Dynamics of ternary liquid mixtures: Generalized collective modes analysis
A parameter-free generalized collective modes (GCM) approach is applied to the study of longitudinal and transverse dynamics in a ternary Lennard-Jones liquid mixture. Spectra of collective excitations are calculated in a wide range of wavenumbers starting from the hydrodynamic region and up to th...
Збережено в:
| Дата: | 2007 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2007
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/118896 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Dynamics of ternary liquid mixtures: Generalized collective modes analysis / T.M. Bryk, I.M. Mryglod // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 481-494. — Бібліогр.: 17 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-118896 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1188962025-02-09T15:43:15Z Dynamics of ternary liquid mixtures: Generalized collective modes analysis Динамiка потрiйних рiдких сумiшей: аналiз узагальнених колективних мод Bryk, T.M. Mryglod, I.M. A parameter-free generalized collective modes (GCM) approach is applied to the study of longitudinal and transverse dynamics in a ternary Lennard-Jones liquid mixture. Spectra of collective excitations are calculated in a wide range of wavenumbers starting from the hydrodynamic region and up to the range of molecular regime. Several simplified dynamical models are used in order to estimate the origin of branches in the spectra of collective excitations. This analysis permits to establish a crossover from a “coherent” to “partial” type of collective dynamics in the dispersion laws of propagating collective modes. The ”coherent” type of collective dynamics is observed in the hydrodynamic region and is represented by sound modes (in the transverse case – shear waves) and propagating optic-like excitations, while in the range of “partial” dynamics (intermediate and large wavenumbers) the dynamical properties are mainly determined by correlated “partial” motions of particles of each species. Within the GCM scheme used the number of propagating optic-like modes, identified in this study, is simply related to the number of components in a multi-component mixture, and this has a close analogy in the dynamics of solids. Пiдхiд узагальнених колективних мод (УКМ), що не мiстить жодних вiльних параметрiв, використано для дослiдження повздовжньої i поперечної динамiки потрiйної рiдкої Леннард-Джонсiвської сумiшi. Розраховано спектри колективних збуджень у широкiй областi змiни значень хвильового вектора – вiд гiдродинамiчної i до областi молекулярного режиму. Для з’ясування механiзмiв формування колективних збуджень, що спостерiгаються у спектрах, використано кiлька спрощених динамiчних моделей. Проведений аналiз дозволив виявити кросовер вiд когерентної до парцiальної динамiки у дисперсiйних залежностях, отриманих для пропагаторних колективних збуджень. Когерентний тип динамiки спостерiгається у гiдродинамiчнiй областi i асоцiюється iз звуковими модами (зсувними хвилями у випадку поперечної динамiки) та пропагаторними збудженнями оптичного типу. В областi парцiальної динамiки (промiжнi i великi значення хвильових векторiв) динамiчнi властивостi визначаються в основному скорельованими парцiальними рухами частинок кожного iз сортiв. В рамках використаного пiдходу УКМ число пропагаторних збуджень оптичного типу, якi є iдентифiкованi у цьому дослiдженi, визначається числом компонент у багатокомпонентнiй сумiшi, що має пряму аналогiю у динамiцi твердотiльних систем. I.M. thanks the Fonds zur F¨orderung der wissenschaftlichen Forschung (Austria) for support under Project No P18592–TPH. 2007 Article Dynamics of ternary liquid mixtures: Generalized collective modes analysis / T.M. Bryk, I.M. Mryglod // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 481-494. — Бібліогр.: 17 назв. — англ. 1607-324X PACS: 05.20.Jj, 61.20.Ja, 61.20.Lc DOI:10.5488/CMP.10.4.481 https://nasplib.isofts.kiev.ua/handle/123456789/118896 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
A parameter-free generalized collective modes (GCM) approach is applied to the study of longitudinal and
transverse dynamics in a ternary Lennard-Jones liquid mixture. Spectra of collective excitations are calculated
in a wide range of wavenumbers starting from the hydrodynamic region and up to the range of molecular
regime. Several simplified dynamical models are used in order to estimate the origin of branches in the spectra
of collective excitations. This analysis permits to establish a crossover from a “coherent” to “partial” type of
collective dynamics in the dispersion laws of propagating collective modes. The ”coherent” type of collective
dynamics is observed in the hydrodynamic region and is represented by sound modes (in the transverse case
– shear waves) and propagating optic-like excitations, while in the range of “partial” dynamics (intermediate
and large wavenumbers) the dynamical properties are mainly determined by correlated “partial” motions of
particles of each species. Within the GCM scheme used the number of propagating optic-like modes, identified
in this study, is simply related to the number of components in a multi-component mixture, and this has a close
analogy in the dynamics of solids. |
| format |
Article |
| author |
Bryk, T.M. Mryglod, I.M. |
| spellingShingle |
Bryk, T.M. Mryglod, I.M. Dynamics of ternary liquid mixtures: Generalized collective modes analysis Condensed Matter Physics |
| author_facet |
Bryk, T.M. Mryglod, I.M. |
| author_sort |
Bryk, T.M. |
| title |
Dynamics of ternary liquid mixtures: Generalized collective modes analysis |
| title_short |
Dynamics of ternary liquid mixtures: Generalized collective modes analysis |
| title_full |
Dynamics of ternary liquid mixtures: Generalized collective modes analysis |
| title_fullStr |
Dynamics of ternary liquid mixtures: Generalized collective modes analysis |
| title_full_unstemmed |
Dynamics of ternary liquid mixtures: Generalized collective modes analysis |
| title_sort |
dynamics of ternary liquid mixtures: generalized collective modes analysis |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2007 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118896 |
| citation_txt |
