Collective excitations and generalized transport coefficients in a molten metallic alloy Li₄Pb
Collective dynamics of a molten metallic alloy Li₄Pb is studied using a combination of analytical multivariable approach of generalized collective modes and molecular dynamics simulations. Dispersion and damping of two branches of propagating collective excitations are analyzed in a wide range of wa...
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Інститут фізики конденсованих систем НАН України
2004
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| Цитувати: | Collective excitations and generalized transport coefficients in a molten metallic alloy Li₄Pb / T. Bryk, I. Mryglod // Condensed Matter Physics. — 2004. — Т. 7, № 2(38). — С. 285–300. — Бібліогр.: 20 назв. — англ. |
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irk-123456789-1189572017-06-02T03:03:09Z Collective excitations and generalized transport coefficients in a molten metallic alloy Li₄Pb Bryk, T. Mryglod, I. Collective dynamics of a molten metallic alloy Li₄Pb is studied using a combination of analytical multivariable approach of generalized collective modes and molecular dynamics simulations. Dispersion and damping of two branches of propagating collective excitations are analyzed in a wide range of wavenumbers. The features in collective dynamics connected with the large difference in species mass are discussed. Generalized k-dependent transport coefficients for Li₄Pb are reported. Колективна динамiка в розплавленому металiчному сплавi Li₄Pb дослiджується комбiнацiєю аналiтичного багатозмiнного пiдходу узагальнених колективних мод та комп’ютерних симуляцiй методом молекулярної динамiки. Дисперсiя та загасання двох вiток пропагаторних колективних збуджень аналiзується в широкiй областi хвильових чисел. Обговорюються особливостi колективної динамiки, пов’язанi з великою рiзницею у масах компонент сплаву. Подано узагальненi k-залежнi коефiцiєнти переносу для Li₄Pb 2004 Article Collective excitations and generalized transport coefficients in a molten metallic alloy Li₄Pb / T. Bryk, I. Mryglod // Condensed Matter Physics. — 2004. — Т. 7, № 2(38). — С. 285–300. — Бібліогр.: 20 назв. — англ. 1607-324X PACS: 05.20.Jj, 61.20.Ja, 61.20.Lc DOI:10.5488/CMP.7.2.285 http://dspace.nbuv.gov.ua/handle/123456789/118957 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Collective dynamics of a molten metallic alloy Li₄Pb is studied using a combination of analytical multivariable approach of generalized collective modes and molecular dynamics simulations. Dispersion and damping of two branches of propagating collective excitations are analyzed in a wide range of wavenumbers. The features in collective dynamics connected with the large difference in species mass are discussed. Generalized k-dependent transport coefficients for Li₄Pb are reported. |
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Bryk, T. Mryglod, I. |
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Bryk, T. Mryglod, I. Collective excitations and generalized transport coefficients in a molten metallic alloy Li₄Pb Condensed Matter Physics |
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Bryk, T. Mryglod, I. |
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Bryk, T. |
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Collective excitations and generalized transport coefficients in a molten metallic alloy Li₄Pb |
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Collective excitations and generalized transport coefficients in a molten metallic alloy Li₄Pb |
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Collective excitations and generalized transport coefficients in a molten metallic alloy Li₄Pb |
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Collective excitations and generalized transport coefficients in a molten metallic alloy Li₄Pb |
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Collective excitations and generalized transport coefficients in a molten metallic alloy Li₄Pb |
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collective excitations and generalized transport coefficients in a molten metallic alloy li₄pb |
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Інститут фізики конденсованих систем НАН України |
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2004 |
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http://dspace.nbuv.gov.ua/handle/123456789/118957 |
| citation_txt |
Collective excitations and generalized transport coefficients in a molten metallic alloy Li₄Pb / T. Bryk, I. Mryglod // Condensed Matter Physics. — 2004. — Т. 7, № 2(38). — С. 285–300. — Бібліогр.: 20 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT brykt collectiveexcitationsandgeneralizedtransportcoefficientsinamoltenmetallicalloyli4pb AT mryglodi collectiveexcitationsandgeneralizedtransportcoefficientsinamoltenmetallicalloyli4pb |
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2025-07-08T14:58:11Z |
| last_indexed |
2025-07-08T14:58:11Z |
| _version_ |
1837091198319198208 |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 2(38), pp. 285–300
Collective excitations and generalized
transport coefficients in a molten
metallic alloy Li4Pb
T.Bryk, I.Mryglod
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitsky Str., 79011 Lviv, Ukraine
Received May 5, 2004
Collective dynamics of a molten metallic alloy Li4Pb is studied using a
combination of analytical multivariable approach of generalized collective
modes and molecular dynamics simulations. Dispersion and damping of
two branches of propagating collective excitations are analyzed in a wide
range of wavenumbers. The features in collective dynamics connected with
the large difference in species mass are discussed. Generalized k-dependent
transport coefficients for Li4Pb are reported.
