Collective dynamics in noble-gas and other very simple classical fluids
Rare gases and their liquids are the simplest systems to study for accurate investigations of the collective dynamics of fluid matter. Much work has been done using different spectroscopic techniques, moleculardynamics simulations, and theoretical developments, in order to gain insight into the mi...
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irk-123456789-1190002017-06-04T03:02:53Z Collective dynamics in noble-gas and other very simple classical fluids Bafile, U. Barocchi, F. Guarini, E. Rare gases and their liquids are the simplest systems to study for accurate investigations of the collective dynamics of fluid matter. Much work has been done using different spectroscopic techniques, moleculardynamics simulations, and theoretical developments, in order to gain insight into the microscopic processes involved, in particular, in the propagation of acoustic excitations in gases and liquids. Here we briefly review the interpretation schemes currently applied to the characterization of such excitations, and recall a few results obtained from the analysis of rare-gas fluids and other very simple systems. Iнертнi гази та їхнi рiдини є найпростiшими системами для точних дослiджень колективної динамiки плинiв. Було пророблено значну роботу з використанням рiзноманiтних спектроскопiчних методик, моделювання методом молекулярної динамiки та теоретичних розробок для розумiння мiкроскопiчних процесiв, що приймають участь у поширеннi звукових збуджень у газах та рiдинах. Ми подаємо короткий огляд iнтерпретацiйних схем, що застосовуються для опису таких збуджень, та нагадуємо декiлька результатiв, отриманих iз аналiзу плинiв iнертних газiв та iнших дуже простих систем. 2008 Article Collective dynamics in noble-gas and other very simple classical fluids / U. Bafile, F. Barocchi, E. Guarini // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 107-118. — Бібліогр.: 66 назв. — англ. 1607-324X PACS: 61.20.Lc, 61.20.Gy, 61.25.Bi DOI:10.5488/CMP.11.1.107 http://dspace.nbuv.gov.ua/handle/123456789/119000 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Rare gases and their liquids are the simplest systems to study for accurate investigations of the collective
dynamics of fluid matter. Much work has been done using different spectroscopic techniques, moleculardynamics
simulations, and theoretical developments, in order to gain insight into the microscopic processes
involved, in particular, in the propagation of acoustic excitations in gases and liquids. Here we briefly review
the interpretation schemes currently applied to the characterization of such excitations, and recall a few results
obtained from the analysis of rare-gas fluids and other very simple systems. |
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Article |
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Bafile, U. Barocchi, F. Guarini, E. |
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Bafile, U. Barocchi, F. Guarini, E. Collective dynamics in noble-gas and other very simple classical fluids Condensed Matter Physics |
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Bafile, U. Barocchi, F. Guarini, E. |
| author_sort |
Bafile, U. |
| title |
Collective dynamics in noble-gas and other very simple classical fluids |
| title_short |
Collective dynamics in noble-gas and other very simple classical fluids |
| title_full |
Collective dynamics in noble-gas and other very simple classical fluids |
| title_fullStr |
Collective dynamics in noble-gas and other very simple classical fluids |
| title_full_unstemmed |
Collective dynamics in noble-gas and other very simple classical fluids |
| title_sort |
collective dynamics in noble-gas and other very simple classical fluids |
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Інститут фізики конденсованих систем НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/119000 |
| citation_txt |
Collective dynamics in noble-gas and other very simple classical fluids / U. Bafile, F. Barocchi, E. Guarini // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 107-118. — Бібліогр.: 66 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT bafileu collectivedynamicsinnoblegasandotherverysimpleclassicalfluids AT barocchif collectivedynamicsinnoblegasandotherverysimpleclassicalfluids AT guarinie collectivedynamicsinnoblegasandotherverysimpleclassicalfluids |
| first_indexed |
2025-07-08T15:03:44Z |
| last_indexed |
2025-07-08T15:03:44Z |
| _version_ |
1837091556664803328 |
| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 1(53), pp. 107–118
Collective dynamics in noble-gas and other very simple
classical fluids
U.Bafile1, F.Barocchi2,3, E.Guarini2,3
1 Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, via Madonna del Piano 10, I–50019 Sesto
Fiorentino, Italy
2 Dipartimento di Fisica, Università di Firenze, via G. Sansone 1, I–50019 Sesto Fiorentino, Italy
3 CNR-INFM CRS-Soft c/o Dipartimento di Fisica, Università di Firenze, via G. Sansone 1, I–50019 Sesto
Fiorentino, Italy
Received November 12, 2007
Rare gases and their liquids are the simplest systems to study for accurate investigations of the collective
dynamics of fluid matter. Much work has been done using different spectroscopic techniques, molecular-
dynamics simulations, and theoretical developments, in order to gain insight into the microscopic processes
involved, in particular, in the propagation of acoustic excitations in gases and liquids. Here we briefly review
the interpretation schemes currently applied to the characterization of such excitations, and recall a few results
obtained from the analysis of rare-gas fluids and other very simple systems.