Dynamics of ternary liquid mixtures: Generalized collective modes analysis / T.M. Bryk, I.M. Mryglod // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 481-494. — Бібліогр.: 17 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT bryktm dynamicsofternaryliquidmixturesgeneralizedcollectivemodesanalysis AT mryglodim dynamicsofternaryliquidmixturesgeneralizedcollectivemodesanalysis AT bryktm dinamikapotrijnihridkihsumišejanalizuzagalʹnenihkolektivnihmod AT mryglodim dinamikapotrijnihridkihsumišejanalizuzagalʹnenihkolektivnihmod |
| first_indexed |
2025-11-27T12:51:50Z |
| last_indexed |
2025-11-27T12:51:50Z |
| _version_ |
1849948019398541312 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 4(52), pp. 481–494
Dynamics of ternary liquid mixtures: Generalized
collective modes analysis
T.M.Bryk1,2, I.M.Mryglod1,3
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str.,
79011 Lviv, Ukraine
2 Institute of Applied Mathematics and Fundamental Sciences, National Technical University of Lviv, 79013
Lviv, Ukraine
3 Institute for Theoretical Physics, Linz University, 4040 Linz, Austria
Received November 13, 2007
A parameter-free generalized collective modes (GCM) approach is applied to the study of longitudinal and
transverse dynamics in a ternary Lennard-Jones liquid mixture. Spectra of collective excitations are calculated
in a wide range of wavenumbers starting from the hydrodynamic region and up to the range of molecular
regime. Several simplified dynamical models are used in order to estimate the origin of branches in the spectra
of collective excitations. This analysis permits to establish a crossover from a “coherent” to “partial” type of
collective dynamics in the dispersion laws of propagating collective modes. The ”coherent” type of collective
dynamics is observed in the hydrodynamic region and is represented by sound modes (in the transverse case
– shear waves) and propagating optic-like excitations, while in the range of “partial” dynamics (intermediate
and large wavenumbers) the dynamical properties are mainly determined by correlated “partial” motions of
particles of each species. Within the GCM scheme used the number of propagating optic-like modes, identified
in this study, is simply related to the number of components in a multi-component mixture, and this has a close
analogy in the dynamics of solids.
Key words: ternary liquid mixtures, collective excitations, optic-like modes, molecular dynamics
PACS: 05.20.Jj, 61.20.Ja, 61.20.Lc
1. Introduction
Collective dynamics in disordered systems is one of the complex and unresolved problems in
statistical physics. The existence of propagating and relaxation processes of different origin and
having distinct characteristic times makes the collective dynamics in liquids extremely difficult to
be analytically treated. Nevertheless, modern theoretical approaches of generalized hydrodynamics
[1] to a certain extent make it possible to correctly describe hydrodynamic and non-hydrodynamic
processes as well as the coupling effects between them in simple [2] and molecular liquids [3,4].
Over the last decade an essential advance was made in the theoretical description of collec-
tive dynamics in simple, binary and molecular liquids within the framework of the approach of
generalized collective modes (GCM), which, like in hydrodynamics, results in decomposition of
relevant time correlation functions into separated mode contributions coming from generalized hy-
drodynamic and non-hydrodynamic (kinetic) processes. The GCM approach consists in treating
the collective dynamics on an extended basis set of Nv dynamical variables, that apart from the
densities of hydrodynamic variables contain their time derivatives, whose purpose is to correctly
describe non-hydrodynamic short-time processes in liquids. Being consistent with the general def-
inition of collective excitations in statistical physics as the poles of relevant Green functions, the
GCM approach permits to calculate the dynamical eigenmodes in the system and to find their
partial contributions to the time correlation functions. All the eigenmodes obtained within the
GCM approach can be divided into two groups: (i) generalized hydrodynamic modes, having cor-
rect hydrodynamic asymptotes in the long-wavelength limit and the lifetime of which scales as
k−2 for k → 0, and (ii) kinetic modes with a finite nonzero lifetime in the long-wavelength limit,
c© T.M.Bryk, I.M.Mryglod 481
T.M.Bryk, I.M.Mryglod
being irrelevant to the dynamics on large spatial and temporal scales. Kinetic collective modes
are important beyond the hydrodynamic region, where they essentially contribute to the shape of
time correlation functions. In the cases of simple and binary liquids, the GCM studies [5–9] permit
us to perform a classification of kinetic collective modes of different physical origin. As examples
of kinetic propagating modes one can mention shear and heat waves, and for binary liquids there
are found optic-like excitations that, by analogy with optic phonons in binary crystals, reflect
the opposite-phase oscillations of the neighbors of different kind. A non-hydrodynamic process of
structural relaxation is the most obvious example of kinetic relaxing modes, which are not taken
into account within the standard hydrodynamics.
In the case of collective dynamics in multi-component liquids (with the number of species more
than two) the theoretical and simulation studies turn out to be very scarce. Even the simplest
dynamic properties like the relation between self- and mutual diffusion coefficients are not well
defined, while the mechanism of formation of collective excitations and their contributions to the
total spectral functions of three-component liquids have not been studied at all. Concerning ternary
liquids, there were no reports on classification of propagating and relaxing modes beyond hydro-
dynamic region. Therefore, the main purpose of our study was to obtain dynamical eigenmodes in
a simple Lennard-Jones three-component mixture using the parameter-free GCM approach as well
as our previous GCM results on binary liquids in order to establish the origin of mode formation
for different branches of propagating and relaxing excitations.
The paper is organized as follows. In section II an application of the method of generalized
collective modes to the three-component Lennard-Jones liquid mixture is reported. In this study
we follow the GCM scheme developed for multi-component fluids in [10]. An analytical treatment of
the longitudinal and transverse mass-concentration fluctuations within some simplified dynamical
models is presented in section III, and Section IV contains conclusions of this study.