Key words: collective dynamics, binary liquid, collective excitation,
generalized transport coefficients
PACS: 05.20.Jj, 61.20.Ja, 61.20.Lc
1. Introduction
A molten metallic alloy Li4Pb has become well-known following the Letter by
Bosse et al [1] in which the existence of “fast sound” collective excitations in dis-
parate mass two-component liquids was suggested. The new collective excitation
in Li4Pb was interpreted as “a propagating lithium density fluctuations in a back-
ground of heavy lead ions, which do not participate, to an essential extent, in the
high-frequency motion” [1]. Moreover, the frequency of “fast sound” excitations was
found from the molecular dynamics (MD) simulations on a model system of 250 par-
ticles to follow the nearly linear law of dispersion in the region of small wavenumbers.
This raised a question whether the fast-sound mode will disappear as the wavenum-
ber decreases or alternatively, whether the fast-sound mode will continuously change
its slope in the transition regime to merge into the ordinary Brillouin peak without
ever disappearing [1].
Collective dynamics of binary liquids is so far well-known only in hydrodynamic
limit when the liquid is treated as continuum. Four local conservation laws permit
c© T.Bryk, I.Mryglod 285
T.Bryk, I.Mryglod
to obtain analytical expressions for hydrodynamic time correlation functions and
dynamical structure factors Sij(k, ω) in the limit k → 0, ω → 0 (k and ω are the
wavenumber and frequency, respectively) [2–4]. It is obvious that due to a very large
mass difference for components in Li4Pb the lithium (light) and lead (heavy) sub-
systems would clearly display different tendencies behind the hydrodynamic region.
Computational studies of molten Li4Pb using molecular dynamics simulations [5,6]
did not shed the light on the problem of how the branches of collective excitations
behave by approaching the hydrodynamic region because the numerical method of
estimating the dispersion was not based in those studies on particular models of
generalized hydrodynamics but simply traced out the maxima positions in relevant
spectral functions. Therefore, the theoretical studies based on generalized hydro-
dynamics are extremely important in collective dynamics of disparate-mass binary
liquids.
A theoretical analysis of collective dynamics in binary fluids, based on the gen-
eralized hydrodynamics, was for the first time applied in the study of another “fast
sound” mixture He-Ne [7]. A five-variable kinetic approach was based on a simultane-
ous treatment of hydrodynamic and short-time (extended) dynamical variables and
enabled one to obtain dynamical eigenmodes in the binary fluids as the eigenvalues
of 5×5 generalized hydrodynamic matrix, as well as to estimate their contributions
to all spectral functions of interest. However, the necessity to know a priori the
transport coefficients of the studied mixture, or alternatively the presence of free
parameters in the generalized hydrodynamic matrix, was a considerable drawback
of the method proposed in [7]. Later, in [8] the parameter-free generalized collec-
tive modes (GCM) approach based on the original concept developed for simple
fluids by de Schepper et al [9], in its parameter-free version [10], was employed in
seven-variable approximation to study the binary mixture He0.65Ne0.35. In contrast
to the study [7] all the elements of the generalized hydrodynamic matrix in [8] were
calculated directly in MD. Following the collective excitation study in He0.65Ne0.35
an attempt was undertaken to calculate generalized (k, ω)-dependent transport co-
efficients in this gaseous binary mixture [11] based on generalized hydrodynamics of
multicomponent liquids formulated in [12] within the GCM approach.
The GCM approach has proved to be applicable in the study of non-hydrodynamic
collective processes in pure [13] and binary liquids [14,15], such as heat waves, trans-
verse optic-like processes and kinetic modes connected with local structural relax-
ation. In [14] the GCM study of transverse dynamics in molten Li4Pb at 1085 K
permitted to identify the high-frequency branch in low-k region as the transverse
optic-like excitations, caused by mass-concentration fluctuations. It was shown that
in long-wavelength limit, the high-frequency branch in Li4Pb tends to a finite fre-
quency and can be described solely by treatment of transverse mass-concentration
current fluctuations.
Thus, our aim is to apply the generalized hydrodynamic GCM approach to the
study of collective dynamics in Li4Pb. So far this approach turns out to be the most
consistent method for analytical and numerical studies of collective kinetic modes,
because it enables us to treat the hydrodynamic and more short-time processes in
286
Collective excitations and generalized transport coefficients in Li4Pb
liquids on the same footing. We will obtain the spectrum of longitudinal collective
excitations and find out the origin of each branch. For the first time we report herein
the calculations of generalized transport coefficients in a “fast sound” liquid alloy.