Key words: dynamics of fluids, collective excitations
PACS: 61.20.Lc, 61.20.Gy, 61.25.Bi
1. Introduction
Rare-gas fluids are the simplest systems, and therefore ideal candidates, for studying collective
dynamical properties at a nanoscopic level.
In dealing with such phenomena one would naturally like to focus on the translational motions
of the constituent particles. However, even in the most simple molecules there exist complicated
static and dynamic correlations due to the rotational and vibrational degrees of freedom. In fact,
even when internal and centre-of-mass motions take place in well-separated energy windows, as it
happens for light molecules, they generally are not independent of each other. If the dynamics is
investigated by means of a spectroscopic probe sensitive to the positions of all atoms of the system,
the detected signal includes contributions from all intra- and intermolecular correlations, weighted
by cross-section factors that are specific to the employed technique. This fact usually makes the data
analysis a difficult task. On the contrary, due to the absence of a molecular structure, monatomic
fluids have the nice property that the position of one particle changes with time only because of
its translational displacement in the environment of the other ones.
That a deep connection exists between dynamical properties and the forces acting at the atomic
level is an obvious fact, but how such a relationship is particularly working is still a fundamen-
tal, unanswered question. Molecular-dynamics (MD) computer simulations can be used to try to
understand which dynamical behaviour is caused by a given interaction potential model [1–4].
Moreover, if realistic and well-modelled interatomic forces are available, the combination of MD
and experimental studies can also shed light on whether specific properties of the true interaction
law can be inferred from an actually observed dynamical behaviour.
In this respect, rare gases offer a clear advantage as well. Their interatomic pair potentials are
very well known functions of the only variable involved, the distance between the two atoms of a
pair [5]. The fundamental assumption that such functions do not depend on the thermodynamic
state of the fluid, never contradicted by experimental evidence, has also made possible accurate
investigations of interatomic forces in dilute gases using a variety of methods. The total potential
c© U.Bafile, F.Barocchi, E.Guarini 107
U.Bafile, F.Barocchi, E.Guarini
energy of a dense many-body system can then be appropriately described by assuming that small
irreducible contributions due to triplets, quadruplets, and so on, can be added to the dominant
pair-additive part. The situation is very different for most molecular systems, whose intermolecular
interactions are either unknown or at least not yet well tested with a sufficient accuracy, even at
the pair level. Moreover, with the exception of very few cases, the anisotropic part of the potential
plays a non-negligible role, making the interaction energy a multi-variable function of distances
and orientation angles.
Although the metallic liquids are also simple monatomic systems, the presence of nearly free
electrons clearly affects the properties of the interaction between atoms [6], making the modelling
of potentials by far a harder task than in insulating fluids. As a result, liquid-metal interaction
energies are not yet known with an accuracy comparable to that of the rare-gas case.
Thus, a theoretical or simulation approach to dynamical problems is far easier in noble-gas
fluids. These, moreover, are often simpler systems from the experimental point of view. For example,
they are easier samples to handle for lack of chemical reactivity, no need of high temperatures, and
usually, at least with liquids, of high pressures.
The above facts have made rare gases and liquids one of the preferred choices for the study of
microscopic collective dynamics. In this paper, where we focus on very simple systems, we shall
obviously be concerned with rare-gas fluids, but we shall also include as examples the case of a
simple molecular liquid and that of a rare-gas binary mixture. Liquid metals [7], on the contrary,
will not be dealt with here.
We will not consider the important case of quantum fluids. Collective excitations in liquid
helium constitute a well-established research field, which would deserve by itself a wholly dedi-
cated treatment [8,9]. Hydrogen and deuterium have also been the subject of several investigations
[10–17]. Their molecular nature is not an obstacle due to the separability of the centre-of-mass
translational dynamics from the rotovibrational motions [18]. However, H2 and, to a lesser extent,
D2 also display quantum behaviour in the liquid phases and will not be treated here. Neon, in-
stead, features a modest quantum behaviour and has been investigated using essentially the same
approach as the one used for truly classical fluids.
In this paper we restrict ourselves to considering collective acoustic excitations, related to the
density fluctuations. The fundamental quantity that reflects this dynamical behaviour is the dy-
namic structure factor S(Q,ω), where Q and ω are the wavevector and frequency of the excitation,
respectively. S(Q,ω) is the time Fourier transform of the intermediate scattering function F (Q, t),
which represents the time autocorrelation of the spatial Fourier component with wave vector Q of
the microscopic local density n(r, t) =
∑N
α=1 δ(r−Rα(t)) [19] for a system of N identical particles
at positions Rα(t). Therefore, S(Q,ω) or F (Q, t) are the right quantities to determine, by means
of various methods, in order to gain direct access to the atomic motions in a many-body system.