2. Static and time correlation functions
2.1. Details of methodology
We have performed MD simulations for a three-component Lennard-Jones liquid AxByCy with
molar ratio 2:1:1 and mass ratio 13.91:8.63:4.63 at temperature T = 116 K and density n =
0.0182 Å−3. Parameters of LJ potentials were taken just as for the atoms Kr:Ar:Ar – hence, such
a three-component system corresponds in the case of mass ratio 13.91:6.63:6.63 to a well-studied,
within the GCM approach, equimolar KrAr liquid mixture [8,11]. Such a correspondence between
the two systems was intentionally set up, because it permits to check most static averages and
generalized thermodynamic quantities of the ternary liquid mixture with the ones of the binary
Lennard-Jones KrAr liquid.
The time evolution of basis dynamical variables has been obtained from production runs of
standard molecular dynamics simulations in microcanonical ensemble for the system of 2000 par-
ticles in a cubic box. Regular production run took over 3 · 105 time steps, while for five lowest
k-values, in order to obtain the desired convergence of relevant static averages and time correlation
functions, the system was simulated over 9 · 105 time steps. Twenty two k-points were sampled in
MD simulations, and the smallest wavenumber reached in our MD study was kmin = 0.131 Å−1.
Additional averages were taken over all the possible directions of wavenumbers corresponding to
the same absolute value. In order to reduce the dimension of relevant quantities, the following
energy, mass, spatial and time scales were used in our simulations: ε = kBT , µ = m̄, σ = k−1
min,
τ = σ(µ/ε)1/2 = 6.1056 ps.
We combined MD simulations with the parameter-free GCM approach to study the longitu-
dinal and transverse collective mode spectra, time correlation functions (TCFs) and separated
contributions to certain TCFs, caused by various collective excitations and relaxation processes.
For each sampled k-point, the shape of MD-derived time correlation functions and the spectrum of
longitudinal eigenvalues were analyzed within the eleven-variable (Nv = 11) GCM approach with
482
Dynamics of ternary liquid mixtures
the following basis set of dynamical variables:
A
(11)(k, t) =
{
nA, nB, nC , JA, JB, JC , ε, J̇A, J̇B, J̇C , ε̇
}
, (1)
where the dynamical variables of partial densities are defined by
nα(k, t) =
1√
N
Nα
∑
j=1
e−ikr
α
j (t) , α = A,B,C, (2)
the partial densities of longitudinal mass-currents are given by
Jα(k, t) =
mα
k
1√
N
Nα
∑
j=1
(kv
α
j )e−ikr
α
j (t) α = A,B,C, (3)
and density of total energy is
ε(k, t) =
1√
N
∑
α
Nα
∑
j=1
eα
j (t)e−ikr
α
j (t) . (4)
Here the eα
j (t) are single-particle energies in the αth species, estimated in a simple way as a sum of
kinetic and potential energy of the j-th particle. In (1) and (3) we have dropped the label indicating
the longitudinal component of partial mass-currents. One can obtain the first time derivatives of
partial mass-currents and energy straightforwardly from the expressions (3) and (4).
In the case of transverse dynamics, the same level of treatment of short-time processes corre-
sponds to a six-variable (Nv = 6) basis set of transverse dynamical variables, namely:
A
(6T)(k, t) =
{
JT
A , JT
B , JT
C , J̇T
A , J̇T
B , J̇T
C
}
. (5)
Here the index T denotes the transverse component of currents and their time derivatives.
We would like to notice that the hydrodynamic set of dynamic variables for the ternary liquid
mixtures includes
A
(hyd)(k, t) = {nA, nB, nC , Jt, ε} , A
(hyd, T)(k) =
{
JT
t
}
, (6)
where longitudinal and transverse components of total mass-current Jt(k, t) in the above adopted
normalization (3) can be simply written in terms of partial mass-currents,
Jt(k, t) = JA(k, t) + JB(k, t) + JC(k, t) . (7)
Comparing (1) with the standard set of the hydrodynamic variables, one can see that within the
GCM approach the partial density fluctuations are treated with the same precision in respect
to the order of frequency sum rules reproduced explicitly as total density fluctuations. This is
connected with the fact observed previously for binary liquids in the GCM treatment [8] and the
numerical analysis of MD data [12] that some branches of propagating collective excitations have a
pronouncedly partial character in the short-wavelength region and can be simply obtained from the
analysis of MD-derived partial current-current spectral functions. Hence, the simplest extension of
the hydrodynamic sets (6) by taking into account just the first time derivatives of hydrodynamic
variables would not permit to correctly describe collective excitations in short-wavelength region.
Therefore, all the partial currents are equitably considered in our theoretical scheme. Hereafter
we shall call the TCFs, constructed on the hydrodynamic variables (6), the hydrodynamic time
correlation functions.
All the static and time correlation functions, needed for the estimation of matrix elements of
the Nv × Nv matrices of time correlation functions F(k, t) and their Laplace transforms F̃(k, z),
483
T.M.Bryk, I.M.Mryglod
were directly evaluated in computer simulations. Eigenvalues and eigenvectors of generalized hy-
drodynamic (otherwise referred to as kinetic) matrix [5]
T(k) = F(k, 0) · F̃−1(k, 0), (8)
were calculated for either of the twenty two k-points, sampled in MD. From the eigenvalues the
spectra of propagating collective excitations were constructed and the k-dependent damping coef-
ficients for the main relaxation processes were found. The eigenvectors were used in the analysis of
mode contributions to time correlation functions of interest in different regions of wavenumbers.
