Atomic dynamics of alumina melt: A molecular dynamics simulation study
The atomic dynamics of Al₂O₃ melt are studied by molecular dynamics simulation. The particle interactions are described by an advanced ionic interaction model that includes polarization effects and ionic shape deformations. The model has been shown to reproduce accurately the static structure factor...
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Інститут фізики конденсованих систем НАН України
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| Cite this: | Atomic dynamics of alumina melt: A molecular dynamics simulation study / S. Jahn, P.A. Madden // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 169-178. — Бібліогр.: 36 назв. — англ. |
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| citation_txt | Atomic dynamics of alumina melt: A molecular dynamics simulation study / S. Jahn, P.A. Madden // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 169-178. — Бібліогр.: 36 назв. — англ. |
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| description | The atomic dynamics of Al₂O₃ melt are studied by molecular dynamics simulation. The particle interactions are described by an advanced ionic interaction model that includes polarization effects and ionic shape deformations. The model has been shown to reproduce accurately the static structure factors S(Q) from neutron and x-ray diffraction and the dynamic structure factor S(Q, ω) from inelastic x-ray scattering. Analysis of the partial dynamic structure factors shows inelastic features in the spectra up to momentum transfers, Q, close to the principal peaks of partial static structure factors. The broadening of the Brillouin line widths is discussed in terms of a frequency dependent viscosity η(ω).
Атомна динамiка розплаву Al₂O₃ дослiджується моделюванням методом молекулярної динамiки. Мiжчастинковi взаємодiї описуються вдосконаленою моделлю iонних взаємодiй, що включає ефекти поляризацiї та деформацiї iонної форми. Показано, що модель акуратно вiдтворює статичнi структурнi фактори S(Q) з дифракцiї нейтронiв та рентгенiвських променiв та динамiчний структурний фактор S(Q, ω) з непружнього розсiяння рентгенiвських променiв. Аналiз парцiальних динамiчних структурних факторiв показує непружнi особливостi у спектрах аж до передачi iмпульсу, Q, близької до головних пiкiв парцiальних статичних структурних факторiв. Збiльшення ширини брiлюенiвського максимуму обговорюється у зв’язку з залежною вiд частоти в’язкiстю η(ω).
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Condensed Matter Physics 2008, Vol. 11, No 1(53), pp. 169–178
Atomic dynamics of alumina melt: A molecular
dynamics simulation study
S.Jahn1∗, P.A.Madden2
1 GeoForschungsZentrum Potsdam, Department 4, Telegrafenberg, 14473 Potsdam, Germany
2 Chemistry Department, University of Edinburgh, Edinburgh EH9 3JJ, U.K.
Received November 2, 2007
The atomic dynamics of Al2O3 melt are studied by molecular dynamics simulation. The particle interactions
are described by an advanced ionic interaction model that includes polarization effects and ionic shape defor-
mations. The model has been shown to reproduce accurately the static structure factors S(Q) from neutron
and x-ray diffraction and the dynamic structure factor S(Q, ω) from inelastic x-ray scattering. Analysis of the
partial dynamic structure factors shows inelastic features in the spectra up to momentum transfers, Q, close
to the principal peaks of partial static structure factors. The broadening of the Brillouin line widths is discussed
in terms of a frequency dependent viscosity η(ω).
Key words: molecular dynamics, collective dynamics, liquid, viscosity, relaxation time, alumina
PACS: 61.20.Ja, 61.20.Lc, 66.20.+d, 61.10.Eq, 62.60.+v
1. Introduction
In recent years, substantial progress has been achieved in understanding the atomic dynamics of
liquids. This is mainly due to two achievements, i.e., the development of inelastic x-ray scattering
(IXS) as complementary technique to inelastic neutron scattering (INS) and the increasing power
of computer simulations. It is well known that the interatomic potential has an increasing effect on
the dynamics when time scales of atomic interactions and length scales of interatomic distances are
approached [1,2]. Experimentally, a detailed information about the interatomic forces can be ob-
tained from measurements of the dynamic structure factor S(Q,ω), which represents the collective
atomic dynamics, using inelastic neutron or x-ray scattering techniques. Q is the magnitude of the
wave vector transfer and ω is the angular frequency, ~ω the respective energy transfer with ~ being
Planck’s constant divided by 2π. Propagating acoustic-like modes have been observed in many
liquids up to the terahertz frequency range, but they are much more pronounced and persistent in
metallic than in molecular systems, which has been shown to be a consequence of different types
of interaction [3].
For a long time the experimental observation of collective excitations in ionic liquids was very
complicated or impossible due to the large sound velocity and kinematic restrictions of INS [4,5].
