Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures
We propose a new «toy» dynamical model that permits us to derive analytical expressions for dispersion of two branches of «bare» propagating collective excitations in binary disordered systems in the whole range of wavenumbers. These expressions are used for the analysis of dependence of dispersio...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
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| Cite this: | Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures / T. Bryk, I.M. Mryglod // Физика низких температур. — 2007. — Т. 33, № 09. — С. 1036–1044. — Бібліогр.: 24 назв. — англ. |
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| citation_txt | Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures / T. Bryk, I.M. Mryglod // Физика низких температур. — 2007. — Т. 33, № 09. — С. 1036–1044. — Бібліогр.: 24 назв. — англ. |
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| description | We propose a new «toy» dynamical model that permits us to derive analytical expressions for dispersion
of two branches of «bare» propagating collective excitations in binary disordered systems in the whole range
of wavenumbers. These expressions are used for the analysis of dependence of dispersion curves on mass ratio
and concentration at fixed density of the system. An effect of hybridization of two branches is discussed
in terms of mode contributions to time correlation functions. This allows us to estimate the regions with
dominant types of coherent or partial dynamics.
|
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Fizika Nizkikh Temperatur, 2007, v. 33, No. 9, p. 1036–1044
Collective excitations in dynamics of liquids: a «toy»
dynamical model for binary mixtures
T. Bryk1,2 and I.M. Mryglod1,2
1
Institute for Condensed Matter Physics of National Academy of Sciences of Ukraine
1 Svientsitskii Str., Lviv 79011, Ukraine
E-mail: mryglod@icmp.lviv.ua
2
Institute of Applied Mathematics and Fundamental Sciences of National Technical University of Lviv
Lviv 79013, Ukraine
Received January 18, 2007
We propose a new «toy» dynamical model that permits us to derive analytical expressions for dispersion
of two branches of «bare» propagating collective excitations in binary disordered systems in the whole range
of wavenumbers. These expressions are used for the analysis of dependence of dispersion curves on mass ra-
tio and concentration at fixed density of the system. An effect of hybridization of two branches is discussed
in terms of mode contributions to time correlation functions. This allows us to estimate the regions with
dominant types of coherent or partial dynamics.
PACS: 05.20.Jj Statistical mechanics of classical fluids;
61.20.Lc Time-dependent properties; relaxation;
61.25.Mv Liquid metals and alloys.
Keywords: collective excitations, dispersion law, binary liquids, molecular dynamics.
1. Introduction
Neutron and x-ray scattering experiments [1,2] are
main experimental techniques that along with molecular
dynamics (MD) simulations are intensively used for ex-
ploration of dynamical processes of crystals, glasses and
liquids. With a subsequent use of mainly oversimplified
models like damped harmonic oscillator (DHO) the ex-
perimental and MD data on spectral functions can be used
for estimation of dispersion law and damping of collec-
tive excitations. Collective excitations in liquids and
glass-forming systems are extremely difficult for theoret-
ical treatment, when the system is considered on spatial
and time scales comparable with specific atomic scales,
i.e beyond the hydrodynamic description of the system as
a continuum media. Therefore any analytical results that
can shed light on dispersion law of propagating modes in
disordered systems are of great interest. For binary sys-
tems the situation is even more sophisticated — in some
studies [3,4] it was stressed, that for binary liquids there
exist regions of wavenumbers with different types of col-
lective dynamics: «coherent» type in long-wavelength re-
gion and «partial» one in the region of intermediate and
large wavenumbers. The existence of two types of collec-
tive dynamics follows from the shape of current spectral
functions C kL T
�� �, ( , ) with sub-index � representing either
total mass current t, or mass-concentration current x, or
partial currents A B, . It appears [3,4], that in long-wave-
length region the spectral functions C ktt ( , )� and
C kxx ( , )� represent two distinct collective excitations,
while C kAA ( , )� and C kBB ( , )� are very similar and usu-
ally have a single-peak shape.
