Structural relaxation in pure liquids: Analysis of wavenumber dependence within the approach of generalized collective modes
Wavenumber dependence of structural relaxation in liquids is studied by the method of generalized collective modes (GCM). A new perturbation approach within the GCM method is proposed and applied in the longwavelenth limit in order to obtain analytical expressions for the wavenumber-dependent stru...
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| Zitieren: | Structural relaxation in pure liquids: Analysis of wavenumber dependence within the approach of generalized collective modes / T. Bryk, I. Mryglod // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 139-154. — Бібліогр.: 26 назв. — англ. |
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irk-123456789-1190032017-06-04T03:03:59Z Structural relaxation in pure liquids: Analysis of wavenumber dependence within the approach of generalized collective modes Bryk, T. Mryglod, I. Wavenumber dependence of structural relaxation in liquids is studied by the method of generalized collective modes (GCM). A new perturbation approach within the GCM method is proposed and applied in the longwavelenth limit in order to obtain analytical expressions for the wavenumber-dependent structural relaxation and sound dispersion within a simplified three-variable dynamical model. Analytical results are compared with numerical study of generalized modes within a more general five-variable dynamical model, which accounts for thermal processes in liquids. Numerical results are presented for four thermodynamic points of Lennard- Jones fluid at the reduced temperature T* = 1.71. We discuss the features of non-hydrodynamic process of structural relaxation in different regions of wavenumbers. Залежнiсть вiд хвильового числа для структурної релаксацiї в рiдинах дослiджується методом узагальнених колективних мод (УКМ). Запропоновано новий пертурбативний пiдхiд в рамках методу УКМ, що застосовано для знаходження залежної вiд хвильового числа структурної релаксацiї та дисперсiї звуку у довгохвильовiй границi в рамках спрощеної тризмiнної динамiчної моделi. Аналiтичнi результати порiвнюються iз числовими дослiдженнями узагальнених мод в рамках бiльш загальної п’ятизмiнної динамiчної моделi, що враховує термiчнi процеси в рiдинах. Представлено числовi результати для чотирьох термодинамiчних точок ленард-джонсiвського плину з приведеною температурою T* = 1.71. Обговорюються особливостi негiдродинамiчного процесу структурної релаксацiї у рiзних областях хвильових чисел. 2008 Article Structural relaxation in pure liquids: Analysis of wavenumber dependence within the approach of generalized collective modes / T. Bryk, I. Mryglod // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 139-154. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 05.20.Jj, 61.20.Lc, 62.60.+v DOI:10.5488/CMP.11.1.139 http://dspace.nbuv.gov.ua/handle/123456789/119003 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Wavenumber dependence of structural relaxation in liquids is studied by the method of generalized collective
modes (GCM). A new perturbation approach within the GCM method is proposed and applied in the longwavelenth
limit in order to obtain analytical expressions for the wavenumber-dependent structural relaxation
and sound dispersion within a simplified three-variable dynamical model. Analytical results are compared with
numerical study of generalized modes within a more general five-variable dynamical model, which accounts
for thermal processes in liquids. Numerical results are presented for four thermodynamic points of Lennard-
Jones fluid at the reduced temperature T* = 1.71. We discuss the features of non-hydrodynamic process of
structural relaxation in different regions of wavenumbers. |
| format |
Article |
| author |
Bryk, T. Mryglod, I. |
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Bryk, T. Mryglod, I. Structural relaxation in pure liquids: Analysis of wavenumber dependence within the approach of generalized collective modes Condensed Matter Physics |
| author_facet |
Bryk, T. Mryglod, I. |
| author_sort |
Bryk, T. |
| title |
Structural relaxation in pure liquids: Analysis of wavenumber dependence within the approach of generalized collective modes |
| title_short |
Structural relaxation in pure liquids: Analysis of wavenumber dependence within the approach of generalized collective modes |
| title_full |
Structural relaxation in pure liquids: Analysis of wavenumber dependence within the approach of generalized collective modes |
| title_fullStr |
Structural relaxation in pure liquids: Analysis of wavenumber dependence within the approach of generalized collective modes |
| title_full_unstemmed |
Structural relaxation in pure liquids: Analysis of wavenumber dependence within the approach of generalized collective modes |
| title_sort |
structural relaxation in pure liquids: analysis of wavenumber dependence within the approach of generalized collective modes |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119003 |
| citation_txt |
Structural relaxation in pure liquids: Analysis of wavenumber dependence within the approach of generalized collective modes / T. Bryk, I. Mryglod // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 139-154. — Бібліогр.: 26 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT brykt structuralrelaxationinpureliquidsanalysisofwavenumberdependencewithintheapproachofgeneralizedcollectivemodes AT mryglodi structuralrelaxationinpureliquidsanalysisofwavenumberdependencewithintheapproachofgeneralizedcollectivemodes |
| first_indexed |
2025-07-08T15:04:16Z |
| last_indexed |
2025-07-08T15:04:16Z |
| _version_ |
1837091588063363072 |
| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 1(53), pp. 139–154
Structural relaxation in pure liquids: Analysis of
wavenumber dependence within the approach of
generalized collective modes
T.Bryk1,2, I.Mryglod1,2
1 Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 Institute of Applied Mathematics and Fundamental Sciences
Lviv Polytechnic National University, 12 Bandery Str., 79013 Lviv, Ukraine
Received January 4, 2008, in final form February 28, 2008
Wavenumber dependence of structural relaxation in liquids is studied by the method of generalized collective
modes (GCM). A new perturbation approach within the GCM method is proposed and applied in the long-
wavelenth limit in order to obtain analytical expressions for the wavenumber-dependent structural relaxation
and sound dispersion within a simplified three-variable dynamical model. Analytical results are compared with
numerical study of generalized modes within a more general five-variable dynamical model, which accounts
for thermal processes in liquids. Numerical results are presented for four thermodynamic points of Lennard-
Jones fluid at the reduced temperature T
∗
= 1.71. We discuss the features of non-hydrodynamic process of
structural relaxation in different regions of wavenumbers.