MD simulation has found in the dynamics of liquids an extremely important field of application
[20], also due to its capability of giving insight into physical properties inaccessible to experiments
or too difficult to be calculated from theoretical considerations. Needless to say, the effectiveness of
MD is strictly bound to the availability of reliable models for the interparticle potentials. However,
even the study of very simple model fluids, such as those ruled by hard-sphere (HS) [21–23],
or Lennard-Jones (LJ) [3,24,25] interactions, has provided extremely useful results for the basic
understanding of dynamical structures.
On the other hand, S(Q,ω) is the natural output of spectroscopic techniques based on the
measurement of the (double-differential) cross-section in scattering experiments where energy ~ω
and momentum ~Q are transferred from the probe to the sample. Inelastic scattering of light,
ultraviolet (UV) radiation, x rays, or thermal neutrons have been used for this purpose and, in
the specific field of acoustic excitations, share nowadays the common denomination of “Brillouin
scattering” that was initially coined for light scattering studies where the well-known Brillouin
lines reflect the existence of propagating sound modes.
The Q range of interest for the investigation of these phenomena goes usually from zero up
to a value which corresponds to wavelengths of the order of the distance travelled between two
consecutive collisions experienced by one atom. Thus, if ` is such a characteristic length (for
108
Collective dynamics in very simple fluids
example, the Enskog mean free path of an equivalent HS fluid [22]), the most frequently studied
range is identified by the condition 0 < Q` ∼ 1. With visible radiation, density fluctuations can
only be explored at very low Q (of the order of 0.01 nm−1), where hydrodynamics applies [19,26].
Non-hydrodynamic regimes can also be studied by means of light scattering, but this requires to
increase ` by reducing the density to that of dilute gases [27,28]. At about the same time as for
light Brillouin spectroscopy, neutron scattering also came into use in order to probe much higher Q
values, of the order of the position Qp of the main peak of the static structure factor S(Q), showing
the persistence of collective excitations with wavelengths comparable with the typical microscopic
distances of a dense fluid [29–31].
More recent technical advances are the extension of neutron scattering to lower Q (down to
Q ∼ 0.2 nm−1) [32–34], the advent of high-resolution x-ray scattering [35–37], widening the energy
window at a given Q due to the absence of the intrinsic kinematic limitations of neutron scattering,
and, finally, the application of UV spectroscopy [38] to cover the Q gap between light scattering
and the other techniques.
The whole Q range covered by a full set of experimental methods is thus very broad. Never-
theless, the theoretical developments that enable a detailed description of the measured spectral
distributions are essentially the same at all the Q values relevant to the investigation of sound
excitations. Thus, a unitary picture has been established in this field.
In section 2 we summarize the basic theoretical concepts, described more in detail in recent
works [39,40], including the commonly used fit models and the criteria for a quantitative assessment
of the excitation frequencies and damping. Sections 3, 4, and 5 are devoted to a short discussion
of a few examples. The conclusions are contained in section 6.
2. Theoretical framework
Here we give a short summary of the ideas and formulas relevant to the description of acoustic
excitations in the simplest case of monatomic one-component fluids. We recall that we are dealing
with classical systems for which S(Q,ω) is an even function of frequency at constant Q. This implies
that, in applying theoretically derived or fit model line shapes to real experimental data, proper
account must be taken of the detailed-balance asymmetry, besides the instrumental resolution
broadening. The theoretical description of the acoustic excitations in fluids, in the greatest part
of the published works, especially in recent years, is based on the use of the concept of memory
function in the time evolution of the intermediate scattering function F (Q, t). One then arrives at
the basic Langevin-type equation [19]
F̈ (Q, t) +
∫ t
0
dt′ M(Q, t − t′) Ḟ (Q, t′) + 〈ω2
Q〉 F (Q, t) = 0, (1)
where the dots denote time derivatives and 〈ωk
Q〉 is the k-th frequency moment of the normalized
dynamic structure factor I(Q,ω) = S(Q,ω)/S(Q). If (1) is solved in terms of the Laplace trans-
forms of F (Q, t) and M(Q, t), denoted by F̃ (Q, z) and M̃(Q, z) respectively, with F (Q, 0) = S(Q)
and Ḟ (Q, 0) = 0, one obtains
F̃ (Q, z)
S(Q)
=
(
z +
〈ω2
Q〉
z + M̃(Q, z)
)
−1
(2)
and, from this, the general expression for the spectrum
I(Q,ω) =
1
π
Re
F̃ (Q, iω)
S(Q)
=
1
π
〈ω2
Q〉M ′(Q,ω)
[ω2 − 〈ω2
Q〉 + ωM ′′(Q,ω)]2 + [ωM ′(Q,ω)]2
, (3)
where M ′(Q,ω) and M ′′(Q,ω) are the real and imaginary part, respectively, of M̃(Q, iω).
Equation (3) would solve the problem fully, were it not for the fact that M(Q, t) is actually
unknown and cannot be determined from theoretical arguments. Therefore, realistic modelling of
109
U.Bafile, F.Barocchi, E.Guarini
the memory effects becomes an essential step towards an accurate accounting of the measured or
simulated line shapes. This has proved to be a very useful approach, because very simple expressions
of M(Q, t) can be shown to give rise to realistic forms of the spectral distribution [19,39].