In this study we will focus on a theoretical treatment of
dispersion law for two types of propagating collective ex-
citations in binary liquids. The current status of theory in
this field is far from being satisfactory. Up to the date
there do not exist analytica1 solutions for theoretical
models with simultaneous treatment of two branches of
collective excitations — only numerical calculations
within the GCM approach [4], memory function formal-
ism [5,6] and numerical analysis of MD data [3,7,8] were
reported for dispersion and damping of two branches of
collective excitations in binary liquids and glasses. The
problems in theoretical treatment are connected with dif-
ferent origin of the two branches of collective excitations:
© T. Bryk and I.M. Mryglod, 2007
hydrodynamic acoustic excitations and kinetic high-fre-
quency excitations. The absence of consistent analytical
solutions for the two branches of collective excitations
causes many confusions in analysis of MD or experimen-
tal data — hence there were reports of existence of exotic
«fast sound» excitations [9] or a hypothetical merger of
dispersion laws of two branches of collective excitations
into a hydrodynamic sound branch by approaching the
hydrodynamic region [10–12]. All these results were
based on some assumptions of absence of coupling be-
tween the two types of collective modes and did not take
into account different asymptotics in long-wavelength re-
gion of contributions from low- and high-frequency exci-
tations, because of absence of relevant analytical results,
which would be a basis for analysis of experimental and
MD data.
There exist in the literature analytical results for a sep-
arated treatment of high-frequency collective excitations
in binary liquids. Based on a three-variable dynamical
model of mass-concentration fluctuations [13] and
two-variable dynamical model of transverse mass-con-
centration current fluctuations [4] it was revealed some
mechanisms of damping of optic-like excitations in liq-
uids. In comparison with the hydrodynamic relaxation
process of mutual diffusion the contribution from op-
tic-like excitations to the mass-concentration time auto-
correlation function contained a pre-factor k 2 and in
long-wavelength limit such a time correlation function
was in complete agreement with hydrodynamic expres-
sion [14].
However, the separated treatment of low- or high-fre-
quency branches in long-wavelength limit cannot explain
many features of dynamics of binary liquids, in particular
a crossover from «coherent» to «partial» types of collec-
tive dynamics by increasing of wavenumbers from hydro-
dynamic region towards the Gaussian regime. Therefore,
the main aim of this study is to propose a simple toy dy-
namical model, which would permit simultaneous analy-
sis of two branches of collective excitations. This allows
us to derive analytical expressions for dispersion laws, to
use them for the study of systems with different mass ratio
and composition, and to analyze on such a basis the cross-
over from «coherent» to «partial» types of dynamics in
dependence on mass ratio of components.
The paper is organized as follows: in the next Section
we discuss our choice of dynamical model and construct a
generalized kinetic matrix needed for subsequent calcula-
tions. In Sec. 3 both analytical and numerical results for
dispersion laws of two branches of propagating collective
excitations are presented and the mass-ratio and concen-
tration dependence of dispersion curves is discussed.
Conclusions of this study are collected in the last Section.
2. Elastic four-variable model for binary disordered
systems
2.1. General definitions
Collective dynamics of binary liquids in a wide range
of wavenumbers is much less studied that in the case of
pure single-component fluids. Partially this is connected
with a fact, that hydrodynamic expressions [15] cannot be
applied for the analysis of MD simulation results, which
clearly indicate the presence of two branches of collective
excitations [4,7]. In order to match hydrodynamic and
MD results Bosse and coauthors [9] proposed existence
of a «fast sound» excitations with a linear dispersion law
in long-wavelength limit, but with propagation speed in
several times higher than for the hydrodynamic acoustic
excitations. Here we are studying the collective propagat-
ing excitations in binary liquids within an approach of
generalized collective modes, based on an eigenmode
analysis of collective processes in a wide region of spatial
and temporal scales.
Let us consider a binary liquid as a mixture of N A par-
ticles with a mass mA and N B particles with mass mB ,
N N NA B� � , which are confined in a volume V and in-
teracting via two-body potentials��� � �( ) , , ,r A Bij � . In-
stantaneous positions and velocities of particles ri t, ( )�
and v i t, ( )� have additional species subindex �. Dynami-
cal variables of partial densities of particles, defined as
n k t
N
i ti
i
N
� �
�
( , ) exp ( ( )),�
�
�1
1
kr , � � A B, , (1)
are connected by continuity equations with longitudinal
components of partial mass-current densities:
dn k t
dt
ik
m
J k tL�
�
�
( , )
( , )� , � � A B, ,
where
J k t
N
m
k
i tL
i
N
i
i� �
�
�
�
( , ) ( ( ))
,
,�
�
�1
1
kv
krexp , � � A B, (2)
are the longitudinal components of partial mass-currents.