Key words: generalized hydrodynamics, structural relaxation, dynamical models, sound excitations, positive
dispersion
PACS: 05.20.Jj, 61.20.Lc, 62.60.+v
1. Introduction
Collective dynamics in liquids is so far well understood mainly on macroscopic length and time
scales, where the hydrodynamic theory (i.e. local conservation laws) is applicable [1,2]. On the spa-
tial scales comparable with mean interatomic distance the microscopic dynamics in liquids is essen-
tially more rich than in hydrodynamic region: there exist non-hydrodynamic collective processes
with finite lifetime, which contribute to all measurable quantities. Examples of non-hydrodynamic
propagating processes are shear waves and heat waves (zero-sound), charge waves in ionic liquids,
while most obvious non-hydrodynamic relaxation process is the structural relaxation. In contrast
to hydrodynamic processes the non-hydrodynamic ones do not survive on macroscopic distances
and times due to finite lifetime or inability of liquids to support the specific propagating process
on macroscopic length scales as this is in case of shear waves. The non-hydrodynamic processes,
sometimes yet called as kinetic ones [3], should have wavenumber dependence of their specific li-
fetimes τi(k), as well as frequencies ωi(k) for propagating kinetic modes, essentially different from
hydrodynamic ones.
The standard inverse lifetimes for collective processes obtained within the hydrodynamic de-
scription are proportional to k2. For pure liquids there exist two hydrodynamic collective relax-
ation processes: thermal and viscous relaxations with inverse lifetimes τ−1
th (k) = λk2/ncV and
τ−1
v (k) = ηk2/ρ, respectively, with λ, η, cV being macroscopic thermal conductivity, shear vis-
cosity and specific heat at constant volume. Herein ρ and n denote mass and numerical densities
of the system, respectively. The single hydrodynamic propagating process, which corresponds to
longitudinal acoustic excitations, has the inverse lifetime Γk2, which essentially is dependent on a
c© T.Bryk, I.Mryglod 139
T.Bryk, I.Mryglod
coupling between viscous and thermal processes. Here the sound attenuation coefficient
Γ =
1
2
[DL + (γ − 1)DT] ,
where the ratio of specific heats γ = cP/cV is a measure of such coupling, DL = (4
3η + ζ)/ρ
is kinematic viscosity, ζ – bulk viscosity, and DT = λ/ncP is thermal diffusivity. For the case
of ν-component liquids there appear in addition other ν − 1 hydrodynamic relaxation processes
connected with mutual diffusivity of species. The inverse lifetimes of hydrodynamic relaxation
processes are connected with relevant transport coefficients. Another important point is that within
the hydrodynamic description the relaxation processes enter to the density-density, density-energy
and energy-energy time correlation functions as exponentially decaying with time terms, that
simply implies definition of their specific lifetimes τi(k) [4].
Non-hydrodynamic processes contribute to all physical quantities, which can be measured on
relevant spatial and temporal scales. Among them are the wavenumber- and frequency-dependent
spectral functions I(k, ω) obtained in INS or IXS experiments, or time correlation functions
Fij(k, t) calculated in molecular dynamics simulations. Outside the hydrodynamic region the non-
hydrodynamic processes should essentially affect the shape of time correlation functions, and there
appears a problem how to extract in a correct way the contributions of non-hydrodynamic pro-
cesses from the calculated Fij(k, t) or measured intensities I(k, ω). There were many attempts to
take into account the existence of non-hydrodynamic processes in fluids for analysis of scattering
experiments [5–7]. But still the issue of non-hydrodynamic processes and their effect on the dis-
persion of the acoustic excitations is among the unsolved problems of the collective dynamics in
liquids [8]. Perhaps the most obvious relaxation process, not reflected in hydrodynamic approach,
is the structural relaxation, which is related to cooperative rearrangements of the local structure
due to density fluctuations. Although the structural relaxation is one of the most studied relax-
ation processes for supercooled liquids and at the glass transition, its role and contribution to
the time correlation functions in the liquid state is not well understood. Moreover, little is known
about the wavenumber dependence of structural relaxation in liquids. One of the first studies of
the wavenumber dependence of structural relaxation in liquids [9] resulted in conclusion, that in
the long-wavelength limit the lifetime of structural relaxation must tend to a finite value.
The mostly used approach in the studies of effects of structural relaxation in liquids was based
on a memory function formalism [2,10] and appeared to be quite successful in explaining the tem-
perature dependence of structural relaxation in liquids [11]. The approach itself was based on
assumption, that the second-order memory function can be separated into different decay chan-
nels, each of which corresponded to a specific collective process: thermal relaxation, structural
relaxation, instantaneous relaxation (for time decay of the memory function on time scales much
faster than the experimentally accessed frequency window), etc. Relaxation processes entered the
second-order memory function as exponentially decaying in time terms, while instantaneous decay
channel was represented via time delta-function [12,13]. Relaxation times of the decay channels
(corresponding factors in exponentials of the second-order memory function) were treated as fitting
parameters. The generalized in this way expression for the dynamical structure factor S(k, ω) after
convolution with a measured instrumental resolution function was used in fitting to experimental
scattering intensities in wide region of wavenumbers and resulted in k-dependent relaxation time
of structural relaxation. Such an approach resulted in very reasonable Arrhenius-type behaviour of
temperature dependence of extrapolated to long-wavelength limit of structural relaxation τstr(T )
[14,15]. However, it is not clear yet, whether the relaxation times associated with the decay channels
of the second-order memory function can be directly related in the whole range of wavenumbers to
relaxation times of physical processes. It was demonstrated in [16], that the hydrodynamic form
of dynamical structure factor can be reproduced by a suitable choice of two decay channels of
the second-order memory function. However, there does not exist a proof of correct wavenumber
dependence of relaxation time of structural relaxation, obtained within the fitting procedure with
additional decay channels in a wide region of wavenumbers.
There exists another theoretical method, known as the approach of Generalized Collective
Modes (GCM), which is closely related to the memory function formalism. However, the essential
140
Structural relaxation in pure liquids
difference of the GCM approach is treatment of non-hydrodynamic processes via extended set of
hydrodynamic variables and orthogonal to them short-time dynamical variables, which is applied
for solving of the generalized Langevin equation in terms of dynamical eigenmodes. The GCM
approach treats hydrodynamic and non-hydrodynamic processes on the same footing and takes
into account local coupling effects between them. All the relaxation processes – hydrodynamic
and kinetic ones - contribute directly to the shape of all time correlation functions of interest via
exponentially decaying in time terms. Hence, the GCM approach can shed light on the problem
of wavenumber dependence of structural relaxation in liquids, because it is straightforward by
analogy with hydrodynamic relaxation processes to associate particular exponential term (with
real eigenvalue) in the relevant time correlation function with the non-hydrodynamic process of
structural relaxation, and it is obvious that the corresponding real eigenvalue will define the inverse
wavenumber-dependent specific time of structural relaxation dstr(k) = τ−1
str (k).