2.2. Generalized thermodynamic quantities
The generalized hydrodynamic theory usually operates with the so-called generalized thermo-
dynamic quantities (see, e.g., [13,14]), being k-dependent functions that tend in the limit k → 0 to
their thermodynamic values observed in real experiments. Here we follow the general expressions
for the generalized thermodynamic quantities, obtained for ν-component mixtures in [10]. In the
case of a ternary liquid one has:
(i) for the generalized isothermal compressibility θ(k),
θ(k) =
1
nkBT
[
3
∑
α,β=1
cαF−1
αβ (k)cβ
]
−1
,
where n = N/V , cα is the concentration of particles in the α-th species, and elements of the
3 × 3 matrix of partial density-density correlations are
Fαβ(k) = 〈nα(k, 0)n∗
β(k, 0)〉 ;
(ii) for the generalized specific heat CV(k) at constant volume per one particle,
CV(k) =
1
kBT 2
〈h(k, 0)h∗(k, 0)〉,
where h(k, t) is defined as follows
h(k, t) = ε(k, t) −
3
∑
α,β=1
〈ε(k, 0)n∗
α(k, 0)〉F−1
αβ (k) nβ(k, t) ; (9)
(iii) for the generalized linear thermal expansion coefficient αP (k),
αP (k) =
1
ik
〈J̇t(k, 0)h∗(k, 0)〉nθ(k)
kBT 2
,
where the static average of longitudinal stress-heat density correlation can be straightfor-
wardly calculated from the definition (9);
(iv) for the generalized ratio of specific heats γ(k),
γ(k) = 1 +
Tα2
P (k)
nθ(k)CV(k)
.
Numerical results, obtained directly in MD simulations as equal time averages, for the gen-
eralized thermodynamic quantities are shown in figure 1. Note, first, that at least three out of
four generalized thermodynamic quantities in figure 1 have well pronounced peaks at k nearby the
position kp ' 1.8Å−1 of the first peak of the total static structure factor. Besides, some useful in-
formation can be extracted from such k-dependent thermodynamic quantities. In particular, from
484
Dynamics of ternary liquid mixtures
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3
α P
(k
)T
k / A-1
1
1.2
1.4
1.6
1.8
2
2.2
0 0.5 1 1.5 2 2.5 3 3.5
k / A-1
γ(
k)
0
0.5
1
1.5
2
2.5
θ(
k)
1.8
2
2.2
2.4
2.6
C
V
(k
)
/ k
B
Figure 1. Generalized k-dependent thermodynamic quantities: generalized isothermal compress-
ibility θ(k), generalized linear thermal expansion coefficient αP (k), generalized specific heat at
constant volume CV(k), and generalized ratio of specific heats γ(k). Open circle at k = 0 in
the upper right frame corresponds to specific heat at a constant volume obtained from the
fluctuation formula during the MD run.
these calculations, considering small k limit, one can predict the value of αT = 0.0043 K−1 for
linear expansion coefficient of such a ternary liquid at the temperature of 116 K.
In the long-wavelength limit, the generalized k-dependent specific heat at constant volume
CV(k) is in agreement with the value of 2.43 kB, independently obtained at k = 0 via the fluctuation
formula during the MD production runs. This value is quite reasonable for Lennard-Jones liquids.
One should also note, that γ(k) is in fact a measure of coupling between the heat and viscous
processes: for γ = 1 one can describe heat and viscous processes separately. That is why the
longitudinal dynamics of liquid metals with the typical value of γ being at the level of 1.1 − 1.2
is described pretty well within the viscoelastic theories. For Lennard-Jones liquids the coupling
between the heat and viscous processes is usually rather strong and, for instance, in our case only
in the restricted range of wavenumbers nearby k ' 1.3 Å−1 the generalized ratio of specific heats
γ(k) is close to unity, so that the viscoelastic treatment could be successfully applied in this region.
2.3. Time correlation functions
In figure 2 the numerical results, obtained at k = 0.371Å−1 for time correlation functions
describing the longitudinal dynamics of the three-component Lennard-Jones liquid in MD simu-
lations, are presented. In the same figure we show our results for these TCFs found within the
parameter-free GCM approach. Hence, one can see how the GCM approach makes it possible to
reproduce the main TCFs of primary interest obtained in MD simulations. Note that the time scale
of τ indicates that the range of time correlations at k = 0.37Å−1 is of the order of 70 ps. For this
reason, to provide good convergence of the tails of time correlation functions, we have simulated
the system three times longer for the smallest wavenumbers.
The GCM replicas were calculated using theoretical expressions for time correlation functions
(see, e.g., [5]), which in this study for the case of basis set (1) could be represented as linear
combinations of the eleven contributions from the dynamical eigenvalues that correspond either to
generalized collective propagating excitations or k-dependent relaxation processes, namely
F
(GCM)
ij (k, t) =
Nv
∑
α=1
Gij
α (k)e−zα(k)t , (10)
where Gij
α (k) are the k-dependent amplitudes, describing the corresponding partial contribu-
tion of the collective mode zα(k) to the shape of Fij(k, t). Both the amplitudes Gij
α (k) and
the eigenvalues zα(k) can be in general the complex quantities and are calculated from the
485
T.M.Bryk, I.M.Mryglod
-1
-0.8
-0.6
-0.4
-0.2
0
0 2 4 6 8 10
t / τ
FεA(k,t)
0
2
4
6
0 2 4 6 8 10 12
t / τ
Fεε(k,t)
-0.3
-0.2
-0.1
0
FAB(k,t)
0
0.2
0.4
0.6
0.8
FCC(k,t)
0
0.1
0.2
0.3
0.4
0.5
FAA(k,t)
MD
GCM
0
0.2
0.4
0.6
0.8
FBB(k,t)
Figure 2. Partial density-density, density-energy and energy-energy time correlation functions
Fij(k, t), obtained in MD simulations (solid lines) and their theoretical replicas (dotted lines),
calculated within the parameter-free GCM approach for k = 0.371Å−1. For partial TCFs a
normalization constant was used in order to correspond to Ashcroft-Langreth partial structure
factors at t = 0. The time scale is τ=6.1056 ps.
eigenvalue problem for the generalized hydrodynamic matrix (8), generated in our case on the
eleven-variable basis set A
(11)(k).