The hydrodynamic set of variables A
( )( , )4hyd k t for bi-
nary liquids contains along with total density, total longi-
tudinal current and heat density also an additional dynam-
ical variable n k tc ( , ) in comparison with the case of
simple fluids,
n k t c n k t c n k tc B A A B( , ) ( , ) ( , )� � , (3)
that describes the concentration fluctuations in the mix-
ture and is expressed via partial densities (1), so that
A
( )( , ) { ( , ), ( , ), ( , ), ( , )}4hyd k t n k t n k t J k t h k tt c t
L� . (4)
Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1037
Analytical expressions for time correlation functions
and dynamical structure factors of binary liquids valid in
the hydrodynamic limit were obtained in [15,16]. An im-
portant feature of binary liquids in hydrodynamic limit is
a purely relaxation behavior of time autocorrelation func-
tions of concentration density, that corresponds to ab-
sence of any side peaks on the shape of concentration dy-
namical structure factor S kcc ( , )� , i.e. hydrodynamic
approach points out the absence of effects of propagating
modes on concentration fluctuations.
A simplest extension of the hydrodynamic model for
binary liquids A
( )( , )4hyd k t within the GCM approach is a
seven-variable dynamical model (1), that contains among
basis variables the first time derivatives of hydrodynamic
variables:
A
( )( , )7 k t �
�{ ( , ), ( , ), ( , ), ( , ), ( , ), � ( ,n k t n k t J k t h k t J k t J kt c t
L
c
L
t
L t h k t), �( , )}.
(5)
Note, that this model takes into account fast longitudinal
current fluctuations only via the first time derivative of
total current fluctuations. However, one can consider
more general dynamical model, that allows us to treat
both species on the same footing and therefore it requires
the same order of time derivatives of partial currents
A
( )( , ) { ( , ), ( , ), ( , ), ( , ), ( ,8 k t n k t n k t J k t J k t h k tA B A
L
B
L� ),
� ( , ), � ( , ), �( , )}J k t J k t h k tA
L
B
L . (6)
If the coupling with the thermal fluctuations could be
neglected one derives from the eight-variable dynamical
model (6) a viscoelastic model [5] of binary liquids
A
( )( , )6 k t �
� { ( , ), ( , ), ( , ), ( , ), � ( , ), �n k t n k t J k t J k t J k t JA B A
L
B
L
A
L
B
L ( , )}k t ,
(7)
in framework of which one can correctly treat cross-cor-
relations between partial dynamical variables.
2.2. Elastic approximation
The eight-variable generalized hydrodynamic (6) and
six-variable viscoelastic (7) dynamical models are still
too complicate for an analytical analysis. That is why we
consider a simplified model of collective dynamics in a
binary liquid, that can be called «elastic» one because it
does not take into account explicitly the slow thermal and
mutual diffusion processes in liquid, however micro-
scopic quantities connected with the forces acting on par-
ticles (and therefore reflecting elastic properties) are pre-
sent in this model. In this case the basis set of dynamical
variables for longitudinal dynamics includes four vari-
ables
A
( ) . .
( , ) ( , ), ( , ), � ( , ), � ( , )4 k t J k t J k t J k t J k tL L L L�
� � � �
�
�
�
,
(8)
where the pair of indexes� �, corresponds to two orthogo-
nal currents, so that � � �J J� � 0. In particular, it is conve-
nient to consider as such pairs of orthogonal currents the
partial mass currents
J v k� � � �
�
�
�
�1
1
N
m t i r t
i
N
i i, ,( ) exp ( ( )), � � A B, , (9)
or the linear combinations of partial currents that describe
the total mass J t k t( , ) and mass-concentration J x k t( , )
currents and are simply related with the partial currents
via relation
J
J
J
J
x
t
B A A
B
x x�
�
��
�
�
�� �
��
�
�
�
�
�
�
�
��
�
�
��1 1
, (10)
where x m N M� � �� / is a mass-concentration, and
M m N m N mNA A B B� � � . Since x xA B� �1, the deter-
minant of transformation matrix to new variables {J t , J x}
is equal to unity.
The basis set of four variables (8) contains only the dy-
namical variables connected with faster processes, which
are defined by velocities and accelerations of particles,
that makes close analogy with the treatment of phonon ex-
citations in solids, where acceleration of particles is de-
fined by effective elastic interactions.