The remaining paper is organized as follows. In the next Section we describe the approach of
Generalized Collective Modes for analysis of collective dynamics in liquids. In Section 3 analyt-
ical solutions for a simplified three-variable model are presented in order to estimate the long-
wavelength asymptote of relaxation mode connected with structural relaxation. Section 4 contains
results of MD simulations and numerical GCM analysis of relaxation processes in a wide range of
wavenumbers, and the last Section contains conclusions of this study.
2. The approach of generalized collective modes
The GCM approach consists in solving the generalized Langevin equation, which is represented
in matrix form generated on a chosen set of dynamical variables, in terms of eigenmodes and
their contributions to the matrix of time correlation functions. The most simple formulation of
the GCM approach is based on Markovian approximation for the highest-order memory functions,
although this is not the limitation and a more sophisticated version of the GCM approach is
available [17]. Usually the basis set of Nv dynamical variables is constructed by extension of
the set of hydrodynamic variables, which describe fluctuations of conserved quantities, by their
time derivatives, which being orthogonal to hydrodynamic slow fluctuations are aimed to describe
correctly more short-time fluctuations, which are not considered within hydrodynamic theory. For
pure liquids the hydrodynamic set of variables for description of longitudinal dynamics consists of
three dynamical variables:
A
(3hyd)(k, t) = {n(k, t), JL(k, t), e(k, t)} , (1)
where
n(k, t) =
1√
N
N
∑
i=1
eikri(t), (2)
is the k-th spatial-Fourier component of particle density of the system composed of N particles
with instantaneous positions ri(t) and velocities vi(t),
JL(k, t) =
m√
N
1
k
N
∑
i=1
kvie
ikri(t) (3)
is the Fourier-component of longitudinal part of mass-current density, and the energy density is
e(k, t) =
1√
N
N
∑
i=1
εie
ikri(t), (4)
where εi is the single-particle energy of i-th particle. The dynamical variables (2)–(4) correspond
to fluctuations of conserved quantities. One can restrict the treatment of dynamical processes
just by a set of equations with only dynamical variables of fluctuating conserved quantities and
141
T.Bryk, I.Mryglod
in Markovian approximation this results to expressions valid on macroscopic distances and time
scales only.
In order to account for non-hydrodynamic processes within the GCM approach one applies for
construction of all matrices an extended set of dynamical variables
A
(Nv)(k, t) = A
(3hyd)(k, t) ⊕
{
J̇L(k, t), ė(k, t), J̈L(k, t), ë(k, t), . . .
}
, (5)
where overdots mean the corresponding order of time derivatives. The chosen set of Nv dynamical
variables is used for construction of the matrix of static correlation functions F(k, t = 0) with
elements
Fij(k, t = 0) = 〈A∗
i (k, t = 0)Aj(k, t = 0)〉 , (6)
where the asterisk means complex conjugation, and matrix of corresponding Laplace-transformed
time correlation functions F̃(k, z). In the simplest Markovian approximation the matrix form of
generalized Langevin equation for Laplace-components of time correlation functions reads
F̃(k, z) = [zI + T(k)]
−1
F(k, t = 0) (7)
with T(k) called as generalized hydrodynamic matrix and expressed via
T(k) = −iΩ(k) + M̃(k, z = 0) ≡ F(k, t = 0)F̃−1(k, z = 0) . (8)
Here Ω(k) and M̃(k, z) are the standard frequency matrix and matrix of memory functions, re-
spectively. The number of successive time derivatives of some hydrodynamic variable in the basis
set defines the order of the highest memory function, which is taken in Markovian approximation.
We stress, that using a large number of successive time derivatives of hydrodynamic variables
(large number of dynamical variables Nv) one applies the Markovian approximation to the very
high-order memory function, that is quite justified, because the high-order memory functions are
rapidly decaying functions of time.
The generalized hydrodynamic matrix can be diagonalized using the solutions of eigenvalue
problem
Nv
∑
j=1
Tij(k)X
α
j (k) = zα(k)Xα
i (k) , (9)
where zα(k) and Xα
i (k) are the α-th eigenvalue and corresponding eigenvector, respectively. In
terms of the eigenvalues and eigenvectors the solution of generalized Langevin equation is repre-
sented as follows:
F̃ij(k, z) =
Nv
∑
α=1
Gα
ij(k)
z + zα(k)
, (10)
which transforming back to time dependence yields
Fij(k, t) =
Nv
∑
α=1
Gα
ij(k)e
−zα(k)t . (11)
The weight coefficients Gα
ij(k) are simply estimated from the relevant eigenvectors [18]. It is impor-
tant, that the time correlation functions are represented as a sum of Nv separated contributions
from different dynamical eigenmodes. The Nv eigenvalues zα(k) of the matrix T(k) correspond
either to propagating modes (complex-conjugated pairs of eigenvalues) or to relaxation processes
(real eigenvalues), which contribute with the strength Gα
ij(k) to the shape of relevant time correla-
tion function Fij(k, t) as oscillating or exponential functions of time, respectively. For convenience
we will further note the purely real eigenvalues as dα(k).
The number of successive time derivatives of hydrodynamic variables in the basis set defines
the order of frequency sum rules fulfilled for the solutions F̃ij(k, z). In general the presence of
s successive time derivatives of hydrodynamic variable Ai(k, t) in the basis set results in 2s + 1
142
Structural relaxation in pure liquids
frequency sum rules fulfilled for the theoretical representation (GCM replica) of relevant F̃ii(k, z).
Several additional advantages of the GCM approach over the traditional memory function method
are the following: (i) an additional sum rule which requires the equivalence of relevant correlation
time with some reference correlation time allows to avoid a priori modeling of memory functions.
The reference correlation time either in analytical form in hydrodynamic regime, or it can be
obtained numerically for a given system as a function of wavenumbers from MD simulations;
(ii) the time correlation functions (and relevant dynamical structure factors) are expressed as a
separable sum over Nv mode contributions from extended hydrodynamic and non-hydrodynamic
modes in the whole range of wavenumbers in analogy as this is in hydrodynamic theory. Each
collective excitation is characterized by a complex eigenvalue zα(k) and corresponding amplitude
of mode contribution Gα
ij(k) to the shape of dynamical structure factor Sij(k, ω).