Comparing the theoretical curves with MD data, shown in figure 2, it is seen that the GCM
approach permits to reproduce the time-dependence of MD-derived TCFs with very good precision
without any fitting parameters. We stress that for each wavenumber the obtained set of eleven
(Nv = 11) dynamical eigenvalues and relevant eigenvectors reproduce simultaneously Nv(Nv +
1)/2 time correlation functions. Moreover, the TCFs of primary interest, describing the density
fluctuations, are reproduced within the GCM scheme used in this study with the precision that
the sum rules are satisfied explicitly up to the fourth order inclusive.
Using the TCFs shown in figure 2, one can define the main correlation times of a ternary liquid
as follows:
τij(k) =
1
Fij(k, 0)
∞
∫
0
Fij(k, t)dt , i, j = A,B,C, ε .
In general, in the case of a ternary mixture one can find ten independent correlation functions,
which in this study were directly estimated from MD-derived TCFs and were subsequently used
in the calculation of the matrix elements of generalized hydrodynamic matrix T(k), generated on
the basis set A
(11)(k, t).
3. Spectra of collective excitations
3.1. Dispersion and damping coefficients for transverse and longitudinal excitations
Let us start from the discussion of transverse dynamics, because due to the absence of cou-
pling with the heat fluctuations, this case turns out to be much simpler for the analysis than the
longitudinal dynamics. In figure 3 we report the imaginary and real parts of complex eigenvalues
zT
α (k) = σT
α (k) + iωT
α (k) ,
486
Dynamics of ternary liquid mixtures
which correspond to the dispersion and damping of propagating collective modes, respectively.
Purely real transverse eigenvalues
Im zT
j (k) = 0, Re zT
j (k) = dT
j (k)
will be marked by symbols dT
j (k). They have the meaning of inverse k-dependent lifetimes of the
relevant relaxation processes. We stress that our approach provides a straightforward and consis-
tent extension of the standard hydrodynamics that allows us to take into account the k-dependent
relaxation processes of non-hydrodynamic origin. Like in the case of hydrodynamic processes, the
non-hydrodynamic relaxation processes make exponential contributions with k-dependent ampli-
tudes to the shape of the relevant hydrodynamic time correlation functions.
0
2
4
6
8
10
12
Im
z
jT
(k
)
/ p
s-1
0
1
2
3
4
5
6
7
0 0.5 1 1.5 2 2.5 3 3.5
R
e
z j
T
(k
),
d
jT
(k
)
/ p
s-1
k / A-1
Figure 3. Imaginary (top) and real (bottom) parts of complex eigenvalues for transverse collec-
tive excitations in a wide region of wavenumbers. In the lower frame the real parts of complex
eigenvalues are shown together with the purely real eigenvalues dT
j (k), describing the relax-
ation processes in transverse dynamics. Dashed lines in small k range are spline-interpolation
for purely real eigenvalues.
In figure 3, purely real eigenvalues are shown by dash-line connected symbols. It is seen that
these eigenvalues occur only in a narrow region for k < 0.4Å−1. The slowest transverse relaxation
process with the smallest eigenvalue, dT
1 (k), gives an infinite relaxation time in the k → 0 limit
and according to the hydrodynamics should behave like
dT
1 (k) =
η
ρ
k2 , when k → 0,
with η and ρ being the shear viscosity and mass density of the mixture, respectively. The estimated
value of the shear viscosity is η = 2.24 · 10−4 kg/s·m. The second type of relaxation process, seen
in figure 3, is of non-hydrodynamic origin and describes the process with a finite lifetime in the
long-wavelength limit, defined mainly by the rigidity modulus G (see also [1])
dT
2 (k) =
G
η
− η
ρ
k2 , k → 0.
For larger k, k > 0.4Å−1, both relaxing modes merge and the pair of propagating collective
modes known as shear waves emerge in the system. Thus, the both relaxing modes dT
j (k) and the
487
T.M.Bryk, I.M.Mryglod
emergence of shear waves are in complete analogy with the results known so far for simple and
binary liquids.
The difference in the transverse dynamics of a ternary mixture in comparison with one- and two-
component cases appears in the number of high-frequency propagating modes. In the case of three-
component liquid mixture within the six-variable GCM treatment we have found two branches of
high-frequency transverse propagating modes existing in the whole k-range considered. In figure 3
one can see the main features of transverse propagating modes in a wide range of wavenumbers:
(i) two high-frequency branches with rather close damping coefficients tend to finite frequencies
in the long-wavelength limit, which implies their origin as optic phonon-like excitations; (ii) all
three branches of propagating modes, obtained in our study, have very close damping coefficients
in the region of 1Å−1 < k < 1.8Å−1; (iii) a legible separation in frequency and damping coefficients
is observed for all three propagating modes in short-wavelength region k > 2Å−1. The origin of
high-frequency branches of transverse excitations will be discussed in the next subsection.
0
2
4
6
8
10
12
14
Im
z
j(k
)
/ p
s-1
0
2
4
6
8
0 0.5 1 1.5 2 2.5 3 3.5
R
e
z j
(k
)
/ p
s-1
k / A-1
Figure 4. Imaginary (top) and real (bottom) parts of the longitudinal complex eigenvalues,
obtained on the eleven-variable basis set A
(11).
In figure 4, the dispersion and damping coefficients of longitudinal propagating modes, obtained
within the same level of treatment of short-time processes as in the transverse case, are shown. In
the whole range of wavenumbers, using the set (1), we have found three branches of propagating
modes (presented in figure 4). Moreover, for k > 1Å−1 there can appear an additional pair of
propagating excitations that correspond to low-frequency heat waves (not shown in figure 4). The
three branches of propagating excitations shown in figure 4 can be easily identified only in long-
wavelength region, where the lowest branch behaves like the longitudinal sound modes having
dispersion ω(k) with obvious positive deviation (positive dispersion) from linear dependence and
the damping coefficient being proportional to k2, as is shown by relevant dotted lines in the small k
domain of a lower frame in figure 4. The estimated values of sound velocity and damping coefficient
are c = 704 m/s and Γ = 1.7·10−7 m2/s, respectively. For a larger k, k > 0.7Å−1, all three branches
have comparable damping, while the frequencies are still well separated. The origin of the other
two branches will be discussed more in detail herein below, where an analysis on some simplified
models of the longitudinal dynamics is presented.