Let us construct a generalized kinetic matrix T
( )( )4 k ,
constructed on the basis set A
( )( , )4 k t for particular pairs
of currents {A B, } and {T X, } introduced above. The ma-
trices of static correlation functions F( , )k t � 0 and zeroth
Laplace-component of time correlation functions
~
( , )F k z � 0 have the following form:
F( , )
� � � �
� �
k t
f
f
f f
f
J J
J J
J J J J
J J
� �0
0 0 0
0 0 0
0 0
0 0
� �
� �
� � � �
� �
f J J� �
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
(11)
and
~
( , )F k z
f
f
f
f
J J
J J
J J
J J
� �
�
�
�
�
�
�
�
�
0
0 0 0
0 0 0
0 0 0
0 0 0
� �
� �
� �
� �
�
�
�
�
�
�
�
�
. (12)
By the definition [18,19] the generalized kinetic matrix
can be obtained via expression
T F F
( )( ) ( , )
~
( , )4 10 0k k t k z� � �� ,
so that one has
1038 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9
T. Bryk and I.M. Mryglod
T
( )( )
( ) ( )
( ) ( )
4
31 32
41 42
0 0 1 0
0 0 0 1
0 0
0 0
k
T k T k
T k T k
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
, (13)
with the matrix elements
T k
f
f
J J
J J
31( )
� �
� � �
� �
, T k
f
f
J J
J J
32( )
� �
� � �
� �
, T k
f
f
J J
J J
41( )
� �
� � �
� �
,
T k
f
f
J J
J J
42( )
� �
� � �
� �
.
The structure of T
( )( )4 k is rather simple for an analytical
treatment.
3. Results and discussion
3.1. Dispersion laws
For the generalized kinetic matrix (13) it is straightfor-
ward to obtain the dynamical eigenmodes, which can
propagate in the system:
z i T T T T T T1
0
31 42 31 42
2
32 41
1
21
2
1
2
4� � � � � ��
��
�
��
�( ) ( )
� � i k�1
0( ),
z i T T T T T T2
0
31 42 31 42
2
32 41
1
21
2
1
2
4� � � � � ��
��
�
��
�( ) ( )
� � i k�2
0( ) .
(14)
Note that these eigenvalues are purely imaginary, because
all the dissipation mechanisms are neglected in the adopt-
ed elastic approximation. For instance, nonzero damping
coefficients (real parts of eigenvalues) can appear if one
includes in addition the coupling of these «bare» propa-
gating modes with slow relaxation processes in liquids,
being connected with structural disorder.
The expressions (14) describe the dispersion of two
branches of collective excitations and can be used for
both cases of orthogonal currents considered, namely for
the partial { , }A B and total with mass-concentration
{T X, }. It is also important that in the whole range of
wavenumbers such a «toy» dynamical model takes into
account in appropriate way the effects of cross-correla-
tions between two propagating processes. Let us consider
now in more detail the k-dependence of frequencies (14).
To do so we need to know the explicit dependence of the
matrix elements T k31( ), T k32( ), T k41( ), T k42( ) on wave-
number k. By the definition for small k one has
� �
� �
�
� �
|
J J
J J
c kt
L
t
L
t
L
t
L k 0
2 2
� , (15)
where c is a high-frequency sound velocity. The static
cross-correlations � �� �J Jt
L
x
L are also functions of k 2 in
the limit k � 0, while the matrix element T k42( ) defined
on the pair of orthogonal currents {T X, }, tends [20] in
long-wavelength limit to a nonzero value �0
2 that has a
sense of square of «bare» frequency for optic-like excita-
tions [13]. Let us look at the behavior of eigenvalues
z k1( ) and z k2( ) when k � 0 retaining under the square
root terms within the precision O k( )2 . One gets for small
k the expressions:
z ic k1
0
� � , z i2
0
0� � � . (16)
This means that in the hydrodynamic limit the elastic ap-
proximation leads to two propagating collective modes
with different dispersion laws, namely: one branch of col-
lective excitations has the linear dispersion law with a co-
efficient being the high-frequency (elastic) speed of
sound and the second branch describes the propagating
optic-like modes with finite frequency.