3. Analytical treatment in long-wavelength region via pert urbation approach
One of the simplest dynamical models, which permits us to find analytical expressions and to
describe a relaxation kinetic process of non-thermal origin is a three-variable dynamical model
with the following dynamical variables:
A
(3)(k, t) =
{
n(k, t), JL(k, t), J̇L(k, t)
}
, (12)
where the first time derivative of longitudinal component of mass-current is
J̇L(k, t) =
1√
N
1
k
N
∑
i=1
[(kFi) + i(kvi)
2]eikri(t) . (13)
Here Fi(t) denotes the force acting on the i-th particle. The only non-hydrodynamic variable among
these three ones is J̇(k, t) and it is connected to Fourier-components of stress tensor σαβ(k, t) via
relation [5]:
d
dt
J(k, t) = ikσ̂(k, t) ,
i.e. in the generalized hydrodynamic description appear quantities from theory of elasticity.
It is quite straightforward to construct the 3× 3 generalized hydrodynamic matrix on the basis
set of non-orthogonal dynamical variables (12) and find its eigenvalues. However, to make the
results and origin of eigenmodes more transparent we will apply a perturbation approach recently
elaborated within the GCM method. The idea is to separate in the generalized hydrodynamic
matrix T(k) a part, T0(k), which permits simple analytical solution in terms of “bare” modes
[19], while the remaining part δT(k) contains local coupling effects between “bare” modes and is
treated as perturbation. If the perturbation δT(k) is essentially smaller than the distance between
“bare” modes, one can expect the rapid convergence of perturbation series.
First of all in order to simplify perturbation series one has to construct the generalized hydro-
dynamic matrix T(k) on orthonormal basis set of dynamical variables. Therefore we can transform
initial basis set (12) as follows:
ñk
J̃L
k
˜̇J
L
k
= L
nk
JL
k
J̇L
k
, (14)
where the matrix of linear transformation
L(k) =
1√
S(k)
0 0
0 1√
mkBT
0
−ik mcT√
ω2
l
(k)−k2c2
T
(k)
0 1√
ω2
l
(k)−k2c2
T
(k)
. (15)
143
T.Bryk, I.Mryglod
Here
c2T(k) =
kBT
mS(k)
is the square of isothermal velocity, and
ω2
l (k) =
〈J̇L
−kJ̇
L
k 〉
〈JL
−kJ
L
k 〉
k→0→ c2∞k
2 , (16)
where c∞ is the high-frequency speed of sound. Now the tilded variables in (14) satisfy the condition
〈ã−k b̃k〉 = δab .
The generalized hydrodynamic matrix T(k), constructed on the set of non-orthogonal variables can
be simply related via linear transformation with the matrix T̃(k), defined on the orthogonalized
variables
T̃(k) = LTL
−1 . (17)
Obviously the eigenvalues of both matrices T(k) and T̃(k) are the same. Hereby, one has to find
eigensolutions of the generalized hydrodynamic matrix
T̃(k) =
0 −ikcT 0
−ikcT 0 −
√
ω2
l (k) − k2c2T
0
√
ω2
l (k) − k2c2T (
ω2
l
(k)
k2c2
T
− 1) 1
τ(k)
, (18)
where the quantity τ(k) is the correlation time associated with the density-density correlations
without coupling to thermal processes. In general one does not know the wavenumber dependence
of τ(k) in the whole range of wavenumbers, therefore within the GCM approach one is required
to use molecular dynamics simulations in order to have numerical dependence of τ(k) as input
quantity. In case of dynamical models with explicit treatment of density and thermal fluctuations
one calculates the generalized correlation times via their definition
τij(k) =
1
Fij(k, t = 0)
∫ ∞
0
Fij(k, t)dt , (19)
where Fij(k, t) are the density-density, density-energy or energy-energy time correlation functions
calculated in MD simulations for given wavenumber k.
The density fluctuations in liquids on macroscopic scale are always constrained to local law
of energy conservation. The three-variable model (12) considered here, however, does not take
implicitly into account coupling between energy and density fluctuations, therefore the correlation
time τ(k) cannot be taken from MD simulations – the density-density time correlation functions for
any k value always have contribution from thermal processes, although for intermediate and large
wavenumbers that contribution is quite small and can be neglected in simplest approximations.
However, in analytical approaches it is possible to remove the contribution from thermal processes
– this is made by setting γ = 1. Therefore, we can use analytical expression for density-density time
correlation functions [4] in the long-wavelength limit and obtain the correlation time in viscoelastic
approximation:
τ(k)
k→0→ DL
c2T
. (20)
We stress, that in contrast to the hydrodynamic correlation times (19) with long-wavelength asymp-
totes k−2 the correlation time τ(k) in viscoelastic theory (i.e. idealized model without coupling to
thermal fluctuations) tends to a constant in k → 0 limit. This is because all three hydrodynamic
time correlation functions of longitudinal dynamics in pure liquids [4] have similar exponential
contributions of the same origin, which come from thermal relaxation.