These restrictions were lifted when IXS was developed [6]. Since then the collective excitations
in S(Q,ω) have been measured, e.g. in molten NaCl [7] and molten Al2O3 [8], the latter in a
containerless levitation experiment. On the other hand, the increasing computing power allows
us to perform computer simulations of a new quality. The latter is essential since, especially in
the field of liquids, the combination of experiment and simulation has always been a very fruitful
symbiosis towards the understanding of atomistic processes.
If a realistic modelling of the atomic dynamics of liquids is attempted, an accurate description
of the interatomic forces is crucial. The simplest representation of atomic interactions via pair
potentials, which gives a good description of molecular systems in a wide range of thermodynamic
states, cannot be easily applied to ionic materials. The reason is that the electronic structure of an
∗E-mail: jahn@gfz-potsdam.de
c© S.Jahn, P.A.Madden 169
S.Jahn, P.A.Madden
ion in a condensed phase strongly depends on the interaction with its neighbours and hence many-
body effects have to be considered. Rigid ion models of Born-Mayer-type, for example, subsume the
many-body effects in some average sense and may be capable of reproducing some of the experi-
mental results. However, they often do not provide a reliable description of both static and dynamic
properties. Other effects such as polarization have to be included into an ionic model to obtain a
more realistic description of disordered ionic systems, which has been shown for halide melts [9] or
amorphous silica [10]. Therefore, various polarizable ion models have been developed in order to
model the properties of ionic solids and melts. However, most of them are optimized for a specific
system and possess just a very limited transferability. Recently, a systematic approach to the con-
struction of more transferable ionic potentials was described [11,12]. The so-called Aspherical Ion
Model (AIM) includes explicitly all components of the interaction that are considered essential. Its
parameters are optimized using first-principles electronic structure calculations [11,12].
The work presented here is inspired by recent experimental studies of the atomic structure
[13–15] and dynamics [8] of Al2O3 melt. Most of the previous simulation studies of alumina melt
(e.g. [16–18]) were concerned with structural properties at ambient pressure. Hoang and coworkers
used a rigid ion model in order to study the self-diffusion [19] and the high pressure behaviour of
the melt [20]. An advanced ionic model of AIM-type [21] was recently used to perfectly reproduce
both the experimental static and dynamic structure factors of alumina melt [22]. Here, we focus
on the relations between the spectra of collective modes, the vibrational spectra and the structural
relaxation processes in the melt.
2. Computational method
2.1. Interaction potential
Details of the AIM potential and its parametrization have been described earlier [21]. The Al2O3
potential is not only applicable to the melt [22] but it also satisfactorily describes the properties of
solid alumina phases [21,23]. The AIM interatomic potential V is constructed from of components
V = V qq + V disp + V rep + V pol . (1)
The first two components, the charge-charge (V qq) and dispersion (V disp) interactions, are a purely
pairwise additive:
V qq =
∑
i6j
qiqj
rij
, (2)
V disp = −
∑
i6j
[
f ij
6 (rij)C
ij
6
/
r6
ij + f ij
8 (rij)C
ij
8
/
r8
ij
]
, (3)
qi is the formal charge on ion i (+3 for Al and –2 for O), Cij
6 and Cij
8 are the dipole-dipole
and dipole-quadrupole dispersion coefficients respectively, and f ij
n are Tang-Toennies dispersion
damping functions [24], which describe short-range corrections to the asymptotic dispersion term.
The overlap repulsion component (V rep) is given by
V rep =
∑
i∈O,j∈Al
[
A−+e−a−+ρij
+ B−+e−b−+ρij
+ C−+e−c−+rij
]
+
∑
i,j∈O
A−−e−a−−rij
+
∑
i∈O
[
D(eβδσi
+ e−βδσi
) + (eζ2|νi|2 − 1) + (eη2|κi|2 − 1)
]
, (4)
with
ρij = rij − δσi − S(1)
α νi
α − S
(2)
αβ κi
αβ , (5)
and summation of repeated indexes is implied. The variable δσi characterizes the deviation of
the radius of oxide anion i from its default value, {νi
α} are a set of three variables describing the
170
Atomic dynamics of alumina melt
Table 1. AIM potential parameters [21]. All values are in atomic units.
A+− 16.635 a+− 1.3850
B+− 75801. b+− 3.8185
C+− 5604.3 c+− 4.1444
A−− 420.17 a−− 2.4438
D 0.42282 β 1.2023
ζ 0.91129 η 3.1961
α 9.3809 C 9.3865
bD 2.1488 cD 1.9869
bQ 2.1429 cQ 1.3404
C+−
6 2.1793 C−−
6 44.372
C+−
8 25.305 C−−
8 853.29
b+−
disp 2.2057 b−−
disp 1.4385
Cartesian components of a dipolar distortion of the ion, and {κi
αβ} are a set of five independent
variables describing the corresponding quadrupolar shape distortions (|κ|2 = κ2
xx + κ2
yy + κ2
zz +
2(κ2
xy + κ2
xz + κ2
yz) with a traceless matrix κ). S
(1)
α = rij,α/rij and S
(2)
αβ = 3rij,αrij,β/r2
ij − δαβ
are interaction tensors. The last summations include the self-energy terms, representing the energy
required to deform the anion charge density, with β, ζ and η as effective force constants. The extent
of each ion’s distortion is determined at each molecular dynamics time-step by energy minimization.