In the opposite limit k � the cross-correlations be-
tween the partial currents in different species can be ne-
glected and one has from (14) the following solutions:
z i
J J
J J
i kA
L
A
L
A
L
A
L1
0
1 2
1
0� �
� �
� �
�
�
�
�
�
�
�
�
� �
� �
( )
/
� ,
z i
J J
J J
i kB
L
B
L
B
L
B
L2
0
1 2
2
0� �
� �
� �
�
�
�
�
�
�
�
�
� �
� �
( )
/
� ,
(17)
i.e. two branches in the limit k � reflect the dynamics
of non-interacting partial densities. One can estimate the
mutual location of two branches in the limit k � . From
expressions for ratio of fourth and second frequency mo-
ments [1,20] it follows, that in this limit
� �
� �
� �J J
J J
k T
m
ki
L
i
L
i
L
i
L
B
i
�
3 2, i A B� , . (18)
so that the ratio of «bare» frequencies is given by
�
�
1
0
2
0
1 2
( )
( )
/
k
k
m
m
B
A
�
�
�
�
�
�
� , (19)
and is the same as the ratio of phonon frequencies of optic
and acoustic branches at the first Brillouin zone boundary
in binary A–B crystals within a harmonic approximation.
The high-frequency branch corresponds to a light parti-
cles in the liquid mixture and low-frequency one to the
heavy particles.
The elastic approximation has several advantages.
First of all, it allows us to estimate the role of coupling to
Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1039
the relaxation processes in different regions of wave-
numbers. Second, the analytical expressions obtained are
valid in the whole region of wavenumbers, i.e. the «bare»
frequencies (14) include already the hybridization effects
between the both branches. Third, as a toy model, the
elastic approximation can be applied for understanding
the tendencies of dispersion law formation in binary sys-
tems with different mass ratio of particles in both species.
Now the question arises how good is the elastic ap-
proximation in comparison with a complete picture of the
collective dynamics of a binary liquid. In Fig. 1 the re-
sults of our study, performed within the generalized col-
lective mode approach with the set of eight-variable (6),
a r e p r e s e n t e d f o r mo l t e n N a I a t T = 1 0 8 0 K .
Eigenfrequencies, calculated within the full treatment of
coupling with thermal fluctuations and other slow relax-
ation processes [21], are shown by symbols «+» con-
nected by lines. Open boxes correspond to the actual ana-
lytic elastic approximation. One can see in Fig. 1 that in
the whole range of wavenumbers considered the frequen-
cies, obtained within the elastic approximation, have a lit-
tle bit higher frequencies comparing to the result of full
treatment, but correctly describe all the main features in
the dispersion laws. This difference can be easily ex-
plained and is mainly caused by the coupling of «bare»
propagating modes with relaxation processes that results
usually to appearance of nonzero damping and reduction
of the «bare» frequencies. It is also seen in Fig. 1 that in
the region k ! 2 2. �
�1 both branches behave almost lin-
early with k, and the ratio of their slopes for k � can be
evaluated as 2.57, while the square root of mass ratio
m mI Na/ � 2.35, that supports the analytical result (19).
3.2. Dispersion laws: dependence on mass ratio and
concentration
Let us consider the dependence of dispersion laws for
both branches of propagating collective excitations on
mass ratio of particles at constant mass-density of the sys-
tem. In Fig. 2 we show results obtained for the dispersion
laws of «bare» collective modes (14) for five systems
with identical static properties. A single distinct parame-
ter, used in our calculations, was the mass ratio R that is
responsible for solely dynamic response of the system.
Lennard–Jones systems with different R were sampled in
our previous molecular dynamics study [22], here we use
only relevant static averages T kij ( ), calculated directly in
molecular dynamics simulations. Our task now is to use
the analytical model developed for the explanation of the
general tendencies in spectra behavior when the mass ra-
tio R is changed.
For the short-wavelength limit we can immediately
use the expressions (18) and (19), showing that the fre-
quencies are inversely proportional to the masses and the
ratio of frequencies scales as R �1 2/ , where R m mh l� / ,
and mh , ml are masses of heavy and light atoms, respec-
tively.
In the long-wavelength region one can see in Fig. 3,
that the square of «bare» frequency of optic-like excita-
tions �
J Jx x
k2 ( ) is a linear function of R, while the square
of «bare» frequency of acoustic-like excitations �
J Jt t
k2 ( )
is independent on R, because it depends only on total den-
sity of the system. Now, let us look at the cross-correla-
tion effects between low- and high-frequency branches in
long-wavelength region as a function of R. Such a
cross-correlation is reflected by the ratio
�
J J
x t
x x
x t
J k J k
J k J k
2 �
� � �
� � �
� ( ) � ( )
( ) ( )
,
1040 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9
T. Bryk and I.M. Mryglod
5
10
15
20
25
30
35
40
45
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
� j
0
0 A
(8)
Im
z
(k
),
z
(k
),
p
s
j
j
–
1
k, �
–1
Fig. 1. Imaginary parts (dispersion) of propagating eigenmo-
des, obtained by the 8-variable treatment of collective dyna-
mics in molten NaI at T = 1080 K (symbols «+» connected by
interpolation line) and as analytical solutions (14) within the
«elastic» four-variable model (open boxes).