144
Structural relaxation in pure liquids
We proceed further with separating the generalized hydrodynamic matrix T̃(k) into two parts:
T̃0(k) + δT̃(k). The first matrix
T̃0(k) =
0 −ikcT 0
−ikcT 0 0
0 0 (
ω2
l
k2c2
T
− 1) 1
τ
(21)
can be immediately analytically solved in terms of eigenvalues and eigenvectors in the whole region
of wavenumbers. Among the eigenvalues in the long-wavelength limit there are a single real number
d(0)(k) ≡ d0 =
c2∞ − c2T
c2T
1
τ
(22)
and a pair of conjugated imaginary numbers
z
(0)
± (k) = ±icTk . (23)
Note, that in this case purely imaginary eigenvalues correspond to propagating waves without any
damping, while purely real eigenvalues di(k) are associated with relaxation process with specific
relaxation time τi(k) = d−1
i (k). The eigenvalues (22) and (23) can be treated as “bare” modes,
which due to coupling between propagating and relaxation processes, considered further as a per-
turbation, will obtain k-dependent corrections. The eigenvectors associated with relevant “bare”
modes are:
ψ
(0)
d =
0
0
1
, ψ
(0)
± =
1√
2
∓1
1
0
. (24)
The first order corrections to both kinds of “bare” modes
δz
(1)
± (k) = ψ
(0)∗
± δT̃ψ
(0)
± = 0
and
δd(1)(k) = ψ
(0)∗
d δT̃ψ
(0)
d = 0
are equal to zero. Here the matrix of perturbation is
δT̃(k) =
0 0 0
0 0 −
√
ω2
l − k2c2T
0
√
ω2
l − k2c2T 0
, (25)
and in long-wavelength limit it is linear in k. The absence of first-order corrections to the eigenvalues
means, that there are no linear in k corrections to the “bare” modes. The first nonzero correction
to the “bare” modes appears in the second order with respect to perturbation. Thus by keeping
the leading in k term one has
δd(2)(k) =
∑
±
(ψ
(0)∗
d δT̃ψ
(0)
± )(ψ
(0)∗
± δT̃ψ
(0)
d )
d(0) − z
(0)
±
= −c2Ttk2 , (26)
and this correction is proportional to k2. Using (20) one obtains
d(k) =
c2∞ − c2T
DL
−DLk
2 . (27)
The negative sign of the second order correction means, that with increasing wavenumbers the
lifetime of kinetic relaxation process increases. The factor (c2∞ − c2T) is associated in the literature
145
T.Bryk, I.Mryglod
with the “strength” of structural relaxation [15], that implies the relevance of the real dynamical
eigenmode d(k) in (27) with the structural relaxation.
For propagating modes one obtains
δz
(2)
± (k) =
(ψ
(0)∗
± δT̃ψ
(0)
d )(ψ
(0)∗
d δT̃ψ
(0)
± )
z
(0)
± − d(0)
=
(c2∞ − c2T)k2(z
(0)
± + d(0))
c2Tk
2 + d2
0
, (28)
that results in
z±(k) =
DL
2
k2 ± icTk(1 +
DL
d0
k2) . (29)
It follows, that due to coupling with the relaxation process d(k) the “bare” propagating modes
obtain damping proportional to k2 and a correction to dispersion law, which can be treated as a
positive deviation from linear dispersion. This positive dispersion is proportional to the kinematic
viscosity and to the long-wavelength limit of structural relaxation time.
For eigenvectors the first-order corrections can be written as follows
δψ
(1)
d = −
√
ω2
l − k2c2T
d2
0 + k2c2T
−ikcT
d0
0
, δψ
(1)
± = − (d0 ± ikcT)
√
ω2
l − k2c2T√
2(d2
0 + k2c2T)
0
0
1
. (30)
Using the properly normalized eigenvectors within the first order in perturbation ψd and ψ± one
can estimate the contributions from relaxation and propagating processes to all time correlation
functions and corresponding spectral functions of interest. For the density-density time correlation
function the model (12) results in three contributions with purely real weight coefficients from
propagating modes Bij(k) and Dij(k):
Fnn(k, t)
S(k)
= Ann(k)e−d(k)t + (Bnn cosω(k)t+Dnn sinω(k)t)e−σ(k)t , (31)
where frequency ω(k) and damping σ(k) of propagating modes are obtained from z±(k, t) as imag-
inary and real parts, respectively. We are mainly interested to look at the long-wavelength asymp-
tote of contribution from kinetic relaxation process to the density-density time correlation function
Ann(k → 0). This weight coefficient can be obtained from the corresponding eigenvectors simply
as follows:
Ann(k) = ψd,1ψ
∗
d,1, (32)
that results in the following asymptotic value
Ann(k → 0) =
c2TDL
d3
0
k4. (33)
Completely in agreement with the concept of kinetic collective processes the contribution from the
structural relaxation to the density-density time correlation function vanishes by approaching the
hydrodynamic region. The special feature of structural relaxation is that the relaxation time for
this mode is finite in the hydrodynamic limit and its contribution decays even faster than k2 when
k is small.
The presented perturbation approach applied within the GCM scheme can be very effective for
finding analytical solutions of more sophisticated dynamical models. It is problematic to solve ana-
lytically the eigenvalue problem for many-variable dynamical models even in the long-wavelength li-
mit. Therefore the presented GCM perturbation approach is very promising for analytical studies of
collective dynamics of binary liquids, disparate mass two-component systems and three-component
liquid mixtures, for which only simplest analytical solutions exist to the date [19–21].
146
Structural relaxation in pure liquids
4. Numerical GCM analysis of collective dynamics: Lennard- Jones fluids
For the study of collective modes of dense liquids in a wide region of wavenumbers one has to
apply within the GCM approach more sophisticated basis sets of dynamical variables that take
into account the main non-hydrodynamic processes within the same level of accuracy. The simplest
set of dynamical variables within the precision of the fourth sum rule for the density-density time
correlation function is the five-variable set
A
(5)(k, t) =
{
n(k, t), JL(k, t), e(k, t), J̇L(k, t), ė(k, t)
}
. (34)
This five-variable dynamical model permits also an analytical GCM solution for collective modes in
long-wavelength limit [22], that makes possible to compare the expressions obtained in the previous
Section within the simplified three-variable dynamical model and perturbation approach. The
dynamical model (34) is able to account for two types of non-hydrodynamic relaxation processes:
structural relaxation mode (d2(k) in notations of [22] in order to distinguish with the lowest
hydrodynamic thermal process dT(k) = d1(k)) and kinetic heat relaxation mode (d3(k)). The
analytical expression for d2(k), obtained in the long-wavelength limit in [22] reads
d2(k) = d0
2 −DLk
2 + (γ − 1)∆k2 , (35)
and can be easily transformed to the obtained above equation (27) by setting γ = 1, i.e. neglecting
coupling with heat fluctuations. In equation (35) the following notations were applied
∆ =
d0
2d
0
3
d0
3 − d0
2
DT
DLc2s
(DT −DL)2 , (36)
where
d0
2 =
c2∞ − c2s
DL
, d0
3 =
cV
mλ
(
Gh − γ − 1
κT
)
, (37)
Gh and κT are the heat rigidity and isothermal compressibility of the liquid, respectively.
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 1.2
R
ed
uc
ed
te
m
pe
ra
tu
re
, T
*
Reduced density, n*
Figure 1. Thermodynamic points on the phase diagram of Lennard-Jones fluids along the isother-
mal line T ∗ = 1.71 used in numerical GCM study.
