Multiple time scales in the microscopic dynamics of simple and complex liquids as studied by radiation scattering
Experimental investigations into the time-scales spanned by the microscopic dynamics of simple (metallic) and molecular liquids, as explored by neutron scattering and muon spin rotation experiments are reviewed.
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Інститут фізики конденсованих систем НАН України
2008
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| Цитувати: | Multiple time scales in the microscopic dynamics of simple and complex liquids as studied by radiation scattering / F.J. Bermejo, F. Fernandez-Alonso, C. Cabrillo // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 95-106. — Бібліогр.: 31 назв. — англ. |
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irk-123456789-1189992017-06-04T03:02:51Z Multiple time scales in the microscopic dynamics of simple and complex liquids as studied by radiation scattering Bermejo, F.J. Fernandez-Alonso, F. Cabrillo, C. Experimental investigations into the time-scales spanned by the microscopic dynamics of simple (metallic) and molecular liquids, as explored by neutron scattering and muon spin rotation experiments are reviewed. Зроблено огляд експериментальних дослiджень часових масштабiв, що охоплюють колективну динамiку простих (металiчних) та молекулярних рiдин, на основi експериментiв по розсiянню нейтронiв та ротацiї спiна мюонiв. 2008 Article Multiple time scales in the microscopic dynamics of simple and complex liquids as studied by radiation scattering / F.J. Bermejo, F. Fernandez-Alonso, C. Cabrillo // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 95-106. — Бібліогр.: 31 назв. — англ. 1607-324X PACS: 61.25.Mv, 66.10.-x, 66.30.Fq, 65.20.+v DOI:10.5488/CMP.11.1.95 http://dspace.nbuv.gov.ua/handle/123456789/118999 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Experimental investigations into the time-scales spanned by the microscopic dynamics of simple (metallic)
and molecular liquids, as explored by neutron scattering and muon spin rotation experiments are reviewed. |
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Article |
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Bermejo, F.J. Fernandez-Alonso, F. Cabrillo, C. |
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Bermejo, F.J. Fernandez-Alonso, F. Cabrillo, C. Multiple time scales in the microscopic dynamics of simple and complex liquids as studied by radiation scattering Condensed Matter Physics |
| author_facet |
Bermejo, F.J. Fernandez-Alonso, F. Cabrillo, C. |
| author_sort |
Bermejo, F.J. |
| title |
Multiple time scales in the microscopic dynamics of simple and complex liquids as studied by radiation scattering |
| title_short |
Multiple time scales in the microscopic dynamics of simple and complex liquids as studied by radiation scattering |
| title_full |
Multiple time scales in the microscopic dynamics of simple and complex liquids as studied by radiation scattering |
| title_fullStr |
Multiple time scales in the microscopic dynamics of simple and complex liquids as studied by radiation scattering |
| title_full_unstemmed |
Multiple time scales in the microscopic dynamics of simple and complex liquids as studied by radiation scattering |
| title_sort |
multiple time scales in the microscopic dynamics of simple and complex liquids as studied by radiation scattering |
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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/123456789/118999 |
| citation_txt |
Multiple time scales in the microscopic dynamics of simple and complex liquids as studied by radiation scattering / F.J. Bermejo, F. Fernandez-Alonso, C. Cabrillo // Condensed Matter Physics. — 2008. — Т. 11, № 1(53). — С. 95-106. — Бібліогр.: 31 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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| first_indexed |
2025-07-08T15:03:31Z |
| last_indexed |
2025-07-08T15:03:31Z |
| _version_ |
1837091546014416896 |
| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 1(53), pp. 95–106
Multiple time scales in the microscopic dynamics of
simple and complex liquids as studied by radiation
scattering
F.J.Bermejo1∗, F.Fernandez-Alonso2,3, C.Cabrillo1
1 Instituto de Estructura de la Materia, C.S.I.C. and Unidad Asociada de Investigación en Fı́sica Aplicable,
Dept. Electricidad y Electónica, Facultad de Ciencia y Tecnologı́a, Universidad del Paı́s Vasco/ Euskal
Herriko Unibertsitatea, P.O.Box 644, E–48080 Bilbao, Spain
2 ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, United Kingdom
3 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT,
United Kingdom
Received October 10, 2007
Experimental investigations into the time-scales spanned by the microscopic dynamics of simple (metallic)
and molecular liquids, as explored by neutron scattering and muon spin rotation experiments are reviewed.