5
10
15
20
25
30
35
40
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
R=2.09
4.65
8.63
12.4
17.1
0
Im
z
(k
),
p
j
s–
1
k,
–1
�
Fig. 2. Frequencies of «bare» propagating collective modes in
dependence on the mass ratio R for Lennard–Jones equimolar
systems with identical mass-density [22]. The dispersion lines,
corresponding to the same mass ratio, are shown by the same
line-connected symbols.
which is shown in the lower frame of Fig. 3 and decrease
linearly with increasing of R. Since�J Jt t
k2 ( ) and�J Jx t
k2 ( )
are functions of k 2 and �J Jx x
k2 ( ) tends to a constant in
small k limit, one can rewrite the expressions (14) for
«bare» frequencies as follows
z k i k
k k
k
J J
J J J J
J J
t t
t x x t
x x
1
0 2
2 2
2
( ) ( )
( ) ( )
( )
" � �
�
�
�
�
�
� �
�
�
�
�
�
1 2/
,
z k i k
k k
k
J J
J J J J
J J
x x
t x x t
x x
2
0 2
2 2
2
( ) ( )
( ) ( )
( )
" � �
�
�
�
�
�
� �
�
�
�
�
�
1 2/
.
(20)
It is seen, that due to the coupling with acoustic branch
the square of «bare» frequency of optic-like modes gets
positive correction ~k 4 , which is proportional to the mass
ratio R. Similar correction, but with opposite sign gets the
dispersion of acoustic-like branch. Hence, one may ex-
pect, that just beyond the hydrodynamic region in the sys-
tems with large R the high-frequency branch should have
a «positive dispersion», while for the low-frequency
branch a «negative dispersion» has to be observed. In Fig.
2 the dispersion curves, calculated for Lennard–Jones
systems with identical static properties but different mass
ratio R, are shown. At small wavenumbers one observes
an increase of «bare» frequencies of optic excitations. It
is seen also that the effects of «positive dispersion» be-
come more pronounced in this case as it was predicted
above. The low-frequency branch in complete agreement
with the analytical treatment displays «negative disper-
sion» effects. For large wavenumbers the distance be-
tween two branches scales as R1 2/ in full agreement with
our predictions. Note that the low-frequency branch
shows a convergence tendency with R due to a constraint
on the constant mass density of systems considered.
About the effect of concentration ch of heavy particles
on the «bare» frequencies one can conclude from Fig. 4.
It is clearly observed a gradually shifting down of the
high-frequency branch when the concentration ch de-
creases. Moreover, the shape of this branch becomes
more similar to the k-dependence of acoustic-like branch
in small k domain.
3.3. Crossover from «coherent» to «partial» type
of dynamics in binary liquids
There is still a lack in simple analytical theory, de-
scribing a crossover from «coherent» to «partial» dynam-
ics in binary disordered systems [3,4], in particular in de-
pendence on mass ratio R. Since our «toy» dynamical
model is rather simple and quite correct in description of
dispersion of two branches in the whole region of wave-
numbers, let us consider the contributions from both
«bare» excitations to different current-current time corre-
lation functions within the model proposed.
Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1041
0
1000
2000
3000
4000
5000
J Jx x
–800
–700
–600
–500
–400
–300
–200
–100
0
2 4 6 8 10 12 14 16 18 20
J Jt t
J Jx t
Mass ratio m /mh l
�
#
Jx
Jx
2
,
2
2
2
�
#
Jt
Jt
�
#
Jx
Jt
2
2
Fig. 3. Dependence of matrix elements of generalized kinetic
matrix on the mass ratio for small wavenumbers. The mass ra-
tio values were sampled for the same systems as in previous
Figure.
5
10
15
20
25
30
35
40
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
ch= 0.50
ch= 0.25
ch= 0.20
ch= 0.10
k, �
–1
Im
z
(k
),
p
j
s–
1
Fig. 4. Frequencies of «bare» propagating collective modes in
dependence on the composition for Lennard–Jones systems
with the same density for the mass ratio R � 17.1. Dispersion
lines, corresponding to the same concentration, are shown by
the same line-connected symbols. Labels ch denote the molar
concentration of heavy component.