Various attempts to theoretically determine the time dependence of the memory function were
made in the past mainly by testing different expressions against dynamics simulations [41,42]. It is
a matter of fact, however, that in the greatest part of the works that appeared in the literature on
the subject, M(Q, t) is given the form of a δ-function and/or a number of exponentials [39]. Then
one can write (3) in the form F̃ (Q, z)/S(Q) = V (z)/W (z), where V and W are polynomials in the
complex variable z having real, Q-dependent coefficients and degrees p − 1 and p, respectively, if
there are p− 2 exponentials in M(Q, t). If, for any Q, the p zeros of W (z) are labelled as zj (with
j = A,B, . . .), one can also write
F̃ (Q, z)
S(Q)
=
∑
j
Ij(Q)
z − zj(Q)
, (4)
where the complex amplitudes Ij can be explicitly calculated [39] and satisfy the sum rule
∑
j Ij =
1. In what follows we will omit the indication of the Q dependence of Ij and zj . In the time domain,
(4) is equivalent to F (Q, t)/F (Q, 0) =
∑
j Ij exp(zjt), from which it is seen that the real part of
all zj ’s must be negative.
The existence of acoustic modes is easily related to the presence of two complex conjugate values
among the zj ’s for which we reserve the subscripts A and B, according to the sign of their imaginary
part. Then, letting zA,B = −zs ± iωs, with ωs > 0, and writing the corresponding amplitudes as
IA,B = Is(1 ∓ ibs), the spectrum is given by
I(Q,ω) =
1
π
[
∑
j=C,D...
−Ijzj
ω2 + z2
j
]
+
1
π
Is
[
zs + bs(ω + ωs)
(ω + ωs)2 + z2
s
+
zs − bs(ω − ωs)
(ω − ωs)2 + z2
s
]
, (5)
where the last two terms represent the Brillouin lines that are the spectral signature of the acoustic
excitations of energy ~ωs (the subscript s standing for “sound”). The remaining terms are central
lines building up the quasielastic part of the spectrum. All lines have a Lorentzian shape with half-
widths given by the negative of the real part of the various roots zj of the equation W (z) = 0. The
side lines, however, are distorted by the presence of asymmetric terms, related to the imaginary
part of IA,B and deriving from the finiteness of 〈ω2
Q〉. The time correlation corresponding to the
spectrum (5) is
F (Q, t)
F (Q, 0)
=
[
∑
j=C,D...
Ij exp(zjt)
]
+ 2Is exp(−zst)
cos(ωst − ϕ)
cos ϕ
, (6)
with the sound modes appearing as one exponentially-damped oscillation with frequency ωs and a
phase shift given by tanϕ = bs.
In the hydrodynamic situation (rigorously, in the limit case Q → 0), where the fluid is probed as
a continuum, I(Q,ω) is given by the Rayleigh-Brillouin (RB) expression derived from linearization
of the Navier-Stokes equations. Such a line shape can also be directly obtained from the assumption
M(Q, t) = 2νQ2δ(t) + (γ0 − 1)〈ω2
0〉 exp(−γ0DTQ2t), (7)
where ν, DT, and γ0 are the kinematic longitudinal viscosity, the thermal diffusivity, and the
specific-heat ratio, respectively, and 〈ω2
0〉 = kBTQ2/mS(0) = c2
sQ
2/γ0 is the Q → 0 limit of 〈ω2
Q〉,
where cs is the adiabatic sound velocity, kB is the Boltzmann constant, T is the temperature, and
m is the mass of the particles [19]. In this case p = 3 and the spectrum I(Q,ω) is then given by
(5) with only one central line, called the “heat” mode.
In the Q range accessed by neutron or x-ray scattering, the RB theory, derived under the
hydrodynamic condition of very-long-wavelength excitations, does not account for the measured
spectral shapes. Therefore the observation of acoustic modes at such Q values requires the models
110
Collective dynamics in very simple fluids
for I(Q,ω) to be valid in non-hydrodynamic regimes. These models have been customarily adopted
as functions to be fitted to constant-Q dynamic structure factor data, with parameters free to
vary with the value of Q. At low enough Q, all models of M(Q, t) are bound to reproduce the
hydrodynamic behaviour, and thus display the Brillouin pair of lines due to the occurrence of two
complex roots of W (z). Here we list the most often used models by simply recalling their respective
memory functions. More details on the properties of the various expressions can be found in [39].
The “generalized RB triplet” (GRB) model keeps the same expression of the RB line shape,
with a memory function written as
M(Q, t) = 2B(Q)δ(t) + (γ(Q) − 1)〈ω2
Q〉 exp[−ΓT(Q)t]. (8)
Another widely used model, called “viscoelastic” (VE), replaces the δ-function with an exponential
term, leading to
M(Q, t) = [ω2
L(Q) − γ(Q)〈ω2
Q〉] exp[−t/τ(Q)] + (γ(Q) − 1)〈ω2
Q〉 exp[−ΓT(Q)t], (9)
with p = 4. The various quantities appearing in (8) and (9) are unknown functions of Q whose
asymptotic behaviours in the limit Q → 0 are to be in agreement with the exact RB theory.