Key words: dynamics, liquids, glasses, neutron scattering, muon spectroscopy
PACS: 61.25.Mv, 66.10.-x, 66.30.Fq, 65.20.+v
1. General overview
Our knowledge of the microscopic details governing atomic and molecular motions in liquids
goes back to the mid 19th century when Stokes [1] considered the motion of a macroscopic spherical
particle of radius a immersed in a continuum of viscosity η. The translational motion of such a
sphere, which was assumed to move with uniform velocity and no slippage of the fluid layer adjacent
to it, was found to be the result of two forces acting on it, namely, f1 = 4πaη resulting from
pressure built in the front of the moving particle and a frictional component f2 = 2πaη parallel
to the particle surface. The resulting friction term yields ξ = 6πaη. Later, Einstein [1] showed
that the mass self-diffusion coefficient for the moving particle is related to such friction term by
DT = kBT/ξ. The beauty of Einstein’s approach was that it enabled direct comparison with
experiment. The underlying probability density associated with the stochastic process governing
the time-dependent position of the Brownian particle also solves Fick’s diffusion equation and
yields a quantitative measure for the mean-square distance 〈u2〉 that a particle moves in a time t,
〈u2〉 = 6DTt.
The Stokes-Einstein (SE) approach is severely simplified since the momentum of the moving
particle is ignored. This leads to an unphysical result for the average velocity of the particle
v =
√
3DT/2t with the obvious consequence of a divergent behavior at short times. As first
suggested by Smoluchowski, the paradox is solved if the particle drift velocity is allowed to depend
on its position, and an approach based upon the Langevin picture is adopted. If the particle feels
the action of a field like gravity, an equivalent expression for the SE relation is replaced by the
Nernst-Einstein (NE) equation DT = kBT/ξNE where ξNE is now a friction coefficient ξNE = ~F/~v
that is now related to the force and terminal velocity of the diffusing particle. The prediction now
made is that at short times the particle velocity should become
√
3kBT/2M where M stands for
the particle mass, which removes the inconsistency mentioned above, and also tells us that at times
longer than the relaxation time for the average momentum τQ = M/ξNE, the expression for 〈u2〉
∗E-mail: javier@langran.iem.csic.es
c© F.J.Bermejo, F.Fernandez-Alonso, C.Cabrillo 95
F.J.Bermejo, F.Fernandez-Alonso, C.Cabrillo
equates that given above. Particle diffusion thus becomes a well-defined quantity for times longer
than τQ and its characteristic times are given in terms of τdiff ∝ 1/DT.
On fully analogous grounds, Debye [2] considered the rotation of a spherical molecule in a
continuum. Within the Stokes approach, this molecule is subjected to a rotational friction coefficient
ξR = 8πa3η. The reorientational motion of such object obeys a rotational diffusion equation, usually
referred to as the Stokes-Einstein-Debye (SED) relationship, with coefficient DR ∝ 1/τR which, in
turn, defines a characteristic timescale for molecular rotations.
Both the SE and SED methods encounter a number of difficulties if applied by brute force to
neat liquids. The first difficulty is to specify the size of a moving molecule. In fact, unphysically
small volumes are usually needed to fit experimental data. Second, for a liquid of like molecules,
the size of the molecules surrounding the tagged particle does matter, and a number of approaches
have been suggested to deal with this added complication [3]. Finally, the data available so far
indicate that while the translational diffusion coefficient senses quite sensitively the effects of
the macroscopic viscosity, those involved in particle reorientations may not do so. To add more
intrigue, for non-spherical molecules, rotational motions about specific axes appear decoupled from
the macroscopic viscosity, while others retain such coupling.