2. The method
The method used in this work for a miscroscopic study of liquid dynamics is a
combination of molecular dynamics simulations and multivariable kinetic approach
for parameter-free analysis of MD-derived time correlation functions.
2.1. Molecular dynamics simulations
Computer simulations were performed in the standard microcanonical ensemble
on a model system of 4000 particles in a cubic box subject to periodic boundary
conditions at the temperature of 1085 K and mass-density of 3556.76 kg/m3. The
pair potentials Φij(r), taken from [16], were the same as the ones used in previous
studies of dynamics in Li4Pb [1,5,6]. The functional form of Φij(r) for molten Li4Pb
was suggested in [17] and consisted of a repulsive soft core term and a screened
Coulomb potential in order to reflect the short-range charge neutrality. The fourth-
order Gear algorithm with the time step δt = 1 · 10−15 s was used to integrate the
equations of motion. Energy conservation in MD turned out to be very good. Over
the length of production run of 3 · 105 steps, the total energy drift was on the level
of 0.1%. The ratio of square roots of variances corresponding to fluctuations of total
and potential energies was on the level of 0.13. The smallest wavenumber reached
in MD simulations was kmin = 0.1414 Å−1.
The main purpose of MD simulations was to obtain the time evolution of hydro-
dynamic and short-time extended dynamic variables, to estimate time correlation
functions and relevant static averages necessary for GCM analysis, and to compare
MD-derived time correlation functions with their GCM-replicas. To reduce the di-
mension of relevant quantities, the following energy, mass, spatial and time scales
were used in our simulations: ε = kBT , µ = m̄ = 7.803 · 10−26 kg, σ = 4.455 Å,
τ = σ(µ/ε)1/2 = 1.017 ps.
2.2. Generalized collective modes approach
In this GCM study of generalized transport coefficients in molten Li4Pb we have
chosen the following eight-variable basis set of dynamic variables:
A(8)(k, t) = {n1(k, t), n2(k, t), JL
1 (k, t), JL
2 (k, t), ε(k, t), J̇L
1 (k, t), J̇L
2 (k, t), ε̇(k, t)} ,
(1)
where the dynamic variables of partial number densities are defined as follows:
nα(k, t) =
1√
N
Nα∑
i=1
eikrα,i(t) , α = 1, 2 .
287
T.Bryk, I.Mryglod
Both partial densities ni(k, t) are hydrodynamic variables and, along with dynamic
variables of total mass-current density Jt(k, t) and energy density ε(k, t), they are
used in the hydrodynamic description of any binary system in liquid state:
Ahyd(k, t) = {n1(k, t), n2(k, t), JL
t (k, t), ε(k, t)} .
These four dynamical variables from the hydrodynamic basis set describe the slowest
miscroscopic processes in binary liquids. All the other dynamical variables from the
basis set A(8)(k, t), which are the time derivatives and linear combinations of hydro-
dynamic variables correspond to faster kinetic-like processes. The time evolution of
dynamical variables from the basis set A(8)(k, t) was obtained in MD simulations.
All the static and time correlation functions necessary for the estimation of matrix
elements of the 8×8 matrices of time correlation functions F(k, t) and their Laplace
transforms F̃(k, z) were directly evaluated in computer simulations. Eigenvalues and
eigenvectors of generalized hydrodynamic matrix [10]
T(k) = F(k, t = 0)F̃−1(k, z = 0),
were calculated for each k-point sampled in molecular dynamics. Thus, in our ap-
proach there was no fitting of free parameters. The set of eigenvalues zα(k) of gen-
eralized hydrodynamic matrix T(k) formed the spectrum of collective excitations.
Any MD-derived time correlation function of interest within the GCM approach has
its GCM replica represented as the sum over mode contributions:
F
(GCM)
ij (k, t) =
8∑
α=1
Gα
ij(k)e−zα(k)t , (2)
where in general complex amplitudes Gα
ij(k) were estimated from the eigenvectors
associated with the relevant eigenvalue zα(k) [10,8].
3. Results and discussion
3.1. Static quantities
Three partial static structure factors Sij(k), i, j = Li, Pb, directly estimated in
MD simulations are shown in figure 1. The difference in the size of components is
clearly seen from the positions of the main peaks of static structure factors. Light
subsystem of small Li atoms has its maximum of SLiLi(k) at approximately 2.4 Å−1,
while the heavy large atoms of Pb have a corresponding maximum of SPbPb(k) at
approximately 1.55 Å−1. The partial static structure factors can be used in calcu-
lating the Bhatia-Thornton “number-concentration” structure factors Sij(k) with
i, j = n, c [2], or Sij(k) with i, j = t, x, where the t − x pair of dynamical variables
corresponds to total density nt(k, t) and mass-concentration density nx(k, t) [18,19].