Moreover, in (9), ω2
L(Q) = 〈ω4
Q〉/〈ω2
Q〉, since the absence of the δ-function in M(Q, t) makes the
fourth frequency moment of I(Q,ω) a finite quantity.
To the above mentioned models, one should also add those obtained from (8) and (9) by deleting
the thermal part of the memory function, i.e., by letting γ(Q) = γ0 = 1. Thus, one obtains,
respectively, the “damped harmonic oscillator” (DHO), where M(Q, t) reduces to the δ-function
term only (p = 2), and the “simple viscoelastic” (SVE) model introduced by Lovesey [43]
M(Q, t) = [ω2
L(Q) − 〈ω2
Q〉] exp[−t/τ(Q)] (10)
with p = 3. Obviously, these simplified versions of the above models may only apply, at low Q, to
the cases where γ0 is close to unity, such as in some liquid metals, but the condition γ(Q) = 1 can
also occur at large Q in other systems. An example of such a behaviour will be given in section 4.
At low Q, both DHO and SVE models produce the Brillouin doublet alone, without the central
line due to thermal fluctuations. Actually, the spectrum corresponding to (10) does have a central
line, corresponding to the so-called structural relaxation process described by the relaxation time
τ(Q), but its amplitude can be shown to vanish in the Q → 0 limit.
The fact that all the mentioned models give rise to spectra in the form (5) should be exploited
to a larger extent than it has been done so far, and we emphasized in [39] the importance of
using the parameters of (5), common to all models, rather than those of the memory functions,
as fit parameters in order to compare the ability of different models to accurately reproduce the
same spectral data. In [39,40] we showed an example of how, in this way, it is possible to detect a
transition that takes place, with increasing Q, from a hydrodynamic-like situation, well accounted
for by the GRB model, to a viscoelastic regime.
Moreover, the common parametrization displayed in (5) has the important advantage of pro-
viding a full understanding of the dispersion curve of the acoustic excitations [39]. At each Q the
acoustic modes follow the dynamical behaviour of a damped harmonic oscillator with characteristic
frequency Ω =
√
zAzB and damping coefficient zs = −(zA+zB)/2 [40]. The propagation state of the
excitation is then determined by whether such an oscillator finds itself in an under- or overdamping
condition, with Ω > zs or Ω < zs respectively, or equivalently whether an oscillatory behaviour of
F (Q, t) takes place or not. In the first case, the right quantity to consider for a correct evaluation
of the excitation frequency is ωs =
√
Ω2 − z2
s , which represents the frequency of the oscillating
part of F (Q, t) (see (6)). For overdamped excitations, instead, ωs = 0, zA and zB become real, and
the spectral shape is modified by the transformation of the Brillouin lines into Lorentzians of half
widths −zA and −zB centered at zero frequency and superimposed onto the quasielastic lines.
Thus the comparison between Ω and zs provides a rigorous answer to the question of what is the
propagation Q-range of acoustic excitations. Apart from qualitative considerations, the only other
quantitative criterion found in the literature [19,43] is based on the properties of the SVE model
111
U.Bafile, F.Barocchi, E.Guarini
line shape, and corresponds to the condition of the existence of two relative maxima of I(Q,ω) at
frequencies different from zero in addition to the presence of a peak at ω = 0. This requirement
leads to the inequality
ω2
L(Q) >
1
τ(Q)
√
2(2ω2
L(Q) − 3〈ω2
Q〉) −
1
τ2(Q)
although a more frequently adopted criterion is the simpler one, not involving τ(Q), given by
ω2
L(Q) < 3〈ω2
Q〉, (11)
which actually expresses a sufficient but not necessary condition for the existence of a three-peak
spectrum. In other words, acoustic modes are said to be propagating if they appear as side peaks
(though possibly largely broadened), but are defined as overdamped, and hence non-propagating,
when they look like bumps, or shoulders, that do not give rise to relative maxima of the spectral
intensity. Such a criterion is not rigorous and is by far too stringent, since in any case such broad
features are located in frequency bands spread around non-zero values and having therefore a
genuine inelastic character. Moreover, (11) has other drawbacks. First of all, it applies only to the
SVE case, and cannot be easily generalized to the other line shape models. Moreover, it is based on
the idea that the damping state of an excitation can depend on its strength. The above mentioned
close mathematical analogy with a harmonic oscillator shows instead [40] that the amplitude of
the sound mode has nothing to do with its being under- or overdamped.