Now we apply the five-variable dynamical model for numerical study of collective modes in
Lennard-Jones fluids. We consider four thermodynamic points with different density, shown by
symbols along the isothermal line T ∗ = 1.71 on the phase diagram of Lennard-Jones fluids in
figure 1. For these thermodynamic points we performed MD simulations with 2000 (for small
wavenumbers) and 1000 particles in order to calculate density-density, energy-density and energy-
energy time correlation functions in the wavenumber region up to 3 Å−1. The majority production
runs were performed over 300000 steps, while for small wavenumbers we simulated the systems
over one million timesteps to obtain good convergence of relevant static averages needed in the
147
T.Bryk, I.Mryglod
GCM analysis. In figure 2 the MD-derived density-density, density-energy and energy-energy time
correlation functions along with their theoretical GCM replicas are shown for the thermodynamic
point with the highest density considered in this study. The GCM replicas were obtained within the
five-variable dynamical model (34) and in general all three functions are equally well represented
by the chosen dynamical model taking into account that no fitting parameters were used in con-
struction of the GCM-replicas. Usually the time correlation functions of the dense liquid states are
described by the generalized hydrodynamic theory worse than the dynamics of fluids with gas-like
densities. Therefore we can accept the five-variable dynamical model as an appropriate one for
analysis of the collective dynamics in a wide region of densities.
0
0.005
0.01
0.015
0.02
0.025
F
nn
(k
,t)
MD
GCM
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
F
ne
(k
,t)
/
re
du
ce
d
un
its
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5
F
ee
(k
,t)
/
re
du
ce
d
un
its
Time / reduced units
Figure 2. Density-density, density-energy
and energy-energy time correlation functions
for the density n∗ = 1.011 and temperature
T ∗ = 1.71: by solid and dashed lines are
shown MD-derived functions and their GCM
replicas, respectively. Time and energy scales
are 3.29ps and kBT , respectively.
In this study we are mainly interested in
modes connected with structural relaxation and
relevant physical quantities. In figure 3 we
show the values of adiabatic and high-frequency
speed of sound estimated from the long-
wavelength asymptotes of MD-derived static av-
erages (γ(k)/S(k))1/2 and ωl(k) [see equation 16],
respectively. The lines in figure 3 correspond to
a fit with a polynomial function (a + bn2) – in
the wide region of density the obtained adia-
batic and high-frequency speeds of sound were
satisfactory fitted by such a dependence. The de-
pendence of adiabatic and high-frequency speed
of sound on density shown in figure 3 implies
that the factor (c2∞ − c2s ), which in the litera-
ture is sometimes called as the “structural re-
laxation strength” [15], increases with the den-
sity. Other factors, that define the behavior of
the generalized relaxation process d2(k) on the
macroscopic length scales (35) are the kinematic
viscosity and ratio of specific heats. In figure 4
the generalized ratio of specific heats γ(k) is pre-
sented for the thermodynamic points considered.
This quantity was estimated using its definition
[3] via the static heat density autocorrelations.
This important generalized quantity reflects the
static coupling between thermal and viscous pro-
cesses on different length scales. On macroscopic
distances, i.e. in the limit k → 0, the functi-
ons γ(k) tend to relevant constants, which decay
with the density. This means, that the closer is
the thermodynamic point to the solid phase the
weaker will be static coupling between thermal
and viscous processes. The generalized ratio of
specific heats γ(k) makes evidence, that for Lennard-Jones fluids one cannot neglect the coupling
of density and thermal fluctuations. One can also see in figure 4 that the generalized ratio γ(k)
depends strongly on k, so that γ cannot be taken in the generalized hydrodynamic approaches as
a simple independent on wavenumber constant with its macroscopic value.
In figure 5 we show another important quantity, that defines the macroscopic lifetime of the
kinetic relaxation process d2(k), the kinematic viscosity DL as a function of density. The closed
boxes in figure 5 represent the values obtained from the direct calculations of the Green-Kubo
integrals [16], while the values shown by open boxes were extracted from the asymptotes of hydro-
dynamic modes taken at the smallest wavenumbers reached in MD simulations for each density.
In general, the dependence of kinematic viscosity implies the divergence of DL by approaching the
148
Structural relaxation in pure liquids
freezing point, as it should be. Looking at the expression (35) for the kinetic relaxation process
one can realize, that at the freezing point the relaxation time of this mode
τ2(k → 0) =
DL
c2∞ − c2s
becomes infinite, that is a feature of the structural relaxation, and namely the alpha-relaxation
with macroscopically large relaxation time. Hence, two facts, discussed above, support the rele-
0
500
1000
1500
2000
2500
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
c ∞
, c
s/
m
s
-1
n*
cs
c∞
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
0 0.5 1 1.5 2 2.5 3
G
en
er
al
iz
ed
r
at
io
γ
(k
)
k / A�°-1
n*=1.011
n*=0.845
n*=0.601
n*=0.357
Figure 3. Infinite and adiabatic speed of sound
along the isothermal line T ∗ = 1.71 for
four different densities of Lennard-Jones flu-
ids. The lines are a fit with a + bn2 function.
Figure 4. Generalized ratio of specific heats
γ(k) as functions of wavenumber for differ-
ent density of Lennard-Jones fluid at T ∗ =
1.71.
vance of the kinetic relaxation mode d2(k) to the structural relaxation, namely, the presence of
the factor called “strength of structural relaxation” and divergence of the relaxation time at the
freezing point. Another interesting consequence follows from the dependence of the relaxation mode
d2(k) on kinematic viscosity (35). In fact this mode restricts the region of wavenumbers, where
hydrodynamic collective modes behave strictly according to the hydrodynamic expressions. The
larger is kinematic viscosity the smaller will be the window of hydrodynamic regime (see [22] for
analytical expressions).
5
10
15
20
25
30
35
40
45
50
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1K
in
em
at
ic
v
is
co
si
ty
D
L
/ 1
0-8
m
2 s-1
Reduced density n*
GK
GCM
Figure 5. Kinematic viscosity as function of density at T ∗ = 1.71. Calculations of Green-Kubo
integrals (GK) and values estimated from long-wavelength asymptotes of generalized modes
(GCM) are shown by plus and cross symbols with errorbars, respectively.