Still within a hydrodynamic description, that is, an extension of thermodynamics that recog-
nizes the dependence of extensive variables on spatial coordinates, we now see what happens when
a particle diffuses, leaving a given volume element due to random microscopic encounters. Such a
mass defect will generate a particle flux to recover the local equilibrium. The solution yields an
expression for the self-correlation function for the density fluctuation caused by diffusion [4] which
contains both algebraic DTt−3/2 and DTt−3 decay terms. This result reveals the complicated dy-
namics which a tagged particle experiences within a fluid as a result of its coupling with the density
and momentum fields. A description of those processes involved in the decay of fluctuations such
as those caused by a diffusing particle is known to require, for a single-component monoatomic
liquid, a minimum set of three density components for the extensive variables such as the mass,
momentum and energy densities. To specify a characteristic time scale for these processes which
explicitly involves collisions with neighboring particles, recourse is made to the simplest case where
an additional timescale τeff is brought in by recourse to the Maxwell picture of a fluid element,
and it is equated to the relaxation time for the viscosity. From here on, theory guides us on how
to calculate the quantities accessible to experiment such as the double Fourier transform of the
distinct density-density correlation function, that is the coherent S(Q,ω) dynamic structure factor
measured with neutrons or X-rays.
Here we review some recent experiments covering several different aspects of liquid dynamics.
Our aim is to illustrate, using whenever possible the same material, the multiplicity of timescales
encompassed by different kinds of molecular motions, going from simple diffusive processes in low
viscosity liquids to the description of the collective response of the fluid within the microscopic
realm.
2. Experimental techniques
2.1. Neutron scattering
Contrary to other experimental probes, inelastic neutron scattering encompasses a wide range of
energy (0.1µeV−1 eV) and momentum transfers (10−2−50 Å−1). In addition, the spin dependence
of the neutron-nucleus cross sections allows us to study, in suitably chosen cases, both single-
particle and collective dynamics in the same set of experiments. Put into different words, neutrons
are capable of sampling particle motions having characteristic times shorter than ≈100 ns, which
for neat liquids in their stable phases comprise both stochastic and finite-frequency processes.
Data reported here for molten K were measured using the full polarization analysis option
of the IN14 cold triple axis neutron spectrometer located at the Institut Laue Langevin (ILL),
Grenoble, France [5]. High-resolution quasielastic neutron spectra for molecular H2, liquid HF and
1-propanol were measured using the IRIS [6] spectrometer at the ISIS pulsed neutron source at the
Rutherford Appleton Laboratory, United Kingdom [7]. Higher-energy data concerning collective
96
Microscopic dynamics of liquids by scattering techniques
excitations for molten K, para-H2 and DF were measured using the MARI [8] chopper spectrometer
at ISIS, and the IN8 thermal triple axis machine at the ILL.
2.2. Muon spectroscopy
These measurements are closely analogous to those of T2 (Trasverse Field relaxation) in con-
ventional nuclear magnetic or electron spin resonance techniques. They are known collectively as
µSR (muon spin relaxation or rotation) – an acronym clearly contrived to resemble ESR (electron
spin resonance). In fact, our µSR studies combine analogies with both ESR and NMR (nuclear
magnetic resonance), since the implanted positive muons can access both paramagnetic and elec-
tronically diamagnetic states. These elementary particles mimic the chemical behavior of protons,
either by picking up an electron to form muonium (Mu = µ+e−) – the analogue of atomic hydrogen
(we refer to this as the paramagnetic fraction) – or else by thermalizing into diamagnetic states. In
alcohols, ROH, this latter species is undoubtedly the positive ion ROHMu+, formed in the manner
of direct protonation. The labelled molecule is neutral and diamagnetic, i.e., a closed-shell species,
and in its intra-molecular motions should closely mimic the host molecule. Data reported here
correspond to measurements carried out using the EMU and MUSR muon spectrometers located
at the ISIS pulsed muon source. Some explicit examples of relaxation spectra for this material are
shown in our previous communication [24].
For additional information on the technique the reader may browse through [9].