We observed the tendency towards nonzero constants in Scc(k) when k → 0, that
is the consequence of the absence of long-range Coulomb two-body potentials in
288
Collective excitations and generalized transport coefficients in Li4Pb
-1
-0.5
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3 3.5 4
P
ar
tia
l s
tr
uc
tu
re
fa
ct
or
s
S
ij(
k)
k / A°-1
Pb-Pb
Li-Pb
Li-Li
Figure 1. Partial static structure factors of Li4Pb obtained directly in MD: Pb-
Pb (open boxes), Li-Pb (cross symbols) and Li-Li (filled boxes). Lines correspond
to cubic spline interpolation.
0
0.4
0.8
1.2
1.6
θ(
k)
0
0.4
0.8
1.2
0 0.5 1 1.5 2 2.5 3 3.5
α T
(k
)T
2.7
2.8
2.9
3.0
3.1
3.2
C
V
(k
)
/ k
B
1.0
1.1
1.2
1.3
0 0.5 1 1.5 2 2.5 3 3.5 4
γ(
k)
k / A°-1
Figure 2. Generalized thermodynamic quantities for molten Li4Pb at T=1085 K:
generalized reduced compressibility θ(k) (upper left frame), generalized thermal
expansion coefficient αT(k) (lower left frame), generalized specific heat at con-
stant volume CV(k) (upper right frame), generalized ratio of specific heats γ(k)
(lower right frame). The filled box at k = 0 in the upper right frame denotes the
specific heat CV obtained from the temperature fluctuations formula during MD
run. Solid lines are the spline interpolation.
289
T.Bryk, I.Mryglod
molten Li4Pb due to the screening by electronic density. Otherwise, the structure
factor Scc(k) would be ∝ k2 in long-wavelength limit as in the case of molten salts.
The isothermal compressibility κT in the case of binary liquids is usually ex-
pressed via a combination of partial structure factors at k = 0 (see [2]). Here we
follow the expressions derived in [8] for reduced generalized isothermal compressibil-
ity θ(k)
θ(k) =
N
V
kBTκT (k) ,
which is shown in figure 2 in the upper left frame. It is seen, that θ(k) behaves
almost the same as SLiLi(k). In figure 2 we also show the k-dependence of general-
ized thermodynamic quantities (see [8] for original expressions): generalized thermal
expansion coefficient αT(k), generalized specific heat at constant volume CV(k) and
generalized ratio of specific heats γ(k). Smooth behaviour of αT(k) at k → 0 made
it possible to estimate the thermal expansion coefficient for Li4Pb at 1085K to be
1.03 · 10−4K−1. The generalized specific heat at a constant volume CV(k), which
is defined via the static value of “heat density-heat density” time correlation func-
tion Fhh(k, t), tends in the k → 0 limit towards the value of 2.87, obtained from
the temperature fluctuations during MD production run. The ratio of specific heats
γ(k) is a very important quantity, which permits to judge about the static coupling
between heat and density fluctuations. It is important to note that γ(k) tends in
the long-wavelength limit towards the value of ≈ 1.15, which is a quite reasonable
value for metallic melts.
3.2. Time correlation functions
Time correlation functions of primary interest are the ones defined on the hydro-
dynamic variables. We will focus on the analysis of the following six time correlation
functions: Fij(k, t), Fεi(k, t), Fεε(k, t), i, j =Li,Pb, which are shown for the smallest
wavenumber in figure 3. The most interesting are the functions describing autocor-
relations of partial densities Fii(k, t), and energy density Fεε(k, t). It is seen from
the behaviour of FLiLi(k, t) and FPbPb(k, t), that even for the smallest wavenum-
ber sampled in MD simulations these functions reflect processes of different time
scales: light Li atoms are responsible for damped oscillating behaviour of FLiLi(k, t),
which would manifest itself as a side Brillouin peak on the partial structure fac-
tor SLiLi(k, ω). Collective correlations of heavy Pb atoms are of a relaxing form,
which, however, cannot be an evidence that Pb atoms do not participate in the
oscillating motion. One can expect that for much smaller wavenumbers so far not
accessible in MD simulations, both functions FLiLi(k, t) and FPbPb(k, t) will display
long-wavelength oscillations with approximately the same frequency reflecting the
hydrodynamic sound propagation in the liquid. The energy-energy autocorrelation
functions Fεε(k, t) are very similar in shape to the FLiLi(k, t) function, which means
that the main contribution to the energy time correlations at small wavenumbers is
determined by the light Li subsystem.