It is to be noted that another argument based on the analysis of intensities rather than frequen-
cies provides the most commonly applied method for the determination of the dispersion curve,
namely the use of the maxima in the spectra of the longitudinal current correlation. While the
use of (11) tends to an underestimation of the propagation Q range, the application of this crite-
rion, which does not have a theoretical justification, has in some sense an opposite effect, since it
always provides a non-zero value for the excitation frequency even when the acoustic modes are
truly overdamped (ωs = 0), thus preventing the detection of Q-ranges for which the acoustic waves
do not possibly propagate. This last remark also applies to other incorrect determinations of the
excitation frequency, such as those based on the use of Ω or of
√
γ(Q)〈ω2
Q〉 [39].
3. Rare gases
As applications of the concepts outlined in section 2, we briefly recall and discuss in the next
three sections a few typical results for the collective acoustic dynamics in very simple fluids.
Argon is the prototype of a monatomic, classical, non-conductive fluid, and has been the object
of many investigations. Moreover, it is an exceptionally convenient sample for neutron studies
due to the very large, totally coherent, scattering cross-section of the 36Ar isotope. Well-known
pioneering neutron works on the collective dynamics of argon are, among others, those of references
[29,30] in the liquid and [44] in the compressed gas phase, while [45] reports on a light scattering
experiment. Though sound modes were in some cases detected, depending on the investigated Q
range, and obviously with particular evidence in the light Brillouin case where Q = 0.017 nm−1
(see figure 3 of [45]), the first quantitative analysis of acoustic excitations was carried out in the
neutron study of reference [46].
The generalized hydrodynamic model (8) (derived from kinetic-theoretical considerations) was
fitted to constant-Q spectra in an extended wavevector range (4 < Q/nm−1 < 38). Asymmetric
Brillouin lines were detected, with one real and two complex conjugate poles of F̃ (Q, z), in the
whole Q range except for 17 < Q/nm−1 < 21, where a better fit was obtained with three real roots.
The authors interpreted this result as the first observation in a real fluid of a “propagation gap”
of sound modes, a phenomenon previously discovered in a HS system [22] and in a GRB analysis
of a simulated LJ dense fluid [3].
In view of the discussion of the previous section, one can easily interpret the occurrence of
such a gap as the one due to the transition from the under- to overdamping condition of the
112
Collective dynamics in very simple fluids
equivalent harmonic oscillator, with Ω < zs, shortly after followed by a reverse transition restoring
the propagating regime. However, the conclusions of reference [46] were criticized [47] on the ground
that it may not be justifiable to speak of collective excitations when the spectral shape does not
show visible inelastic structures, in agreement with the meaning of criterion (11). Apart from
similar results obtained in other thermodynamic states of liquid argon and in liquid neon [48],
where a propagation gap was also reported, this issue has not been dealt with in later works on
fluid systems, because in the Q range where a gap may possibly occur, dynamical data have been
customarily analyzed with models not having ωs and zs as explicit parameters. We will, however,
come back to this point in section 4.
Acoustic excitations of argon were also studied in the gas phase [33] at room temperature and
much lower densities than in the liquid. Low values of Q` (in the range 0.1 6 Q` 6 1) could then
be attained with small-angle inelastic scattering of cold neutrons. The GRB model was well fitted
to the data, which is in close agreement with MD results [49]. At the lowest Q` values, the spectra
agree with the hydrodynamic RB line shape, while they begin to deviate from it with increasing
Q and/or decreasing density. Thus, the onset of non-hydrodynamic behaviour could be evidenced.
From figure 1, where we display Ω, zs and ωs as obtained from the GRB fit to the 200 bar data,
a slightly stronger upward curvature of Ω(Q) and a slightly smaller damping zs are found with
respect to the predictions of RB theory. Then ωs =
√
Ω2 − z2
s stays above the RB dispersion curve,
whose downward bending is effectively compensated, and a linear Q-dependence of ωs is found.
This example also shows that the difference between Ω, ωs, and other frequency parameters should
be taken into account if the presence or absence of the so-called positive dispersion has to be
correctly assessed.
Figure 1. GRB analysis of neutron (full symbols) and MD (empty symbols) data on gaseous
36Ar at 200 bar [33,49]. Solid lines are the corresponding calculations with RB theory. Upper
frame: Ω (circles) and zs (squares). Lower frame: ωs (circles). In both frames, the straight line
csQ is shown by dashes.
Besides argon, neon has also been the object of studies for the collective dynamics. Here too,
neutron [31,50] and light [45] scattering experiment can be traced back far in the past. More
recently, as mentioned above, liquid neon was investigated in parallel with the neutron experiment
on argon [48], and an overall similarity of the dynamical behaviour of the two liquids was found
(see figure 12 of [48]), so that a corresponding-state principle was claimed to hold.
Neon has also been extensively studied by means of inelastic x-ray scattering. Experimental
data in the room-temperature gas, at a pressure of 0.3 GPa and a density close to that of the
liquid phase, were found in agreement with LJ-simulated spectra, whose collective modes were
then analyzed in terms of the GRB model [51] in the range 1 < Q/nm−1 < 25, showing the
absence of positive dispersion. This marks a difference with respect to the liquid [48], though the
densities are similar.