The propagating zj(k) and relaxing dj(k) dynamical eigenmodes in the wide region of wavenum-
bers for the thermodynamic point with the highest density considered in this study are shown in
figure 6. For the chosen five-variable dynamical modes we obtained five eigenvalues for each k-point
149
T.Bryk, I.Mryglod
sampled in MD simulations. In the long-wavelength region we have obtained as the eigenvalues one
complex-conjugated pair and three real numbers. Looking at their wavenumber-dependence and
using the asymptotes of hydrodynamic modes with the values cs, γ, DL, DT, estimated beyond
the GCM framework as in figures 3 and 4 and from Green-Kubo integrals, one can see in figure 6
a nice agreement with the quadratic dependence on k of the sound damping Rezs(k) and thermal
diffusivity mode dT(k), while the dispersion Imzs(k) of acoustic excitations has a positive deviation
called “positive dispersion”. We show by dashed lines the expected hydrodynamic linear dispersion
law csk and the dispersion law given by equation (29) obtained within a simplified three-variable
model, which actually describes local coupling effects between acoustic excitations and structural
relaxation. One can see that the analytical expression equation (29) correctly reproduces the pos-
itive deviation in the small-wavenumber region.
0
2
4
6
8
10
12
14
16
18
Im
z
j(k
)
/ p
s
-1
csk
Im(z) Eq.29
0
2
4
6
8
10
12
R
e
z j
(k
)
/ p
s
-1
Γk2
0
2
4
6
8
10
12
0 0.5 1 1.5 2 2.5 3
d j
(k
)
/ p
s
-1
k / A�°-1
DTk2
d0-DLk2
Figure 6. Imaginary and real parts of generalized
collective modes for the density n∗ = 1.011 and
temperature T ∗ = 1.71. For convenience the
purely real eigenvalues are noted as dj(k) and
shown in the bottom frame. Propagating modes
are shown by symbols connected by solid line
and relaxation modes are represented by symbols
connected by dash-dotted line. By dashed lines
in the upper frame the linear dispersion law with
adiabatic speed of sound and positive dispersion
of sound branch that follows from equation (29)
are shown. Dashed line in middle frame shows
quadratic long-wavelength asymptotes of sound
damping, and in the bottom frame – thermal
relaxation mode. The dotted line in the bottom
frame corresponds to the analytical result for the
structural relaxation without coupling to thermal
processes represented by equation (27).
In the region k > 0.6 Å−1 there ap-
pears another branch of propagating exci-
tations, which corresponds to kinetic prop-
agating heat excitations (heat waves). We
stress that the majority of the generalized
hydrodynamic studies of collective dynam-
ics in liquids usually restrict treatment of
the thermal fluctuations just by the most
slow ones, while correct treatment requires
to account for the same level of short-time
processes in viscous and thermal description
of dynamics. The heat waves appear in com-
plete analogy with shear waves, which are
not supported by liquid systems on macro-
scopic distances. For more detailed informa-
tion on heat waves and convergence of rele-
vant eigenmodes with increasing number of
dynamical variables we refer to [23]. Hence,
the GCM approach predicts, that the ther-
mal relaxation processes on the nanoscale in
Lennard-Jones liquid are unstable with re-
spect to emergence of propagating heat exci-
tations, which can contribute to heat trans-
port at the nanoscale. For the generalized
mode treatment this means, that instead
of two real eigenmodes shown in bottom
frame of figure 6 by open circles and asteri-
sks one obtains for k > 0.6 Å−1 another pair
of complex-conjugated eigenvalues with the
frequency smaller than of the generalized
acoustic excitations (high-frequency branch
in the top frame of figure 6).
The most interesting is the behavior of
relaxation process d2(k) as the function of
the wavenumber (open boxes connected by
dash-dotted line in the bottom frame of fi-
gure 6). The predicted by the analytical
model wavenumber dependence (27) shown
in figure 6 by dotted line implies, that the
lifetime of this relaxation process becomes
larger with increasing k and beyond the hy-
drodynamic region the mode d2(k) becomes
150
Structural relaxation in pure liquids
smaller than dT(k), which though having small negative deviation tends to larger values. The
coupling with thermal processes according to (35) increases the d2(k), that is in agreement with
numerical results. But the most interesting fact is that the mode d2(k), connected with structural
relaxation, becomes the main relaxation process in the region of intermediate and large wavenum-
bers. One can make additional checks with different subsets of dynamical variables as performed
in [23,24] in order to discard any relation between the mode d2(k) and thermal processes in that
range of wavenumbers.
The wavenumber dependence of hydrodynamic relaxation and propagating modes in the wide
k-region is tightly connected with generalized transport coefficients. For example, the negative
deviation of the thermal relaxation mode dT(k) can be associated with decreasing of generalized
wavenumber-dependent thermal diffusivity DT(k). The most correct and transparent way of intro-
duction of the generalized transport coefficients is via the matrix of lowest order (or hydrodynamic)
memory functions [25,26]. For the case of pure liquids the 3 × 3 hydrodynamic matrix of lowest
memory functions is directly connected to the generalized transport coefficients L̃ij(k, ω):
M̃(k, ω) = k2V kBT L̃ (k, ω)F(k, t = 0) , (38)
where L̃ (k, ω) is the 3× 3 matrix of generalized wavenumber- and frequency-dependent transport
coefficients, and V is the volume of the system. As an example we show in figure 7 the generalized
static kinematic viscosity DL(k, ω = 0) and thermal diffusivity DT(k, ω = 0). The general tendency
is the monotonic decay of these quantities with increasing wavenumber.
Figure 7. Generalized wavenumber-dependent kinematic viscosity DL(k) and thermal diffusivity
DT(k) for n∗ = 0.357. At k = 0 the values of relevant Green-Kubo integrals are shown.
The most important issue in the study of non-hydrodynamic processes in liquids is the con-
tribution from the kinetic processes to the observable dynamical structure factor S(k, ω), or its
time-Fourier counterpart the density-density time correlation function. In figure 8 we show the
mode contributions to the Fnn(k, t)/S(k) represented in the form suggested in [24]. Hydrodynamic
theory predicts, that for pure fluids there exist only two contributions to the density-density time
correlation functions: from the thermal diffusivity and sound excitations with the weights (γ−1)/γ
and 1/γ, respectively. In figure 8 at k = 0 the corresponding contributions are shown by open and
closed circles. The GCM approach predicts, that when k increases there appear additional con-
tributions from the kinetic processes and all the weight coefficients must depend strongly on k.
