulk diffusivity of the lithium
2
ions
evaluated
as limcorrelation
∆r /6t and compare
it against
the (scaled)
string diffusivity,
as a function
of ions
temperature.
the scaled
string
bulk string
diffusivities
is quite
tas
→∞lim Δr 2 6t between
evaluatedThe
and compare
it against
theand
(scaled)
diffusivity,
as astriking.
function As
tcorrelation
→∞
P
P
of shown
temperature.
The
between
the
scaled
string
and
bulk
diffusivities
is
quite
striking.
in the inset of Figure 8, the average string displacement δ at τ is only slightly more than
the
of
temperature.
The
correlation
between
the
scaled
string
and
bulk
diffusivities
is
quite
striking.
As than
P
P
Asclosest
showncation-cation
in the inset separation
of Figure 8,distance,
the average
string
displacement
δ at
τ is only slightly
more
and they
differ
little across
the
temperatures.
Thus
the
string
shown in the inset of Figure 8, the average string displacement δP at τP isPonly slightly more than the
thediffusivity
closest cation-cation
separation
distance,
they differ lifetime
little across
the temperatures.
Thus the
is dominated
by the peak
stringand
participation
τ . The
close correspondence
closest cation-cation separation distance, and they differ little across the temperatures.
Thus the string
P
between
the bulk
and string by
diffusivities
emphasizes
the influence
of τ
low-dimensional
string-like
string
diffusivity
is dominated
the peak string
participation
lifetime
. The close correspondence
diffusivity is dominated by the peak string participation lifetime τP. The close correspondence
structures
in
the
superionic
diffusion
processes.
between
the
bulk
and
string
diffusivities
emphasizes
the
influence
of
low-dimensional
string-like
between the bulk and string diffusivities emphasizes the influence of low-dimensional string-like
structures
in
the
superionic
diffusion
processes.
structures in the superionic diffusion processes.
10
(
)
)
10
Bulk Diffusivity
String
Diffusivity
Bulk Diffusivity
1
String Diffusivity
0.1
P
0.1
P
1
Average
δ (Å)
Average
δP (Å)
at τPat τ
Scaled
Diffusivities
Scaled
Diffusivities
(
0.01
0.01
3
3
2
2
1
Lithium ions
1
0
0.8
0
0.8
0.8
0.8
Lithium ions
0.9
1.0
0.9
1.0 T1.1
/T 1.2
λ
0.9
0.9
1.1
Tλ/T
1.2
1.3
1.3
1.0
1.0
1.4
1.4
1.1
1.1
T /T
1.2
1.2
1.3
1.3
Tλλ/T
Figure
8. Bulk and string diffusivities, scaled to the corresponding magnitudes at Tλ, for different
Figure 8. Bulk and string diffusivities, scaled to the corresponding magnitudes at Tλ, for different
Figure
8. Bulk and
stringshows
diffusivities,
scaled to
the corresponding
magnitudes
at Tλ , forlifetime
different
P at peak
temperatures.
displacement
string
participation
P) )
temperatures.The
Theinset
inset showsthe
the mean
mean string
string displacement
(δ(δ
at peak
string
participation
lifetime
P
temperatures.
The inset shows the mean string displacement (δ ) at peak string participation
).P).
(τP(τ
lifetime (τP ).
Entropy 2017, 19, 227
9 of 11
4. Discussion and Conclusions
Early neutron diffraction investigations have focused on determining the most probable defect
positions for the disordered state in fluorites [11,37] and antifluorites [10]. While several quasi-static
defect cluster models have been proposed before [38], none of them has the ability to identify realistic
ionic conduction or diffusive pathways. The goal of this paper has been to investigate the dynamics
of Li ions in the superionic state using atomistic simulations and propose a mechanism that can
potentially pinpoint the origin of enhanced diffusivity of Li ions in Li2 O. Our first endeavor has been
to define the extent of the superionic state of Li2 O. Using the reported neutron diffraction [10] and
ionic conductivity [2] data, we have established that the disorder among Li ions is first manifested at
a characteristic temperature Tα (~1000 K), which is below the λ transition temperature Tλ (1200 K).
The disorder estimated from the atomistic simulations shows excellent agreement with the neutron
diffraction data and has provided an additional confirmation for the order-disorder transition or the
onset of superionicity at Tα .
The hopping of Li ions across the native tetrahedral sites has been reported before, as well as
in this investigation. We have examined the kinetics of hopping over longer timescales and have
determined that Li ions form string-like dynamical structures with a finite lifetime. We have then
shown that the string diffusivity, which is based on peak participation of Li ions in strings, depicts a
remarkable correlation to the bulk diffusivity of the system. Thus, a plausible atomistic mechanism is
proposed that accounts for both the disorder, which is inherently dynamic, and the enhanced ionic
diffusion in the superionic state of Li2 O.
Acknowledgments: The authors acknowledge funding from the US Department of Energy (DOE) through the
Nuclear Energy University Program (NEUP). Insightful observations from Professors Juan Garrahan and Srikanth
Sastry are gratefully acknowledged.
Author Contributions: Ajay Annamareddy conceived the project, performed the simulations, and analyzed the
data. Ajay Annamareddy and Jacob Eapen jointly wrote the paper. Both authors have read and approved the
final manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
The parameters for the inter-ionic potential are shown in Table A1.
Table A1. Inter-ionic potential parameters [24].
z+ (|e|)
z− (|e|)
A++ (eV)
A−− (eV)
A+− (eV)
ρ+− (Å)
0.790909
−1.581818
0.0
0.0
1425.4833
0.236303
C++ = C−− = C+ − = 0.
Appendix B. Comparison of Wolf and Ewald Summation Methods for Coulombic Interactions
The short range Wolf method [30] is an attractive choice for the evaluation of electrostatic
interactions in ionic systems. We have verified that the accuracy of the Wolf method is comparable
to that of the Ewald sum method, which is typically used in ionic molecular simulations. The Ewald
sum simulations are performed using LAMMPS [39] while the Wolf method is implemented using
our in-house code—qbox. As delineated in Figure A1, both methods predict nearly identical structure
(radial distribution function) and dynamics (mean square displacement). The methods are tested on
identical systems with 6144 ions with periodic boundary conditions applied along all three orthogonal
directions. A timestep of 1 fs is employed for all the simulations. For the Ewald sum calculations,
a relative error of 10−7 is assigned (for forces), while for the Wolf method, the damping parameter (α)
is chosen as 0.3. Our earlier simulations [13,21–23] on superionic UO2 and CaF2 also showed excellent
agreement between these methods.
Entropy 2017, 19, 227
Entropy 2017, 19, 227
2.0
Li-Li g(r)
10 (b)
Ewald sum
Wolf method
(a)
MSD of Li ions (Å2)
2.5
10 of 11
10 of 11
900 K
1.5
1200 K
1.0
0.5
0.0
0
2
4
6
r (Å)
8
10
Ewald sum
Wolf method
1200 K
1
900 K
0.1
0.01
0.1
1
10
Time (ps)
Figure A1. Structure and dynamics of lithium ions in Li2O using Ewald sum and Wolf methods.
Figure A1. Structure and dynamics of lithium ions in Li2 O using Ewald sum and Wolf methods.
(a) Radial distribution function, g(r) of lithium ions at temperatures on either side of the superionic(a) Radial distribution function, g(r) of lithium ions at temperatures on either side of the superionic-onset
onset temperature (Tα ~ 1000 K). (b) Mean square displacement of the lithium ions for the same
temperature (Tα ~1000 K); (b) Mean square displacement of the lithium ions for the same
temperature range.
temperature range.
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