Purely real eigenvalues, obtained from the description of the longitudinal dynamics within the
eleven-variable dynamical model (1), are shown in figure 5 for small wavenumbers k. One can easily
488
Dynamics of ternary liquid mixtures
0
1
2
3
4
5
6
7
8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
d j
(k
)
/ p
s-1
k / A-1
d5
d4
d3
d2
d1
Figure 5. Longitudinal purely real eigenvalues, obtained within the eleven-variable GCM treat-
ment in long-wavelength region.
see two kinetic relaxation modes, which tend to nonzero constants when k → 0, while the damping
coefficients for hydrodynamic relaxation processes should behave like k2 in the long-wavelength
limit. In our case there exist two heat relaxing modes: (i) hydrodynamic one (shown as d3(k) in
figure 5) that displays the standard k-dependence
d3(k) = DTk2 , when k → 0,
and (ii) kinetic heat relaxing mode (marked as d5(k) in figure 5), which according to [7] should
behave (without taking into account the coupling with the viscous processes) approximately like
d5(k) =
nGh
mDT
− DTk2 , when k → 0,
where Gh is a heat rigidity modulus.
Another kinetic relaxing mode d4(k), which has a finite lifetime in long-wavelength limit and de-
cays much faster in time in comparison with the hydrodynamic relaxing processes, is of viscoelastic
origin and according to [15] behaves in the long-wavelength limit roughly as
d4(k) =
c2
∞
− c2
s
DL
− DLk2 ,
where cs, c∞ are adiabatic and infinite-frequency sound velocities, respectively, and DL is a kine-
matic viscosity.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
d 1
(k
),
d
2(
k)
/
ps
-1
k / A-1
d2
d1
Figure 6. Long-wavelength asymptotes of the two lowest real eigenvalues corresponding to the
relaxation processes of mutual diffusivity.
The two lowest relaxing modes d1(k) and d2(k) in small k domain describe the relaxation
processes connected with mutual diffusion of species. In figure 6 their long-wavelength behavior
489
T.M.Bryk, I.M.Mryglod
with hydrodynamic asymptotes proportional to k2 is shown. From this figure one can see that
for both diffusive modes there exists a negative deviation from the hydrodynamic asymptotes
when k increases. This corresponds to a decrease with k of the generalized k-dependent mutual
diffusion coefficients Dx1(k) and Dx2(k), which is known from our previous studies of generalized
transport coefficients in binary liquids [16]. From these asymptotes we can easily estimate the
corresponding values of damping coefficients for the ternary mixture studied: Dx1 = 2.75·10−9 m2/s
and Dx2 = 3.1 ·10−9 m2/s. In general, one can say that the heat relaxing processes for k > 0.4Å−1
have much smaller lifetime comparing with the viscoelastic processes that reflect the structural
relaxation phenomena mainly in intermediate and short-wavelength region.
3.2. Simplified models of transverse dynamics
In order to study more in detail the origin of mode formation for different branches in the
spectrum of collective excitations we proceed as in our previous GCM studies of collective dynamics
in simple and binary liquids. Namely, let us try to find the separated subsets of dynamical variables,
which would reproduce well the spectra (or its parts) of eigenvalues in some regions of wavenumbers.
In such a case one can suppose that the coupling between the actual subset and other dynamical
variables is really small and can be neglected. Similar approach allows us to identify the heat waves
in liquid metals [7] and optic-like excitations in binary liquids [8,9].
0
1
2
3
4
5
6
7
8
9
10
0 0.5 1 1.5 2 2.5 3 3.5
Im
z
jT
(k
)
/ p
s-1
k / A-1
t-t
x1-x1
x2-x2
A-A
B-B
C-C
Figure 7. Imaginary parts of the transverse complex-conjugated eigenvalues, obtained for the
six-variable basis set A
(6T) (symbols) and calculated on the separated subsets of dynamical
variables (lines) A
(2T)
i and A
(2T)
α for small and intermediate k values, respectively.
Let us split the six-variable dynamical set (5), used for the case of transverse dynamics, into
three separated two-variable dynamical subsets, describing collective transverse processes only
within the species:
A
(2T)
α (k, t) =
{
JT
α , J̇T
α
}
, α = A,B,C. (11)
Then we have used these subsets in order to generate three different 2 × 2 generalized kinetic
matrices T
(2T)
α (k). This resulted in three complex-conjugated pairs of eigenvalues shown in figure 7
by corresponding lines for k > 0.5Å−1. One can see that the results obtained on the separated
subsets, describing partial dynamics, match perfectly with the results found for the six-variable
dynamical model. This means that the partial dynamics dominates in this region, k > 0.8Å−1, and
therefore we conclude that the collective dynamics becomes essentially of “partial” type and can
be well described in terms of “partial” branches of collective excitations for larger k. For smaller
wavenumbers, the results obtained for the two-variable dynamical models (11) differ essentially
from the results found for the general six-variable dynamical model (5). Hence, it is obvious that
in a small k domain, the collective dynamics is determined by another type of collective behavior.
In the case of binary liquids it was rather easy to construct other subsets of transverse dynamical
variables, reflecting collective properties: it was obvious that one of them should be based on the
total transverse mass-current (the hydrodynamic variable), while another subset was constructed
490
Dynamics of ternary liquid mixtures
for the variable being orthogonal to the total mass-current as a combination of partial currents.