Since the «bare» excitations represent free oscillations
in the system, it seems that their contributions to different
spectral functions do not have much sense, because the
damping effects are not taken into account. In the
time-domain the corresponding diagonal current-current
time correlation functions, found analytically within the
elastic dynamical model for the set A
( )( , )4 k t , have the
form:
F k t F kJ J J J� � � �
( , ) / ( , )0 �
� �B k t B k t�� ��� �1
1
0 2
2
0( ) cos { } ( ) cos { } . (21)
The pre-factors or mode amplitudes B k
j
��( ), describing
the harmonic contributions, reflect all the hybridization
effects between two branches of «bare» excitations and
can be very helpful for understanding of crossover be-
tween «coherent» and «partial» regions of collective dy-
namics in real binary liquid systems [3,4]. These ampli-
tudes correspond to solely symmetric contributions,
defined in Ref. 23, and are given by the expressions:
B k
T k
B k
T
��
��
��
���
� �
1 2
0 2
31
2
0 2
1
0 2
2 31( )
( ) ( )
( ) ( )
, ( )�
�
�
�
( ) ( )
( ) ( )
k �
�
�
� �
1
0 2
2
0 2
1
0 2
,
(22)
so that the sum of both amplitudes is equal to unity as it
should be. The upper indices in the matrix elements
T k31
�� ( ) correspond to the different choices of currents
from the sets {A B, } or {T X, }, used previously for estima-
tion of the matrix element. For the case of a KrAr Len-
nard–Jones fluid with the mass ratio R � 2.09 the cal-
culated values of both amplitudes, describing the mode
contributions to the spectral functions C ktt ( , )� and
C kArAr ( , )� , are shown in Figs. 5 and 6, respectively. One
can see in Fig. 5 that in the long-wavelength region the
contribution to the total current autocorrelation function
comes completely from the low-frequency acoustic-like
branch — its mode amplitude B ktt
l ( ) is almost equal to
unity, while the contribution from the high-frequency
branch is very small. For larger k the contributions from
both branches become comparable (see Fig. 5). The same
tendency was also observed for the mass-current auto-
correlation function with the only difference, that in the
long-wavelength region the high-frequency branch deter-
mines almost completely its shape. From the other side,
an opposite situation is observed in the case of partial cur-
rents autocorrelation functions. For instance, as it is seen
in Fig. 6, for the light subsystem (Ar particles) at
k ! 0 5. �
–1
this function can be reasonably described by
the contribution from high-frequency branch only, and
for k ! 2 5. �
–1
this is exactly correct. Hence, the cross-
over from «coherent» to «partial» type of dynamics in bi-
nary disordered systems can be explained in terms of the
mode contributions from «bare» eigenmodes of our «toy»
elastic model. In order to rationalize the mass ratio of
components affects the region of this crossover we show
in Fig. 7 the amplitudes of all contributions B kj
�� ( ) with
� � t x A B, , , calculated at two values of mass ratio R �
� 2.09 (solid lines) and R � 17.1 (dashed lines). It is seen
that for the larger mass ratio the cross-point of amplitudes
B k B ktt
h
AA
h( ) ( )cross cross� , where A denotes a light com-
ponent of the mixture, is at the same wavenumber as the
cross-point of the amplitudes B k B ktt
l
BB
l( ) ( )cross cross�
and in this case the value kcross is smaller, than similar
cross-point for the mass ratio R � 2.09. For k k! cross one
can accept, that the «partial» type of collective dynamics
prevails, while for k k$ cross collective dynamics can be
well decsribed in terms of acoustic- and optic-like «bare»
collective excitations, representing the «coherent» type
of dynamics [3,24]. Note also that for lager k, as it was
shown above, the main mechanism responsible for mode
formation is connected with the properties of partial cur-
rents (see (17)). Hence, we may use the condition for a
1042 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9
T. Bryk and I.M. Mryglod
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
M
o
d
e
st
re
n
g
th
s
B
(k
),
tt
B
(k
)
tt
low
high
1
2
k,
–1
�
Fig. 5. Mode contributions from the low- and high-frequency
branches of «bare» propagating collective modes to the spec-
tral function C ktt ( , )� .