The polarization part of the potential incorporates dipolar and quadrupolar contributions [25],
V pol =
∑
i,j∈O
(
(
qiµj
α − qjµi
α
)
T (1)
α +
(
qiθj
αβ
3
+
θi
αβqj
3
− µi
αµj
β
)
T
(2)
αβ +
(
µi
αθj
βγ
3
+
θi
αβµj
γ
3
)
T
(3)
αβγ
+
θi
αβθj
γδ
9
T
(4)
αβγδ
)
+
∑
i∈O,j∈Al
(
qjµi
α
[
1 − gD(rij)
]
T (1)
α +
θi
αβqj
3
[
1 − gQ(rij)
]
T
(2)
αβ
)
+
∑
i∈O
(
k1|~µi|2 + k2µ
i
αθi
αβµi
β + k3θ
i
αβθi
αβ + k4|~µi · ~µi|2
)
, (6)
where k1 = 1
2α
, k2 = B
4α2C
, k3 = 1
6C
, k4 = −B2
16α4C
, α, B and C the dipole, dipole-dipole-quadrupole
and quadrupole polarizabilities of the oxygen ion, respectively, and Tαβγδ = ∇α∇β∇γ∇δ . . . 1
rij
are
the multipole interaction tensors [26]. The instantaneous values of these moments are obtained by
minimization of this expression. The charge-dipole and charge-quadrupole cation-anion asymptotic
functions include terms which account for penetration effects at short-range by using Tang-Toennies
damping functions [24] of the form,
gD(rij) = cDe−bDrij
4
∑
k=0
(bDrij)k
k!
, (7)
gQ(rij) = cQe−bQrij
6
∑
k=0
(bQrij)k
k!
, (8)
with D and Q standing for dipolar and quadrupolar parts. While the parameters bD and bQ
determine the range at which the overlap of the charge densities affects the induced multipoles,
the parameters cD and cQ determine the strength of the ion response to this effect. Only oxygen
ions are considered polarizable. A compilation of the potential parameters is given in table 1.
171
S.Jahn, P.A.Madden
2.2. Molecular dynamics simulation
Due to the “electronic” degrees of freedom of the AIM potential, the molecular dynamics simula-
tions include additional annealing in the spirit of Born-Oppenheimer dynamics. Induced multipoles
and ionic deformations are obtained at each time step by conjugate gradient minimization of the
respective contributions to the total energy before forces acting on individual ions are calculated.
The molecular dynamics simulations are performed with cubic simulation cells containing 2160
ions (432 formula units) and a constant temperature T of 2500 K. The cell size is significantly
larger than in the previous study [22] to access smaller Q and to make better contact to the IXS
experimental data. The smallest Q from the simulation is determined by the simulation box length,
here it is 0.21 Å−1. In the simulation, the system is first equilibrated for 10 ps in the NPT ensemble
using an isotropic barostat coupled to a Nosé-Hoover thermostat [27]. Thereafter the barostat is
switched off and simulations continue in the NVT ensemble. The temperature is still controlled
by the Nosé-Hoover thermostat [28] in order to correct for a small energy drift at long simulation
times. Trajectories of the production runs are collected for 200 ps.
3. Results and discussion
3.1. Vibrational spectra and self-diffusion coefficients
Single particle time correlation functions provide information regarding the distribution of
vibrational frequencies in the melt. The velocity autocorrelation functions
〈
vX
1 (0)vX
1 (t)
〉
for X =
Al and O are shown in figure 1. After a fast initial decay the typical oscillatory behaviour is
observed. The correlation is essentially lost after a few tenth of picoseconds. Fourier transformation
of the velocity autocorrelation functions defines a power spectrum of vibrational frequencies
zX(ω) =
1
2π
∫ ∞
−∞
dt
1
3
〈
vX
1 (0)vX
1 (t)
〉
exp(−iωt). (9)
Both zAl(ω) and zO(ω) show a broad distribution with vibrational energies up to about 200 meV
(see figure 1). Besides the principal peak at about 35 meV, there are two shoulders at around 75
and 100 meV. Around those frequencies, weakly dispersive peaks are observed at low Q in the
transverse and longitudinal charge current correlation spectra, respectively. They are reminiscent
of optic modes and seem to make a significant contribution to the total vibrational density of
states. The zero frequency limit of zX(ω) yields the self-diffusion coefficient, DX , divided by π for
species X. The obtained values for Al (DAl = 1.2 × 10−5 cm2/s) and O (DO = 1.3 × 10−5 cm2/s)
compare well with our earlier simulation results using the slope of the mean square displacements
to determine DX . Experimental data from tracer diffusion measurements suggest a slightly larger
diffusion coefficient (D = 2.65×10−5 cm2/s at 2475 K) [29], which is still in reasonable agreement,
considering that tracer and self-diffusion may not be exactly equivalent.