In all frames of figure 3 the six GCM replicas F
(GCM)
ij (k, t) (2) are shown by solid
lines. One can see that MD-derived functions (bold symbols “plus”) and their GCM
290
Collective excitations and generalized transport coefficients in Li4Pb
replicas are almost identical, which indicates a very good quality of the applied
eight-variable approximation and the correct correspondence of the main collective
dynamical processes to the dynamical eigenmodes obtained for these wavenumbers,
which are beyond the hydrodynamic region considered.
0
0.01
0.02
0.03
0.04
0.05
F
P
bP
b(
k,
t)
MD
GCM
-0.2
-0.1
0
F
P
bE
(k
,t)
/
ε
-0.6
-0.4
-0.2
0
0.1
F
P
bL
i(k
,t)
*1
02
-0.1
0
0.1
0.2
0.3
F
Li
E
(k
,t)
/
ε
0
0.01
0.02
0.03
0 0.5 1 1.5 2 2.5 3
F
Li
Li
(k
,t)
Time / τσ
-1
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5 3
F
E
E
(k
,t)
/
ε2
Time / τσ
Figure 3. MD-derived time correlation functions at k = 0.1414 Å−1 (plus sym-
bols) and their GCM replicas (lines).
3.3. Spectrum of collective excitations in molten Li 4 Pb
The set of eight complex and purely real eigenvalues at different k points sampled
in MD represents the spectrum of collective excitations. For the sake of simplicity
we will mark the purely real eigenvalues which correspond to relaxing processes as
dj(k), while the complex eigenvalues
zj(k) = σj(k) ± iωj(k)
are represented by pairs of complex conjugated numbers corresponding to the pro-
cesses propagating in opposide directions with frequency ωj(k) and damping σj(k).
For the case of basis set A(8)(k, t) the spectrum of collective excitations in molten
Li4Pb consisted of two pairs of propagating modes and four relaxing processes in
the whole k-region. Note that for the case of Li4Pb we did not obtain solutions of
the heat waves type, which we observed earlier in GCM studies of pure liquids [13],
Lennard-Jones binary mixtures [19] and molten salts.
The imaginary and real parts of complex eigenvalues for Li4Pb are shown in
figure 4. An interesting behaviour of dispersion of two branches is observed on the
291
T.Bryk, I.Mryglod
top frame: the branches correspond to the processes well separated in frequency.
The high-frequency branch shown by open boxes has a pronounced minimum at the
position of the main maximum of partial static structure factor SLiLi(k) and tends
to a nonzero frequency in the long-wavelength region, which is a specific feature
of optic-like excitations in binary liquids [19]. The low-frequency branch is even
more interesting. It has a similar pronounced minimum at k ≈ 1.45 Å−1, which
is at the main maximum location of SPbPb(k). However, this branch has another
minimum in dispersion law at k ≈ 0.25 Å−1. Such a minimum is never observed
in binary liquid mixtures with comparable masses of species and concentrations.
However, it is obvious that for the disparate-mass liquid mixtures there should be
a qualitative difference in spectra of collective excitations between two cases of
comparable concentrations and an impurity limit of the heavy component. In the
latter case, one would expect the absence of low-frequency branch in the spectrum of
collective excitations in long-wavelength region, while the high-frequency branch of
light subsystem would continuously transform into hydrodynamic sound in the limit
k → 0. A more detailed GCM study of mass ratio and concentration dependence of
spectra of collective excitations in binary liquids will be reported elsewhere. For the
case of Li4Pb, we see that for k < 0.25 Å−1 the spectrum of propagating excitations
assumes the features of regular binary liquids, where in a long-wavelength region two
branches of propagating excitations correspond to hydrodynamic sound and optic-
like excitations caused by fast mass-concentration fluctuations. This means that in
Li4Pb, the heavy subsystem plays an important role in the collective dynamics and
hence the small-k region, where the dispersion of low-frequency branch displays a
minimum, can be treated as a crossover region between partial character of collective
dynamics (k > 0.25 Å−1) and intrinsic collective behaviour in terms of hydrodynamic
sound and optic-like excitations in the region of smaller k.
The real parts of propagating eivenvalues which correspond to the excitation
damping (or inversed lifetime) are shown in the lower frame in figure 4. A well
pronounced maximum in σ(k) for each branch is observed at the main maximum
position of the corresponding partial static structure factor Sii(k), i =Li,Pb, that
again supports the partial character of branches in the region of intermediate and
large wavenumbers. For k < 0.75 Å−1, the wavenumber dependence of damp-
ing coefficients has an opposite tendency for high- and low-frequency branches.