In the case of a ternary mixture we have proceeded in a similar manner and found two dynamical
variables being orthogonal between themselves as well as to the total mass-current. They can be
defined as follows
JT
x1(k, t) = xAJT
B(k, t) − xBJT
A(k, t) , (12)
and
JT
x2(k, t) = (xA + xB)JT
C (k, t) − xC(JT
A(k, t) + JT
B(k, t)) , (13)
where the dynamical variable JT
x1(k, t) describes the transverse component of mutual current of
the most heavy and medium-weight components, while the dynamical variable (13) corresponds to
mutual current of the most light component against two heavier ones.
On this basis we have considered another three separated subsets of dynamical variables
A
(2T)
i (k, t) =
{
JT
i , J̇T
i
}
, i = t, x1, x2, (14)
and performed new calculations of collective mode spectra. This allows us to reproduce very nicely
and separately all the branches of propagating collective excitations in long-wavelength region. The
results are shown in figure 7 by corresponding lines in small k domain. Hence, by analogy with
the case of binary liquids we conclude that both high-frequency branches of kinetic propagating
modes correspond to optic-like excitations, formed by fluctuations of mutual mass-currents (12)
and (13). Since the three subsets (14) describe intrinsic collective fluctuations, one may refer to
the long-wavelength region as the region of coherent dynamics, just as pointed out by Hafner in
1983 in the study of a binary magnetic glass [17].
3.3. Simplified models of longitudinal dynamics
The analysis of longitudinal spectrum is much more sophisticated due to the coupling between
viscoelastic and thermal fluctuations. Let us start our consideration from the simplest dynamical
models, based on three-variable subsets that consist of the partial dynamical variables only:
A
(3)
α (k, t) =
{
nα, Jα, J̇α
}
, α = A,B,C. (15)
All these models result in a relaxing mode and in a pair of propagating collective excitations with
complex-conjugated eigenvalues, shown in the lower frame of figure 8 by corresponding lines. Once
again, comparing with the results found for the eleven-variable model in the short-wavelength
region, we see that all three models of partial dynamics permit to reproduce perfectly and sepa-
rately all the propagating collective excitations. In the opposite range of small k they produce the
dispersion laws that tend to relevant nonzero frequencies at k → 0. Comparing with the results
found for a more sophisticated model (1) we note that the dispersion laws for the upper lying
modes are rather well reproduced up to the smallest k reached in our study, while for the lower
pair of propagating modes one can see a significant deviation in small k domain. Thus, in complete
agreement with the results found for the transverse dynamics, we conclude that the “partial” type
of collective behavior dominates in the range of intermediate and large wavenumbers, k > 0.8Å−1,
and determine the dynamical properties of the mixture in this region.
In order to separately describe the branches of longitudinal collective excitations in the region
of “coherent” dynamics let us consider another class of simplified dynamic models that consists of
the following basic variables:
(i) the total density
nt(k, t) = nA(k, t) + nB(k, t) + nC(k, t) ; (16)
(ii) two mass-concentration densities
nx1(k, t) =
mA
m̄
xBnA(k, t) − mB
m̄
xAnB(k, t) , (17)
491
T.M.Bryk, I.M.Mryglod
0
2
4
6
8
10
12
14
Im
z
j(k
)
/ p
s-1
t-t
x1-x1
x2-x2
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5
Im
z
j(k
)
/ p
s-1
k / A-1
A-A
B-B
C-C
Figure 8. Imaginary parts of the longitudinal complex-conjugated eigenvalues, obtained for the
eleven-variable basis set A
(11) (symbols) and calculated on the separated subsets of dynamical
variables (lines) A
(3)
α (bottom) and A
(3)
i (top).
and
nx2(k, t) =
1
m̄
{
(xA + xB)mCnC(k, t) − xC [mAnA(k, t) + mBnB(k, t)]
}
, (18)
being the linear combinations of partial densities. It is seen from (17) and (18) that the mass-
concentration densities have simple relation to the relevant longitudinal mass-concentration cur-
rents Jxi(k, t), namely,
∂nxi(k, t)
∂t
=
ik
m̄
Jxi(k, t) , i = 1, 2 , (19)
constructed in the same manner as in the case of transverse dynamics (see (12) and (13)). It is easy
to check that both mass-concentration currents Jx(k, t) are mutually orthogonal dynamic variables
and both are orthogonal to the total mass-current density Jt(k, t). This makes the theoretical
treatment of dynamical processes in small k region especially convenient, where the “coherent”
type of the dynamics prevails, because it allows us to perform the analysis in terms of t − x
variables (see [9]).
In the upper frame of figure 8 we show by corresponding lines the results for propagating
collective modes, obtained within the simplified dynamical models that are defined on the separated
subsets A
(3)
i (k, t) with i = t, x1, x2 (compare with (15)). One can see that for k < 0.5Å−1 the
propagating modes, calculated for the subsets A
(3)
x1 (k, t) and A
(3)
x2 (k, t), perfectly reproduce both
high-frequency longitudinal branches. This makes an evidence of the origin of these collective
excitations as optic phonon-like modes. It is also seen in figure 8 that the low-frequency branch is
well described in the long-wavelength region on the separated subset A
(3)
t (k, t) which includes only
the dynamical variables representing total mass-density and its two time derivatives (total current
and longitudinal projection of stress tensor). We note that there exists a close analogy between the
model A
(3)
t and the so-called viscoelastic model of a fluid [1,2]. Besides one purely real eigenvalue,
both these models result in the appearance of a pair of propagating sound modes with the linear
k-dependent dispersion in long-wavelength limit with propagation velocity equal to c∞. Only the
correct treatment of coupling with the heat fluctuations bends down the dispersion curve for sound
492
Dynamics of ternary liquid mixtures
modes in the long-wavelength region and changes the propagation velocity to the correct value cs,
known from the standard hydrodynamics.
We may conclude herein, that in both cases of the transverse and longitudinal dynamics our
analysis indicates an evident crossover between the “coherent” and “partial” types of collective
behavior in the ternary liquid mixture studied. It is also established that this crossover is observed
in the region of 0.5Å−1 < k < 1Å−1 in both cases.