0 0.05 0.1 0.15 0.2
t [ps]
-2e-07
0
2e-07
4e-07
6e-07
8e-07
<
v 1(t
)v
1(0
)>
[
a.
u.
]
X=Al
X=O
0 50 100 150 200
ω [meV]
0
1e-05
2e-05
3e-05
4e-05
5e-05
zX
(ω
)
[c
m
2 /s
]
X=Al
X=O
Figure 1. Velocity autocorrelation functions and partial frequency spectra zO(ω) and zAl(ω) at
T = 2500 K. The zero frequency limit yields the corresponding self-diffusion coefficients DX/π.
172
Atomic dynamics of alumina melt
3.2. Collective dynamics and viscosity
The collective atomic dynamics is best represented by the double Fourier transform of the van
Hove correlation function [30], which is the dynamic structure factor S(Q,ω). For a multicomponent
system, partial dynamic structure factors, SXY (Q,ω), may be defined to represent correlations
between different species (X and Y ). SXY (Q,ω) is obtained via the time correlation function of
density fluctuations in reciprocal space δnX(Q, t), which yields the partial intermediate scattering
functions (NX being the number of particles of species X)
FXY (Q, t) =
1√
NXNY
〈
δnX,∗(Q, 0)δnY (Q, t)
〉
. (10)
Its time Fourier transform is the partial dynamic structure factor
SXY (Q,ω) =
1
2π
∫ ∞
−∞
dt exp(−iωt)FXY (Q, t). (11)
The spectral quantity accessible experimentally in inelastic neutron or x-ray scattering experiments
is a sum of partials weighted by the respective concentrations and coherent scattering lengths. Since
Al3+ and O2− contain the same number of electrons, the x-ray scattering (weighting) factors are
almost equal at low Q and the x-ray dynamic structure factor probes the fluctuations in the
ion number density. Comparison of the total S(Q,ω) calculated from the MD results using the
respective x-ray scattering factors for Al and O with the experimental S(Q,ω) at Q = 0.21 Å−1
from inelastic x-ray scattering [8] shows excellent agreement and confirms the quality of the AIM
potential (see figure 2). A similarly good agreement was achieved in our earlier study [22] using
the smaller 640 ion cell. In that case the smallest accessible Q was 0.31 Å−1.
0
0.001
0.002
0.003
0.004
S(
Q
,ω
)
[1
/m
eV
]
-20 0 20
ω [meV]
0
0.05
0.1
0.15
ω
2 S
(Q
,ω
)
[m
eV
]
0
0.002
0.004
0.006
0.008
0.01
OO
AlO
AlAl
total
-50 0 50
ω [meV]
-1
0
1
2
Q=0.21 Å
-1
Q=2.1 Å
-1
Figure 2. Top: Constant Q projections of the partial and total x-ray dynamic structure factor at
T = 2500 K. For comparison with experimental data from Sinn et al. [8] (symbols) the calculated
spectra are convoluted with a Gaussian resolution function with a width of 1.8 meV FWHM.
Bottom: The same dynamic structure factors but not convoluted and multiplied by ω2, which
is proportional to the partial and total spectra of the longitudinal current correlation functions,
respectively.
Generally, the total (x-ray or neutron weighted) S(Q,ω) is difficult to interpret and the partials
cannot easily be extracted from the experimental data. However, at small Q (usually substantially
173
S.Jahn, P.A.Madden
smaller than 1 Å−1) models of simple liquids, such as generalized hydrodynamic theory [1,2,31],
have been successfully applied to experimental data of binary liquids, including alumina melt
[8]. Figure 2 shows that in the low-Q region all SXY (Q,ω) look very similar, which justifies the
treatment as a one-component system.