Later, more x-ray data at a number of thermodynamic states of gaseous and liquid neon were
113
U.Bafile, F.Barocchi, E.Guarini
analyzed using, instead, the SVE model [52]. Here, a positive dispersion present in the liquid also
tends to disappear in approaching supercritical temperatures. However, since the fitted model was
not expressed in the form (5), the excitation frequencies were estimated through the use of various
parameters different from ωs, preventing the possibility of a comparison with previous studies or
other fluids.
We recall from [39] that, if any of the model shapes (5) can be well fitted to the data, a positive
dispersion can be attributed to either an initial decrease of S(Q) at low Q or to a reduction, with
increasing Q, of the central line-width(s) with respect to the values given by the lowest order in
a Q-power series expansion. Such a damping reduction, expressed by the renormalization function
r(Q) defined in [39,40], plays an evident role in shaping the dispersion curve, and the explanation
of its connection with the microscopic interactions would be of high interest. Unfortunately, this
property of fluid systems is still to be adequately investigated.
4. Methane
Notwithstanding the obvious differences between monatomic and molecular systems, methane
behaves very similarly to the rare-gas fluids if the centre-of-mass dynamics is considered, as it
will be shown below. The comparison between neutron scattering [53] and simulation data has
recently provided the possibility of selecting one among different site-site interaction potentials for
methane [54], and, therefore, of validating MD results for the experimentally inaccessible partial
carbon-carbon dynamic structure factor SCC(Q,ω), which is the molecular analogue of S(Q,ω) of
monatomic systems. The centre-of-mass collective motions of liquid CD4 were then analyzed in a
very wide Q range (up to Q ∼ 3.7 Qp) [40].
For 2 < Q/nm−1 < 15 (∼ 0.8 Qp) we applied both the GRB and VE fit models, obtaining
practically indistinguishable results for Q smaller than about 5 nm−1, while at higher Q the
presence of two quasielastic lines in the VE fit function permits a more accurate description of the
central peak [39]. The viscoelastic model turns out to account well for the simulated spectral shape
at larger Q as well, up to Q = 67 nm−1. However, for Q > 37.5 nm−1, the thermal contribution to
the dynamics vanishes, and the simpler viscoelastic model (SVE) becomes sufficient.
Figure 2. VE analysis of MD data for the centre-of-mass SCC(Q, ω) of liquid CD4 [40]. Upper
frame: ωs; middle frame: Ω (circles) and zs (full dots); lower frame: Is (full dots). For Q >
37.5 nm−1 the SVE model (with γ = 1) has been used.
The shape of the dispersion curve, reported in figure 2, shows a strong similarity to that of
liquid argon [46]. In particular, methane also displays a propagation gap in a narrow Q interval
around Qp. Such a transition to an overdamped regime is clearly related to the Q dependence of Ω
and zs, and, in particular, to the decrease of Ω, caused by the presence of a high peak in SCC(Q),
in coincidence with an increased damping. However, Ω becomes only slightly smaller than zs, so
114
Collective dynamics in very simple fluids
that only a weak overdamping is obtained. This fact suggests that an arrest of the propagation of
sound modes can only occur in strongly structured fluids. Indeed, the whole Q-dependence of all
the quantities plotted in figure 2 directly reflects the oscillatory behaviour of SCC(Q). This shows
that a strong relationship exists at all Q values between the whole dynamics and the structural
correlations. Figure 2 also shows that the amplitude Is of the acoustic part of the intermediate
scattering function rapidly decays towards zero for Q larger than Qp [40]. One can then conclude
that acoustic excitations are present with a propagative nature up to very high Q, where their
relevance is limited by a fading intensity rather than by a vanishing frequency.
5. Rare-gas mixtures
Rare-gas fluids have also played an important role in the investigation of a phenomenon which
characterizes the collective dynamics of binary mixtures, namely the existence of the so-called fast
and slow sounds. In the hydrodynamic regime, such systems behave as homogeneous fluids and
display a regular acoustic mode propagating with the adiabatic sound velocity of the mixture. At
higher Q, a dynamical decoupling of the two components takes place, more marked for higher mass
ratios between the two species. A high-frequency propagation is supported by the light atoms only,
while the heavy ones are incapable of following the fast oscillatory motion. On the other hand, a
slow propagation is attributed to oscillations of the heavy particles.
Fast sound in gas mixtures was predicted to be observable through neutron scattering at inter-
mediate densities [55,56], and through light scattering in the dilute case [57]. The presence of a fast
sound is related to the existence of acoustic excitations in the partial dynamic structure factor of
the light component but not in that of the heavy one. Then the visibility of the effect in the total
S(Q,ω) depends on the concentrations of the two species and on their cross-sections relevant to
the employed spectroscopic technique (scattering lengths for neutron and polarizabilities for light
scattering).