The low-frequency branch shows a slow change towards damping of hydrodynamic
sound, which should be proportional to k2 in the long-wavelength limit. Instead,
the high-frequency branch becomes more overdamped when k → 0, which is a spe-
cific feature of kinetic propagating modes [8,15,18]. The large difference in damp-
ing coefficients for high- and low-frequency branches explains why in the numer-
ical approach to the estimation of dispersion of collective excitations via maxima
positions of partial spectral functions, a merger of two branches was observed in
small-k region for Li4Pb [5,6]. It is obvious that the contribution from strong-
ly overdamped high-frequency branch cannot be well observed in spectral func-
tions on the background of a pronounced maximum coming from the low-frequency
branch.
292
Collective excitations and generalized transport coefficients in Li4Pb
0
20
40
60
80
Im
z
j(k
)
/ p
s-1
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5 3 3.5 4
R
e
z j
(k
)
/ p
s-1
k / A°-1
Figure 4. Imaginary and real parts of complex eigenvalues zj(k) corresponding
to frequency and damping of propagating collective excitations, respectively.
The behaviour of relaxation processes existing in the binary liquid on different
spatial and time scales is reflected by the wavenumber dependence of purely real
eigenvalues dj(k) shown in figure 5. In the whole k-region studied we observed two
pairs of complex conjugated eigenvalues and four purely real (relaxing) modes. The
highest relaxing eigenmode corresponds to extremely short-time processes and does
not considerably affect the collective dynamics of the studied system. Therefore, in
figure 5 we show the k-dependence of three main relaxing processes, which reflect
the heat relaxation (filled circles) and relaxing processes in light (filled triangles)
and heavy (open triangles) subsystems for k > 0.25 Å−1. The way we identified k-
dependence of each relaxing process among the four real eigenvalues was the same as
we used in our previous study of liquid Lennard-Jones mixture Kr-Ar [19] applying
separated subsets of dynamical variables. In figure 5 by dotted and dash-dotted
lines there is shown the behaviour of the lowest real eigenvalues, obtained from the
2× 2 and 3× 3 generalized hydrodynamic matrices generated on the separated sets
A(2h) = {h(k, t), ḣ(k, t)} and A(3Pb) = {nPb(k, t), JL
Pb(k, t), J̇L
Pb(k, t)} respectively. It
is clearly seen that for k > 0.25 Å−1 the relaxing process shown by open triangles is
completely defined by the heavy subsystem of Pb atoms. For smaller wavenumbers,
this relaxing mode continuously changes into a kinetic relaxing process of structural
relaxation with the inversed relaxation time, which in the case of pure liquids has
the following k-dependence [19]:
dstr(k) =
c2
∞ − c2
s
DL
− DLk2 − (γ − 1)Ak2 ,
293
T.Bryk, I.Mryglod
where DL is longitudinal viscosity, c∞ and cs are the high-frequency and hydro-
dynamic speed of sound, respectively, and A is some constant dependent on the
coupling between thermal and viscous processes. In figure 5 we show how the k2
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2 2.5 3 3.5 4
d j
(k
)
/ p
s-1
k / A°-1
A(2h)
A(3Pb)
a-bk2
Figure 5. Purely real eigenvalues dj(k) in molten Li4Pb corresponding to relax-
ation processes on different spatial scales. By dotted line is shown the lowest
real eigenvalue obtained on the two-variable set A(2h) for the treatment of heat
fluctuations. The dash-dotted line represents the only real eigenvalue from the
treatment of solely partial dynamics of the heavy subsystem by the set A(3Pb).
dependence of dstr(k) matches the relaxing behaviour of a heavy subsystem. The
existence of different relaxation processes in light and heavy subsystems is reflected
by well pronounced minima in k-dependence of corresponding dj(k) at the positions
of main maxima in partial static structure factors.
In a long-wavelengh region there exist two relaxing hydrodynamic processes: the
mutual diffusion and thermal diffusion processes, which must have k2 dependence.
One can see how the relaxing process in the light subsystem of Li atoms (filled
triangles) transforms into the mutual diffusion mode when k → 0. Hence, we may
conclude that in the disparate-mass molten alloy the mutual diffusion is connected,
on the departure from hydrodynamic region, with the light subsystem, while the
kinetic process of structural relaxation on the boundary of hydrodynamic region is
defined mainly by the subsystem of large and heavy Pb atoms.