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
M
o
d
e
st
re
n
g
th
s
B
(k
),
A
rA
r
B
(k
)
A
rA
r low
high2
1
k,
–1
�
Fig. 6. Mode contributions from the low- and high-frequency
branches of «bare» propagating collective modes to the spec-
tral function C kArAr( , )� .
cross-point of corresponding mode amplitudes and to de-
rive an analytical expression for kcross taking into account
the long-wavelength asymptotes of relevant frequency
moments. One gets
k
c x a
J J
tx
x x
cross �
�
�2
2
1
0
2
( )
, (23)
where x1 is the mass-concentration of heavy particles and
atx is a constant taken from long-wavelength asymptote
of �J J txx t
k a k2 2( ) � . We have calculated the dependence
of kcross on the mass ratio, and the results are shown in
Fig. 8. As it follows from Fig. 8, one can conclude that the
region of «coherent» dynamics reduces with the increas-
ing mass ratio R. Note, however, that kcross tends to a non-
zero value even in the limit R� , because of the con-
straint put on the total mass density of the systems
considered in our study.
4. Conclusions
Within the generalized collective mode approach, for
description of dispersion of propagating collective exci-
tations in binary disordered systems, we have proposed
and solved analytically in the whole range of wavenum-
bers a new four-variable «toy» dynamical model. The
most important results obtained are the following:
(i) we have obtained analytical expressions for the
«bare» frequencies of propagating collective excitations
in binary systems that can be applied in the whole range
of wavenumbers;
(ii) short-wavelength asymptote of the ratio of «bare»
frequencies scales as square root of mass ratio, similarly
as it is on the Brillouin zone boundary in a binary A–B
crystal. In small k limit the «bare» low-frequency eigen-
values follow linear dispersion law c k with a coefficient
being high-frequency (elastic) sound velocity c , while
the high-frequency branch tends to a nonzero frequency
�0 in complete analogy with optic-like phonon excita-
tions in solids;
(iii) the proposed model allows us to describe the
crossover from «coherent» to «partial» dynamics in bi-
nary liquids in terms of mode contributions to different
current autocorrelation functions;
(iv) it is shown that the crossover region between the
«coherent» and «partial» types of dynamics reduces when
the mass ratio R of particles in different species increases.
This supports, in particular, our recent numerical re-
sults [22].
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Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1043
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8C
o
n
tr
ib
u
ti
o
n
s
B
(k
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B
(k
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B
(k
),
B
(k
)
tt
x
x
A
A
B
B
R=2.09
R=17.1
j
j
j
j
k,
–1
�
Fig. 7. Crossover region between the «coherent» and «partial»
dynamics for two values of the mass ratio.
0.25
0.30
0.35
0.40
0.45
0.50
2 4 6 8 10 12 14 16 18 20
Mass ratio R
k
,
cr
o
ss
�
–
1
Fig. 8. Dependence of crossover region kcross on the mass ra-
tio R.
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1044 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9
T. Bryk and I.M. Mryglod
|
| id | nasplib_isofts_kiev_ua-123456789-120937 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T15:52:04Z |
| publishDate | 2007 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Bryk, T. Mryglod, I.M. 2017-06-13T10:42:30Z 2017-06-13T10:42:30Z 2007 Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures / T. Bryk, I.M. Mryglod // Физика низких температур. — 2007. — Т. 33, № 09. — С. 1036–1044. — Бібліогр.: 24 назв. — англ. 0132-6414 PACS: 05.20.Jj, 61.20.Lc, 61.25.Mv https://nasplib.isofts.kiev.ua/handle/123456789/120937 We propose a new «toy» dynamical model that permits us to derive analytical expressions for dispersion of two branches of «bare» propagating collective excitations in binary disordered systems in the whole range of wavenumbers. These expressions are used for the analysis of dependence of dispersion curves on mass ratio and concentration at fixed density of the system. An effect of hybridization of two branches is discussed in terms of mode contributions to time correlation functions. This allows us to estimate the regions with dominant types of coherent or partial dynamics. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures Article published earlier |
| spellingShingle | Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures Bryk, T. Mryglod, I.M. International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" |
| title | Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures |
| title_full | Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures |
| title_fullStr | Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures |
| title_full_unstemmed | Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures |
| title_short | Collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures |
| title_sort | collective excitations in dynamics of liquids: a «toy» dynamical model for binary mixtures |
| topic | International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" |
| topic_facet | International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120937 |
| work_keys_str_mv | AT brykt collectiveexcitationsindynamicsofliquidsatoydynamicalmodelforbinarymixtures AT mryglodim collectiveexcitationsindynamicsofliquidsatoydynamicalmodelforbinarymixtures |