The Q-range in which side peaks or shoulders are visible in S(Q,ω) of liquids strongly depends
on the type of interaction [3]. While in simple ionic liquids, such as alkali halide melts they are
barely visible even at the lowest Q accessible to IXS [7], they seem to be much more pronounced
in more complex ionic melts as in the present case of alumina melt. It is interesting to note that
shoulders are still clearly visible in SOO(Q,ω) at Q = 2.1 Å−1, which is close to the principal
peak position of the respective SOO(Q) at 2.6 Å−1. The total S(Q,ω) in figure 2 suggests that
these ’collective’ modes cannot be observed by IXS in alumina melt due to the cancellation with
the partial Al–O spectrum. However, designing an experiment with increased weighting factor of
SOO(Q,ω) compared to the Al–O and Al–Al partials could make these shoulders visible. Perhaps
this could be a challenge for inelastic neutron scattering.
0 0.5 1 1.5 2
Q [Å
-1
]
0
20
40
60
80
ω
, Γ
[
m
eV
]
ω
m
(Q) (OO)
ω
m
(Q) (AlAl)
Γ
s
(Q)
ω
s
(Q)
V
s
= 6800 m/s
Γ ∝ Q
2
Figure 3. Dispersion relations ω(Q) (full symbols) and line widths Γ(Q) (open symbols) obtained
from the maxima of the partial longitudinal current correlation spectra (circles) and from fits of
the total x-ray S(Q, ω) with the Lorentzian model (triangles). The initial slope of the dispersion
relations assuming linear behaviour corresponds to a sound velocity of 6800 m/s. The Q2-
dependence of the linewidth Γ at small Q is indicated.
Since clear peaks in SXY (Q,ω) are only visible at small Q, we can use the maxima ωm of the
spectra of the partial longitudinal current correlation functions JXY
l (Q,ω) = (ω2/Q2)SXY (Q,ω)
(see figure 2) to study the dispersion relations of the collective excitations. As shown in figure 3,
the dispersions from JOO
l (Q,ω) and JAlAl
l (Q,ω) are very similar in the Q-range up to at least
2.1 Å−1. Alternatively, the dynamic structure factor is often fitted by generalized hydrodynamic
models, which in the limit of low Q can be written as a sum of three Lorentzians [31]:
S(Q,ω) = Ac
Γc
ω2 + Γ2
c
+ As
(
Γs + b(ω + ωs)
(ω + ωs)2 + Γ2
s
+
Γs − b(ω − ωs)
(ω − ωs)2 + Γ2
s
)
(12)
Ac and As are the amplitudes of the central and inelastic lines, respectively. Γc and Γs are the
corresponding linewidths and ωs is the excitation frequency. ωs(Q) is consistent with the two ωm(Q)
dispersions at the lowest Q (see figure 3). However, the Lorentzian model does not describe well
the spectra at Q larger than about 0.6 Å−1. Assuming a linear dispersion at low Q, an estimate
of the longitudinal sound velocity is obtained from the initial slope. The value of 6800 m/s as
indicated in figure 3 is in excellent agreement with the IXS data [8]. The damping term Γs(Q)
increases approximately with Q2, which will be discussed in the following.
In the hydrodynamic limit (low Q), the inelastic (Brillouin) peak broadening is expected to be
quadratic in Q with the proportionality constant related to the longitudinal viscosity ηl and to the
174
Atomic dynamics of alumina melt
thermal diffusivity DT [31]
Γs =
1
2
[
ηl
ρ
+ (γ − 1)DT
]
Q2 (13)
with γ = Cp/Cv, the ratio of the specific heats at constant pressure and constant volume, estimated
as ∼ 1.08 [8]. This dependence on Q is actually observed experimentally [8] and also in the present
simulations (see figure 3), despite the relatively large Q values at which the observations are
made. However, this should not be taken as conclusive evidence that the hydrodynamic picture
of viscously damped propagating modes is appropriate – even disregarding the probably small
thermal contribution (second term in equation (13)) the proportionality constant is much smaller
than the measured macroscopic viscosity. Knowing the mass density ρ of the melt, the kinematic
viscosity ηl = 4/3η + ηb may be obtained. η and ηb are the shear and bulk viscosities, respectively.
From the simulation we have ρ = 2.83 g/cm3 and proportionality constants ηl/2ρ (figure 3) of
9.6 × 10−7 m2/s, which gives a kinematic viscosities of 5.4 mPa·s. This is reasonably consistent
with the analysis of the experimental data, where the damping constant yields a kinematic viscosity
of about 3.5 mPa·s at T = 2323 K [8]. These values are considerably (here about one order of
magnitude) lower than the macroscopic viscosities [32]. If the macroscopic viscosity is used to
estimate the Brillouin linewidth, through equation (13) it would predict that the Brillouin lines
were heavily overdamped.
0.0001 0.001 0.01 0.1 1 10
t [ps]
0
5e-11
1e-10
1.5e-10
2e-10
<
σ αβ
(t
)
σ αβ
(0
)>
[
a.
u.