One can see in figure 8, that the contribution from generalized acoustic excitations and thermal
relaxation mode are decaying with the increasing wavenumber, while the corresponding contribu-
tion from the kinetic relaxation process d2(k) increases and becomes dominant for representing the
relaxational behavior of Fnn(k, t)/S(k). The kinetic heat waves almost do not contribute to the
shape of Fnn(k, t)/S(k). The increase of the lifetime of the process d2(k) and its contribution to the
density-density time correlation function in the region k ≈ 2.0−2.2 Å−1, where the main maximum
of the static structure factor is located, means that the structural relaxation mode d2(k) is respon-
sible for the increasing of the long-lasting tails of the density-density time correlation functions
and corresponding narrowing of the central peak of S(k, ω) known as the de Gennes narrowing [2].
151
T.Bryk, I.Mryglod
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3
M
od
e
co
nt
rib
ut
io
ns
to
th
e
F
nn
(k
,t)
k / A�°-1
sound
th.relax.
heat waves
str.relax.
-0.5
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3
C
on
tr
ib
ut
io
ns
to
S
(k
)
k / A�°-1
S(k)
th.relax.
str.relax.
sound
Figure 8. Wavenumber-dependent mode con-
tributions to the density-density time correla-
tion functions from different propagating and
relaxation collective modes for the density
n∗ = 1.011 and temperature T ∗ = 1.71. The
closed and open circles at k = 0 show pre-
dicted by hydrodynamics values for contribu-
tions from sound excitations and thermal re-
laxation mode, respectively.
Figure 9. Zero-time contributions from gener-
alized collective modes to the shape of static
structure factor S(k). Leading contribution
to the shape of S(k) from the slow mode
shown by spline interpolated short-dashed
line proves its relation to the wavenumber-
dependent structural relaxation.
Formally one can use the mode contributions to the density-density time correlation function
Fnn(k, t) and represent them at t = 0 as the contributions to the static structure factor S(k) from
different collective processes. Intuitively this means, that each instantaneous configuration one can
represent as some deviations from a reference configuration due to sound propagation, structural
relaxation, etc. In figure 9 we show, that the contributions from generalized modes to Fnn(k, t) at
t = 0 have the only leading term, which comes from the relaxation mode d2(k), that once more
proves its origin as the mode of structural relaxation, because it is directly related to the main
structural quantity of the system – its structure factor.
5. Conclusions
In this study we applied the approach of Generalized Collective Modes to estimation of wavenum-
ber dependence of structural relaxation in pure liquids. The GCM approach has several advantages
over the standard memory function treatment of generalized hydrodynamics. In particular, it repre-
sents the contributions to the time correlation functions from hydrodynamic and non-hydrodynamic
processes via exponential functions of time thus making very transparent estimation of relaxation
time for non-hydrodynamic relaxation processes. In contrast to the standard memory function
treatment of structural relaxation on introducing additional decay channels in the second order
memory function, the GCM approach systematically takes into account non-hydrodynamic pro-
cesses via additional non-hydrodynamic variables allowing to estimate the relevant combinations
for each non-hydrodynamic collective mode in different regions of wavenumbers.
In this study we have shown, how using the GCM approach one can separate a contribution
from the structural relaxation to the density-density time correlation functions in a wide range of
wavenumbers and estimate the wavenumber dependence of the corresponding relaxation time. The
GCM approach predicts several features in the wavenumber dependence of structural relaxation
in pure liquids, namely, it is found that:
(i) in the long-wavelength limit the structural relaxation tends to a finite relaxation time defined
by the kinematic viscosity and a factor called in the literature as “strength of structural relax-
ation” [15];
(ii) at small wavenumbers this relaxation time increases, being in opposite tendency with hydro-
152
Structural relaxation in pure liquids
dynamic thermal relaxation process with decreasing relaxation time;
(iii) by decreasing temperature or increasing density the width of hydrodynamic region, i.e. the
wavenumber range where collective modes behave strictly according to the hydrodynamic expres-
sions, becomes more narrow and the non-hydrodynamic excitations play essential role in wider
range of k;
(iv) for intermediate and large wavenumbers the structural relaxation almost completely defines
the shape of the central peak of dynamical structure factor, while thermal processes do not have
much effect on the central peak. The relevance of the slow relaxation process at intermediate
and large wavenumbers to the structural relaxation is demonstrated via its contribution for the
instantaneous configurations to the structure factor S(k).
This study was aimed to estimate the role of non-hydrodynamic process of structural relax-
ation in collective dynamics of pure liquids and to look at the effect of liquid density on different
parameters, that define the structural relaxation in liquids. The study of temperature dependence
of structural relaxation within the GCM approach will be reported elsewhere.
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153
T.Bryk, I.Mryglod
Структурна релаксацiя у простих рiдинах: аналiз залежностi
вiд хвильового числа в рамках пiдходу узагальнених
колективних мод
Т.Брик1,2, I.Мриглод1,2
1 Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, Україна
2 Iнститут прикладної математики та фундаментальних наук, Нацiональний унiверситет “Львiвська
полiтехнiка”, 79013 Львiв, Україна
Отримано 4 сiчня 2008 р., в остаточному виглядi – 28 лютого 2008 р.
Залежнiсть вiд хвильового числа для структурної релаксацiї в рiдинах дослiджується методом уза-
гальнених колективних мод (УКМ). Запропоновано новий пертурбативний пiдхiд в рамках методу
УКМ, що застосовано для знаходження залежної вiд хвильового числа структурної релаксацiї та дис-
персiї звуку у довгохвильовiй границi в рамках спрощеної тризмiнної динамiчної моделi. Аналiтичнi
результати порiвнюються iз числовими дослiдженнями узагальнених мод в рамках бiльш загальної
п’ятизмiнної динамiчної моделi, що враховує термiчнi процеси в рiдинах. Представлено числовi ре-
зультати для чотирьох термодинамiчних точок ленард-джонсiвського плину з приведеною темпера-
турою T
∗
= 1.71. Обговорюються особливостi негiдродинамiчного процесу структурної релаксацiї у
рiзних областях хвильових чисел.
Ключовi слова: узагальнена гiдродинамiка, структурна релаксацiя, динамiчнi моделi, звуковi
збудження, позитивна дисперсiя
PACS: 05.20.Jj, 61.20.Lc, 62.60.+v
154
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