4. Conclusions
We have performed the analysis of generalized collective mode spectra for a ternary Lennard-
Jones liquid mixture. A parameter-free GCM approach has been used to reach the goal of this
study. Our main conclusions can be formulated as follows:
(i) For a ternary liquid mixture we have found two additional high-frequency optic-like propa-
gating modes in both cases of the longitudinal and transverse dynamics. Within the GCM
scheme used in this study the number of optic-like modes is simply related to the number
of components in a multi-component mixture, and this has a close analogy with a solid-like
picture;
(ii) Another specific feature in the longitudinal dynamics of a ternary mixture in comparison
with a binary liquid is the appearance of an additional relaxation process connected with
mutual diffusion, i.e. in the three-component liquid mixture we have observed two relaxation
processes with different damping coefficients connected with diffusion coefficients;
(iii) Two linear combinations of partial currents have been found that permit to separately repro-
duce the optic-like modes of a ternary liquid in the long-wavelength limit. One of these two
combinations describes mutual current of the most heavy and medium-weight components,
while another one reflects mutual current of the most light component against two heavier
ones;
(iv) An analysis, performed on the separated subsets of dynamical variables, permits us to estab-
lish a crossover from a “coherent” to “partial” type of collective dynamics in the dispersion
laws of propagating collective modes. The “coherent” type of collective dynamics is observed
in a small k region and is represented by sound modes (in the transverse case – shear waves)
and propagating optic-like excitations, while in the range of “partial” dynamics (interme-
diate and large k) the dynamical properties are determined mainly by “partial” motions of
particles of each species;
(v) We have calculated the generalized thermodynamic quantities for the ternary liquid mixture.
This seems to be the first example of such calculations performed for three-component liquids.
Acknowledgements
I.M. thanks the Fonds zur Förderung der wissenschaftlichen Forschung (Austria) for support
under Project No P18592–TPH.
493
T.M.Bryk, I.M.Mryglod
References
1. Boon J.-P., Yip S. Molecular Hydrodynamics, New-York: McGraw-Hill, 1980.
2. Balucani U., Zoppi M. Dynamics of the liquid state, Oxford: Clarendon, 1983.
3. Chong S.-H., Hirata F., Phys. Rev. E, 1998, 57, 1691.
4. Chong S.-H., Hirata F., Phys. Rev. E, 1998, 58, 6188.
5. Mryglod I.M., Omelyan I.P., Tokarchuk M.V., Mol. Phys., 1995, 84, 235.
6. Mryglod I.M., Omelyan I.P., Mol. Phys., 1997, 90, 91.
7. Bryk T., Mryglod I., Phys. Rev. E, 2001, 63, 051202.
8. Bryk T., Mryglod I., J. Phys.: Cond. Matt., 2000, 12, 6063.
9. Bryk T., Mryglod I., J. Phys.: Cond. Matt., 2002, 14, L445.
10. Mryglod I.M., Condens. Matter Phys., 1997, No 10, 115.
11. Bryk T., Mryglod I., Condens. Matter Phys., 2004, 7, 15.
12. Anento N., Padro J.A., Phys. Rev. B, 2000, 62, 11428.
13. de Schepper I.M., Cohen E.G.D., Bruin C., van Rijs J.C., Montfrooij W., de Graaf L.A., Phys. Rev.
A, 1988, 38, 271.
14. Bryk T., Mryglod I., Kahl G., Phys. Rev. E, 1997, 56, 2903.
15. Bryk T., Mryglod I., Condens. Matter Phys., 2004, 7, 471.
16. Bryk T., Mryglod I., Condens. Matter Phys., 2004, 7, 285.
17. Hafner J., J. Phys. C, 1983, 16, 5773.
Динамiка потрiйних рiдких сумiшей: аналiз узагальнених
колективних мод
Т.М.Брик1,2, I.М.Мриглод1,3
1 Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, вул. Свєнцiцького, 1
2 Iнститут прикладної математики i фундаментальних наук,
Нацiональний унiверситет “Львiвська полiтехнiка”, 79013 Львiв, Україна
3 Iнститут теоретичної фiзики, Лiнцський унiверситет, 4040 Лiнц, Австрiя
Отримано 13 листопада 2007 р.
Пiдхiд узагальнених колективних мод (УКМ), що не мiстить жодних вiльних параметрiв, використано
для дослiдження повздовжньої i поперечної динамiки потрiйної рiдкої Леннард-Джонсiвської сумiшi.
Розраховано спектри колективних збуджень у широкiй областi змiни значень хвильового вектора
– вiд гiдродинамiчної i до областi молекулярного режиму. Для з’ясування механiзмiв формування
колективних збуджень, що спостерiгаються у спектрах, використано кiлька спрощених динамiчних
моделей. Проведений аналiз дозволив виявити кросовер вiд когерентної до парцiальної динамiки у
дисперсiйних залежностях, отриманих для пропагаторних колективних збуджень. Когерентний тип
динамiки спостерiгається у гiдродинамiчнiй областi i асоцiюється iз звуковими модами (зсувними
хвилями у випадку поперечної динамiки) та пропагаторними збудженнями оптичного типу. В областi
парцiальної динамiки (промiжнi i великi значення хвильових векторiв) динамiчнi властивостi визна-
чаються в основному скорельованими парцiальними рухами частинок кожного iз сортiв. В рамках
використаного пiдходу УКМ число пропагаторних збуджень оптичного типу, якi є iдентифiкованi у
цьому дослiдженi, визначається числом компонент у багатокомпонентнiй сумiшi, що має пряму ана-
логiю у динамiцi твердотiльних систем.
Ключовi слова: потрiйнi рiдкi сумiшi, колективнi збудження, моди оптичного типу, молекулярна
динамiка
PACS: 05.20.Jj, 61.20.Ja, 61.20.Lc
494
|