]
τ
2
=0.55 ps
τ
1
=0.013 ps
0 50 100 150 200
ω [meV]
0.01
0.1
1
10
η(
ω
) [
m
Pa
.s
]
MD
fit (total)
fit (fast)
fit (slow)
MD ω=0
10 meV
Figure 4. Left: average stress tensor autocorrelation function (solid line) fitted with two stretched
exponentials (dashed line). Right: corresponding spectrum (frequency dependent viscosity). The
zero frequency value of η(ω) = 25 mPa.s, which corresponds to the macroscopic shear viscosity,
is indicated by a full circle. At 10 meV, η(ω) has dropped by more than a factor of ten. Also
shown are the spectra of the fitted curve and its fast and slow components.
A representation of the data with three generalized Lorentzians is also suggested by a generalized
hydrodynamic approach of Sinn et al. [8]. In the Mori-Zwanzig formalism [33] the width parameter
Γs would be frequency dependent and related to the spectrum of a memory function. Since the
information on the width parameter comes from fitting the spectrum in the vicinity of the Brillouin
lines (i.e. close to ωs), Γs should be thought of as reflecting the liquid dynamics at frequencies
close to ωs, which is itself proportional to Q. The memory function of interest in the present
context should closely resemble the correlation function of the elements of the stress tensor [34]
〈σαβ(t)σαβ(0)〉, where σαβ is an off-diagonal element of the stress tensor of the simulation cell.
For an isotropic system, an average over different elements α, β = x, y, z can be performed. The
resulting time correlation function is shown in figure 4. The time dependence of 〈σαβ(t)σαβ(0)〉
suggests two different relation channels, which are modelled by fitting two stretched exponential
functions of the form A exp(−(t/τ)β). The relaxation times τ differ by more than one order of
magnitude. While the fast relaxation channel has a relaxation time of only 13 fs, the slower process
decays in about 0.55 ps.
175
S.Jahn, P.A.Madden
The spectrum of this function defines a frequency-dependent viscosity
η(ω) =
V
kBT
∫ ∞
0
dt eiωt 〈σαβ(t) · σαβ(0)〉, (14)
which is also shown in figure 4, together with the spectra of the two stretched exponentials to
indicate how the two relaxation channels contribute to the spectral shape. The spectrum associated
with the fast component is seen to resemble the vibrational density of states. It extends over a
similar frequency range to the density of states shown in figure 1 and exhibits similar features at
around 100 meV. The slow component dominates the low frequency spectrum and is associated
with the structural (configurational) relaxation of the fluid. The zero frequency viscosity, η(0) =
25 mPa·s (marked by a full circle in figure 1), is the macroscopic viscosity, which is again in
excellent agreement with experimental data [21,32]. We see that its value will be dominated by
this structural relaxation component. The temperature dependence of the structural relaxation
time determines the temperature dependence of the viscosity. In a glass, the structural relaxation
becomes indefinitely slow, and the spectrum η(ω) reduces to a vibrational spectrum very similar
to the vibrational contribution to the liquid spectrum.
The Brillouin frequencies probed in the 0.2−1.0 Å−1 Q-range will sample this spectrum between
about 10 and 50 meV, where the height of the spectrum is much lower than at ω=0. We can
therefore understand the order of magnitude difference between the “viscosities” extracted from the
Brillouin linewidths and the macroscopic value. In this frequency range, η(ω) has a weak frequency
dependence, which the spectral decomposition suggests should be associated with the tail of the
slow structural relaxation component with a flat background associated with the vibrational term.
We note that Sinn et al. [8] fitted their IXS data with a three Lorentzian model which allowed
for a frequency-dependent viscosity and extracted a structural relaxation time of τ ≈ 0.5 ps at
T = 2500 K, which agrees well with our calculated value. The fact that we still see a contribution
from the structural relaxation at 2500 K could be the reason why our calculated Brillouin widths
are larger than those reported experimentally, as noted above. The experimental values pertain to
T =2323 K and it is possible that at this lower temperature, the structural relaxation spectrum
has narrowed and no longer makes as large a contribution to the frequency-dependent viscosity in
the range of Brillouin frequencies as at 2500 K.
This analysis suggests that in Al2O3 at 2500 K the damping of the high-Q acoustic modes
observed in IXS is crossing over from the hydrodynamic, viscous damping associated with structural
relaxation of the fluid, to damping associated with the vibrational dynamics of the atoms about
some disordered configuration in which the atoms are trapped on the timescale of the acoustic
oscillations. As the temperature is raised, the damping due to the structural relaxation should
increase resulting in a broadening of the Brillouin lines. At lower temperatures, it would be more
appropriate to think of the fluid as responding like a disordered solid, and expect the Brillouin
linewidth to be only weakly temperature dependent. In this régime the Brillouin linewidth reflects
the relaxation by dephasing [35] of density fluctuations of wavevector Q. Schirmacher et al. [36] have
recently shown how this mechanism can also lead to a Q2-dependence of the Brillouin linewidth.