3.4. Generalized k-dependent transport coefficients in molten Li4Pb
The approach of GCM makes it possible to generalize transport coefficients of a
liquid onto different spatial and time scales which are reflected in (k, ω)-dependence
of the coefficients. For the case of a simple Lennard-Jones fluid, the generalized
(k, ω)-dependent transport coefficients obtained within the GCM approach were
reported in [20]. The case of binary and multicomponent fluids is much more com-
plicated for the study of generalized transport coefficients. The simplest generaliza-
tion of mutual diffusion coefficients D(k, ω) is mainly known in the literature, while
even the standard Green-Kubo calculations of thermal conductivity λ or longitudi-
nal viscosity ηL are extremely scarce for binary liquids. This is mainly connected
294
Collective excitations and generalized transport coefficients in Li4Pb
0
1
2
3
4
5
Lo
ng
itu
di
na
l v
is
co
si
ty
(
r.
u.
) DL
0
1
2
3
4
5
6
7
0 0.5 1 1.5 2 2.5 3 3.5 4
T
he
rm
al
c
on
du
ct
iv
ity
(
r.
u.
)
k / A°-1
λT
Figure 6. Generalized longitudinal viscosity DL(k) and thermal conductivity
λ(k)T in molten Li4Pb.
with the problems of unambiguous definition of generalized transport coefficients
in multicomponent systems. The theory of generalized (k, ω)-dependent transport
coefficients in multicomponent liquids based on GCM approach was developed in
[12]. An advantage of N -variable approach in solving the system of N generalized
transport equations was shown in [12] in the form of recurrence relations for the ma-
trix of memory functions of different order. For the case of binary liquids, the final
4 × 4 hydrodynamic matrix of the lowest memory functions is directly connected
with the generalized transport coefficients L̃ij(k, ω):
φ̃(k, ω) = k2V kBT L̃(k, ω)F(k, t = 0) , (3)
where the hydrodynamic basis set of four dynamical variables used in defining the
transport coefficients consists of two partial densities of particles ni(k, t), i = 1, 2,
total longitudinal momentum density JL
t (k, t) and heat density h(k, t). As it was
shown in [12], the matrix of hydrodynamic memory functions defines the following
transport coefficients in the limit ω → 0, k → 0: longitudinal viscosity DL = ηL/ρ,
thermal conductivity λ, mutual diffusion coefficients Dij , i, j = 1, 2 and thermal
diffusion coefficients Ki, i = 1, 2. It is important to note that the diffusion and
thermal diffusion coefficients derived in such a way within the GCM approach satisfy
the following conditions that follow from the momentum conservation law [12]:
2∑
i=1
miDij(k, ω) =
2∑
j=1
Dij(k, ω)mj ≡ 0
295
T.Bryk, I.Mryglod
0
1
2
3
4
5
6
D
iff
us
iv
ity
(
r.
u.
)
D
-0.02
0
0.02
0.04
0.06
0.1
0 0.5 1 1.5 2 2.5 3 3.5 4
T
he
rm
al
d
iff
us
iv
ity
(
r.
u.
)
k / A°-1
K
Figure 7. Generalized irreducible coefficients of mutual diffusivity D(k) and ther-
mal diffusivity K(k) in molten Li4Pb.
and
2∑
i=1
miKi(k, ω) ≡ 0 .
In figures 6-8 we report generalized k-dependent transport coefficients for the
disparate-mass metallic molten alloy Li4Pb. The frequency dependence of general-
ized transport coefficients in Li4Pb will be reported elsewhere. In figure 6 the k-
dependent generalized longitudinal viscosity DL(k) and thermal conductivity λ(k)
display similar behaviour. In the large-k region they behave almost like ∝ k−1, while
in the long-wavelength limit, the behaviour resembling DL(k) = DL−Ak2 is expect-
ed. The behaviour of wavenumber-dependent diffusion and thermal diffusion is more
interesting. Note that in figure 7 we show the irreducible coefficients D(k) and K(k),
which are connected to Dij(k) and Ki(k) as follows:
Dij(k) = (−1)(i+j) D(k)
mimj
, Ki(k) = (−1)(i+1) K(k)
mi
.
The striking difference of both irreducible transport coefficients from the k-dependence
of longitudinal viscosity DL(k) and from thermal conductivity λ(k) is the nonmono-
tonic behaviour in the region of intermediate wavenumbers 0.7 Å−1 < k < 3 Å−1,
where we observe a maximum at k ≈ 2.3 Å−1 for generalized diffusivity D(k) and
a minimum for generalized thermal diffusivity K(k) at k ≈ 1.6 Å−1. These both
features imply the effects of the light Li subsystem on the mutual diffusion and of
the heavy Pb subsystem onto the thermal diffusivity on different spatial scales.
296
Collective excitations and generalized transport coefficients in Li4Pb
0
0.1
0.2
0.3
0.4
0.5
0.6
D
-V
c
ro
ss
c
oe
ffi
ci
en
t (
r.
u.