Several studies were devoted to the exploration of these excitations. Considering only the most
studied case of HeNe mixtures with helium concentration ranging from 0.65 to 0.80 and total
number densities around 15 nm−3, the first neutron observation of fast sound [58,59] showed for
Q > 4 nm−1 an excitation propagating with a speed very close to that of pure helium at the
same total density. MD simulations [60] and theoretical calculations [61,62] were also carried out
on the same system. Nevertheless, the question of how this fast sound mode is connected to the
hydrodynamic one has eluded for some time the efforts of the investigators. Based on the kinetic
theory it was predicted [59] that the fast branch would disappear at Q ∼ 0.7 nm−1, in agreement
with MD results [60]. However, a later neutron experiment showed a linear dispersion with the
hydrodynamic sound speed up to Q = 1.8 nm−1 [63], so that the Q range where the transition
from hydrodynamic to fast sound takes place was identified with the still unexplored interval
between those spanned by the two neutron measurements. Such a transition, shown in figure 3,
later turned out to be a smooth one in an MD simulation in agreement with both experimental
data sets [64]. The same simulation did not show evidence of a slow sound, which, on the contrary,
appears clearly at a much higher density.
It should also be mentioned that the detection and the correct location of inelastic mode
frequencies in the spectra of fluid mixtures is even more dependent than in pure systems on the
adoption of adequate criteria for the assignment of the excitation energies. A clear example of this
problem is given in [64], where it was shown that the identification of mode frequencies with the
position of maxima in the Ne-Ne partial longitudinal-current spectra leads to a wrong dispersion
curve for the slow sound.
We finally note that, in the case of mixtures, the extension of the GRB line shape to the
two-component systems is practically the only model applied so far to the description of collective-
dynamics spectra, and only little work has been done [65,66] in order to define or test other spectral
models.
115
U.Bafile, F.Barocchi, E.Guarini
Figure 3. Dispersion curve of the acoustic mode in a HeNe gaseous mixture. The neutron results
from [63] (full dots) and from [58] (circles) are shown together with MD data from [64] (solid
line). The straight lines corresponding to the sound speed of the mixture (cs = 374 m/s) and of
pure helium (cs = 544 m/s) are shown as dashes.
6. Conclusions
Even a quick glance at the very recent literature on the subject of collective modes in fluid
systems is sufficient to realize that this research field is a very active one. New classes of liquids
have come under the scrutiny of experimentalists who take advantage of impressive advances in the
capabilities of innovative instrumentation and techniques and in the computing power available for
simulation studies. Besides the traditionally investigated rare-gas fluids and liquid metals, growing
attention has been paid to molecular liquids, liquid alloys, metal vapours, molten salts and oxides,
metals in solutions, and complex macromolecular systems.
From the viewpoint of the analysis and the interpretation of collective-dynamics spectra,
progress has also been made, with the refinement of model line shapes allowing for more de-
tailed descriptions of the dynamical behaviour in wide energy and momentum ranges, and with
the definition of rigorous criteria for the quantitative assessment of the nature of collective acoustic
modes.
Due to such advances, reliable tools are nowadays available for a full characterization of the
collective excitations in a large variety of fluids. It turns out to be a quite general result that the
whole spectral distribution is essentially determined by the microscopic damping processes that
take place in the fluid and shape both the quasielastic and the propagating modes [39]. One can
thus envisage, as a further step in the understanding of these processes, the direct tackling of
a far more fundamental question, namely the deep connection between the emergence of such a
dynamical behaviour and the underlying microscopic forces among the individual particles. The
way in which the details of the potential functions affect the Q dependence of the various relaxation
mechanisms associated to collective motions is still very little investigated and deserves dedicated
efforts. In this sense, the study of very simple fluids such as rare gases and simple molecules is
likely to still have much to reveal.
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Колективна динамiка в благородних газах та iнших дуже
простих класичних плинах
У.Бафiле1, Ф.Бароккi2,3, Е.Ґуарiнi2,3
1 Iнститут складних систем, Нацiональна Рада Дослiджень, Сесто Фiорентiно, Iталiя
2 Фiзичний факультет, Унiверситет Флоренцiї, Сесто Фiорентiно, Iталiя
3 Нацiональна Рада Дослiджень, Центр SOFT п/а Фiзичний факультет, Унiверситет Флоренцiї, Сесто
Фiорентiно, Iталiя
Отримано 12 листопада 2007 р.
Iнертнi гази та їхнi рiдини є найпростiшими системами для точних дослiджень колективної динамiки
плинiв. Було пророблено значну роботу з використанням рiзноманiтних спектроскопiчних методик,
моделювання методом молекулярної динамiки та теоретичних розробок для розумiння мiкроскопi-
чних процесiв, що приймають участь у поширеннi звукових збуджень у газах та рiдинах. Ми подаємо
короткий огляд iнтерпретацiйних схем, що застосовуються для опису таких збуджень, та нагадуємо
декiлька результатiв, отриманих iз аналiзу плинiв iнертних газiв та iнших дуже простих систем.
Ключовi слова: динамiка плинiв, колективнi збудження
PACS: 61.20.Lc, 61.20.Gy, 61.25.Bi
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