4. Conclusions
A realistic modelling of alumina melt is achieved by using an advanced ionic interaction po-
tential and molecular dynamics. Atomic structure and dynamics as well as transport coefficients
are quantitatively reproduced. The modelling approach provides a deeper insight into the relation
between atomic-scale and macroscopic properties and facilitates the interpretation of experimental
data. Inelastic features are visible in SOO(Q,ω) up to relatively large Q close to the principal
peak of S(Q). The visibility and broadening of Brillouin peaks seem to be closely related to the
frequency dependent viscosity, which is in agreement with recent theoretical developments.
176
Atomic dynamics of alumina melt
References
1. Lovesey S.W. Theory of neutron scattering from condensed matter. Oxford University Press, 1984.
2. Balucani U., Zoppi M. Dynamics of the liquid state. Oxford University Press Inc., New York, 1994.
3. Suck J.-B., Int. J. Mod. Phys. B, 1993, 7, 3003.
4. Price D.L., Copley J.R.D., Phys. Rev. A, 1975, 11, 2124.
5. McGreevy R.L., Mitchell E.W.J., Margaca F.M.A., Howe M.A., J. Phys. C: Solid State Phys., 1985,
18, 5235.
6. Burkel E., Rep. Prog. Phys., 2000, 63, 171.
7. Demmel F., Hosokawa S., Lorenzen M., Pilgrim W.-C., Phys. Rev. B, 2004, 69, 012203.
8. Sinn H. et al., Science, 2003, 299, 2047.
9. Wilson M., Madden P.A., J. Phys.: Condens. Matter, 1993, 5, 6833.
10. Wilson M., Madden P.A., Hemmati M., Angell C.A., Phys. Rev. Lett., 1996, 77, 4023.
11. Aguado A., Bernasconi L., Jahn S., Madden P.A., Faraday Discuss., 2003, 124, 171.
12. Madden P.A., Heaton R., Aguado A., Jahn S., J. Mol. Struc. (Theochem), 2006, 771, 9.
13. Ansell S. et al., Phys. Rev. Lett., 1997, 78, 464.
14. Landron C. et al., Phys. Rev. Lett., 2001, 86, 4839.
15. Krishnan S. et al., Chem. Mater., 2005, 17, 2662.
16. San Miguel M.A., Sanz J.F., Alvarez L.J., Odriozola J.A., Phys. Rev. B, 1998, 58, 2369.
17. Hemmati M., Wilson M., Madden P.A., J. Phys. Chem. B, 1999, 103, 4023.
18. Gutierrez G., Belonoshko A.B., Ahuja R., Johansson B., Phys. Rev. E, 2000, 61, 2723.
19. Hoang V.V., Phys. Rev. B, 2004, 70, 134204.
20. Hoang V.V., Oh S.K., Phys. Rev. B, 2005, 72, 054209.
21. Jahn S., Madden P.A., Wilson M., Phys. Rev. B, 2006, 74, 024112.
22. Jahn S., Madden P.A., J. Non-Cryst. Solids, 2007, 353, 3500.
23. Jahn S., Madden P.A., Wilson M., Phys. Rev. B, 2004, 69, 020106(R).
24. Tang K.T., Toennies J.P., J. Chem. Phys., 1984, 80, 3726.
25. Wilson M., Madden P.A., Costa-Cabral B.J., J. Phys. Chem., 1996, 100, 1227.
26. Stone A.J. The theory of intermolecular forces. Oxford University Press, Oxford, 1996.
27. Martyna G.J., Tobias D.J., Klein M.L., J. Chem. Phys., 1994, 101, 4177.
28. Nosé S., Klein M.L., Mol. Phys., 1983, 50, 1055.
29. Anisimov Y.S., Mitin B.S., Izv. Akad. Nauk SSSR. Neorg. Mater., 1977, 13, 1442.
30. van Hove L., Phys. Rev., 1954, 95, 249.
31. Boon J.P., Yip S., Molecular Hydrodynamics. McGraw-Hill, New York, 1980.
32. Urbain G., Rev. Int. Hautes Temp. Refract., 1982, 19, 55.
33. Mori H., Prog. Theor. Phys., 1965, 33, 423.
34. Hansen J.-P., McDonald I.R. Theory of simple liquids. Academic Press, London, 2006.
35. Ribeiro M.C.C., Wilson M., Madden P.A., J. Chem. Phys., 1998, 108, 9027.
36. Schirmacher W., Ruocco G., Scopigno T., Phys. Rev. Lett., 2007, 98, 025501.
177
S.Jahn, P.A.Madden
Атомна динамiка розплаву оксиду алюмiнiю: дослiдження
методом молекулярної динамiки
С.Ян1, П.А.Мадден2
1 Центр геодослiджень Потсдаму, Потсдам, Нiмеччина
2 Хiмiчний факультет, Унiверситет Едiнбурґу, Едiнбурґ, Великобританiя
Отримано 2 листопада 2007 р.