) ξ
0
0.02
0.04
0.06
0.08
0.1
0 0.5 1 1.5 2 2.5 3 3.5 4
V
-T
c
ro
ss
c
oe
ffi
ci
en
t (
r.
u.
)
k / A°-1
ζ
Figure 8. Purely imaginary generalized cross-correlation transport coefficients:
cross diffusivity-viscosity coefficient ξ(k) and cross viscosity-heat coefficient ζ(k)
in molten Li4Pb.
The approach of GCM permits to introduce additional transport coefficients for
the case of binary liquids, which for the static case ω = 0 are purely imaginary num-
bers and describe the dynamical cross-correlations in diffusivity-viscosity ξ(k) and in
viscosity-heat ζ(k) transport properties. Imaginary parts of these cross-coefficients
are shown in figure 8. Note that in the k → 0 limit both cross-coefficients should
vanish. The behaviour of their k-dependence in a small-wavenumber region shows
the right tendency. Both transport cross-coefficients display a well pronounced max-
imum in their k-dependence at k ≈ 0.6− 0.7 Å−1 which indicates very strong cross-
correlations between microscopic processes describing diffusivity, viscosity and heat
transport. Another characteristic is observed on the k-dependence of ξ(k) cross-
coefficients which may be connected with the characteristic in mutual diffusivity
D(k) due to the light Li subsystem. We would like to stress that the coefficient ξ(k)
does not have an analogy in the case of pure liquids, where only cross-correlation
between viscosity and heat transport can be represented via an additional transport
coefficient ζ(k) [20].
4. Conclusions
In this study we focused on the calculations of the spectrum of collective ex-
citations and generalized k-dependent transport coefficients in the disparate-mass
metallic molten alloy Li4Pb using a combination of molecular dynamics simula-
297
T.Bryk, I.Mryglod
tions and analytical eight-variable generalized collective modes approach. The main
conclusions of this study are as follows. Our analytical analysis of different time
correlation functions and the obtained dispersion curves for the propagating col-
lective excitations permit us to conclude that in the region k ≈ 0.25 Å−1 there
exists a crossover from the “partial” picture of collective dynamics in terms of light
and heavy subsystems to the “intrinsic collective” behaviour of the liquid alloy for
k < 0.25 Å−1. In this long-wavelength limit, two branches of propagating excita-
tions assume the specific features of hydrodynamic sound (low-frequency branch)
and optic-like (high-frequency) branch. The damping of high-frequency optic-like
branch in a long-wavelength region is much higher than that of hydrodynamic sound
excitations, which does not permit one to see the high-frequency excitations as a
well-defined maximum on relevant spectral functions on the background of the con-
tribution coming from the hydrodynamic sound. This is perhaps the main reason
why the numerical approaches of spectrum estimation reported in [5,6] resulted in
a merger of two high- and low-frequency branches in the long-wavelength region. In
the case of disparate-mass metallic alloy, the relaxation processes distinctly show
a “partial” origin in the region of intermediate wavenumbers, where the structural
relaxation can be represented as a combination of relaxations in the light and heavy
subsystems in contrast to our recent results for a Lennard-Jones liquid mixture [19],
where “intrinsic collective” character of structural relaxation was observed. For the
first time we have calculated the generalized k-dependent transport coefficients in
molten Li4Pb. The results show an interesting effect of light and heavy subsystems
onto the generalized mutual diffusivity and thermal diffusivity. We also showed for
the first time how the cross-correlation transport coefficients behave on different
spatial scales in the disparate-mass metallic molten alloy.
Acknowledgements
I.M. is thankful to the Fonds zur Förderung der wissenschaftlichen Forschung
(Austria) for financial support under Project No. P15247.
298
Collective excitations and generalized transport coefficients in Li4Pb
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299
T.Bryk, I.Mryglod
Колективнi збудження та узагальненi коефiцiєнти
переносу в розплавленому металiчному сплавi
Li4Pb
Т.Брик, I.Мриглод
Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 5 травня 2004 р.
Колективна динамiка в розплавленому металiчному сплавi Li4Pb до-
слiджується комбiнацiєю аналiтичного багатозмiнного пiдходу уза-
гальнених колективних мод та комп’ютерних симуляцiй методоммо-
лекулярної динамiки. Дисперсiя та загасання двох вiток пропагатор-
них колективних збуджень аналiзується в широкiй областi хвильових
чисел. Обговорюються особливостi колективної динамiки, пов’язанi
з великою рiзницею у масах компонент сплаву. Подано узагальненi
k-залежнi коефiцiєнти переносу для Li4Pb.
Ключові слова: колективна динамiка, бiнарна рiдина, колективнi
збудження, узагальненi коефiцiєнти переносу
PACS: 05.20.Jj, 61.20.Ja, 61.20.Lc
300
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