Атомна динамiка розплаву Al2O3 дослiджується моделюванням методом молекулярної динамiки.
Мiжчастинковi взаємодiї описуються вдосконаленою моделлю iонних взаємодiй, що включає ефе-
кти поляризацiї та деформацiї iонної форми. Показано, що модель акуратно вiдтворює статичнi
структурнi фактори S(Q) з дифракцiї нейтронiв та рентгенiвських променiв та динамiчний структур-
ний фактор S(Q, ω) з непружнього розсiяння рентгенiвських променiв. Аналiз парцiальних динамi-
чних структурних факторiв показує непружнi особливостi у спектрах аж до передачi iмпульсу, Q,
близької до головних пiкiв парцiальних статичних структурних факторiв. Збiльшення ширини брiлю-
енiвського максимуму обговорюється у зв’язку з залежною вiд частоти в’язкiстю η(ω).
Ключовi слова: молекулярна динамiка, колективна динамiка, рiдина, в’язкiсть, релаксацiйний час,
оксид алюмiнiю
PACS: 61.20.Ja, 61.20.Lc, 66.20.+d, 61.10.Eq, 62.60.+v
178
|
| id | nasplib_isofts_kiev_ua-123456789-119005 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-01T11:26:34Z |
| publishDate | 2008 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Jahn, S. Madden, P.A. 2017-06-03T03:47:54Z 2017-06-03T03:47:54Z 2008 Atomic dynamics of alumina melt: A molecular dynamics simulation study / S. Jahn, P.A. Madden // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 169-178. — Бібліогр.: 36 назв. — англ. 1607-324X PACS: 61.20.Ja, 61.20.Lc, 66.20.+d, 61.10.Eq, 62.60.+v DOI:10.5488/CMP.11.1.169 https://nasplib.isofts.kiev.ua/handle/123456789/119005 The atomic dynamics of Al₂O₃ melt are studied by molecular dynamics simulation. The particle interactions are described by an advanced ionic interaction model that includes polarization effects and ionic shape deformations. The model has been shown to reproduce accurately the static structure factors S(Q) from neutron and x-ray diffraction and the dynamic structure factor S(Q, ω) from inelastic x-ray scattering. Analysis of the partial dynamic structure factors shows inelastic features in the spectra up to momentum transfers, Q, close to the principal peaks of partial static structure factors. The broadening of the Brillouin line widths is discussed in terms of a frequency dependent viscosity η(ω). Атомна динамiка розплаву Al₂O₃ дослiджується моделюванням методом молекулярної динамiки. Мiжчастинковi взаємодiї описуються вдосконаленою моделлю iонних взаємодiй, що включає ефекти поляризацiї та деформацiї iонної форми. Показано, що модель акуратно вiдтворює статичнi структурнi фактори S(Q) з дифракцiї нейтронiв та рентгенiвських променiв та динамiчний структурний фактор S(Q, ω) з непружнього розсiяння рентгенiвських променiв. Аналiз парцiальних динамiчних структурних факторiв показує непружнi особливостi у спектрах аж до передачi iмпульсу, Q, близької до головних пiкiв парцiальних статичних структурних факторiв. Збiльшення ширини брiлюенiвського максимуму обговорюється у зв’язку з залежною вiд частоти в’язкiстю η(ω). en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Atomic dynamics of alumina melt: A molecular dynamics simulation study Атомна динамiка розплаву оксиду алюмiнiю: дослiдження методом молекулярної динамiки Article published earlier |
| spellingShingle | Atomic dynamics of alumina melt: A molecular dynamics simulation study Jahn, S. Madden, P.A. |
| title | Atomic dynamics of alumina melt: A molecular dynamics simulation study |
| title_alt | Атомна динамiка розплаву оксиду алюмiнiю: дослiдження методом молекулярної динамiки |
| title_full | Atomic dynamics of alumina melt: A molecular dynamics simulation study |
| title_fullStr | Atomic dynamics of alumina melt: A molecular dynamics simulation study |
| title_full_unstemmed | Atomic dynamics of alumina melt: A molecular dynamics simulation study |
| title_short | Atomic dynamics of alumina melt: A molecular dynamics simulation study |
| title_sort | atomic dynamics of alumina melt: a molecular dynamics simulation study |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/119005 |
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