3. Monatomic liquids
One of the pending issues of the physics of simple alkali-metal liquids concerns the origin of
the quasielastic part of the spectrum of density fluctuations Sc(Q,ω). Here we consider the case
of molten K which, as most alkali metals, is regarded as the closest physical realization of a hard-
sphere liquid. Previous results regarding the experimental determination of the full Sc(Q,ω) [10]
have shown the inadequacy of the hydrodynamic approach, valid at low momentum transfers, and
which predicts a central, relaxing thermodiffusive mode with a line width given up to second order
in Q by DλQ2, where Dλ stands for the thermal diffusion coefficient. Beyond the hydrodynamic
realm, our knowledge of the low-frequency part of Sc(Q,ω) mostly relies on results from kinetic
theory [11] which tells that close to Qp, that is, where the static S(Q) structure factor shows its
maximum, spectral line widths may be approximated by [11],
∆ωc =
2DEQ2
S(Q)[1 − j0(Qσ) + 2j2(Qσ)]
, (1)
where DE stands for a hard-sphere diffusion coefficient as given by the Enskog prescription, σ
characterizes the size of the fluid particles and jx are spherical bessel functions of order x. Data
given in [11] for liquified rare gases and molten rubidium show a reasonable agreement between line
widths determined by experiment and equation (1), as also does more recent data on more complex
liquids such as Ga, molecular D2 or molten Ni [11,12]. Such results conform to the prediction of
equation (1) provided that the value for DE is set to that given by the mass-diffusion coefficient.
As a result, the quasielastic part of Sc(Q,ω) is understood as a self-diffusion-like process of the
liquid particles that enables the relaxation of density fluctuations.
In contrast, the Ss(Q,ω) spectra comprising the single-particle dynamics is known to follow,
within the hydrodynamic realm, an exponential relaxation process, as dictated by Fick’s Law with a
decay constant given in terms of the self-diffusion coefficient Ds which in frequency space translates
into a lorentzian line shape of width DsQ
2 and peak height proportional to 1/DsQ
2. Beyond the
hydrodynamic limit, usually located at momentum transfers of the order of 10−2 Å−1, the behavior
of Ss(Q,ω) is now understood quantitatively [13,14], mostly as a consequence of developments of
kinetic theories of the mode-coupling family. These have provided us with predictive capabilities
of accounting for the shape and characteristic parameters (i.e. wave vector dependence of its
line width and amplitude) of the single-particle spectrum. The picture that emerges from these
97
F.J.Bermejo, F.Fernandez-Alonso, C.Cabrillo
theoretical approaches portrays motions within the kinetic regime, that is, for time-length scales
comparable with inter-particle separations and microscopic times mediated by collective modes.
For temperatures close to melting, coupling of particle diffusion to a longitudinal mode results in
a retardation of diffusive motions if compared to the hydrodynamic prescription [14]. In contrast,
diffusive dynamics at higher temperatures is thought to couple to transverse modes.
Some recent measurements of the quasielastic parts of both Sc(Q,ω) (coherent) and Ss(Q,ω)
(self) have been performed by means of fully polarized neutron quasielastic scattering, in order to
compare the characteristic times involved in both single-particle and collective-diffusion processes
as well as the relationship between the Q dependence of their corresponding line widths ∆ωc and
∆ωs. Although the ratio of coherent to incoherent neutron scattering for K is about 6.18, the
coherent intensity, especially at low Q and low frequencies, is too weak to be separated by means
of model fits to the spin-independent neutron double differential cross section. Such a separation
can, however, be achieved by means of scattering experiments using polarized neutrons.
0.2
0.4
0.6
0.8
1
1.2
∆∆∆∆ ωωωω
s (
Q
)
(m
eV
)
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5 3
∆∆∆∆ ωωωω
c (
Q
)
(m
eV
),
<
ωωωω
02 >1/
2
(m
eV
)
Q (Å-1)
0
50
100
150
200
250
300
350
400
0 0.5 1 1.5 2
I s (
Q
, ωωωω
)
(a
rb
. u
ni
ts
)
Q (Å-1)
Figure 1. (Color online) The upper frame
shows ∆ωs(Q) (symbols) together with a
fit to DeffQm. The dash-dot line depicts the
best fit achieved using a quadratic, Fickian
behavior. The inset displays the measured
intensity IS(Q, ω = 0) (symbols) and the
solid line displays a fit to ∝ 1/DeffQm
prediction (see text). The lower frame
shows ∆ωc(Q) (symbols) together with a
comparison to the prediction made from
equation (1) using σ=4.7 Å and Deff=
0.24 meV Å2 (solid line). The dashed line
depicts the estimated wavevector depen-
dence of the reduced second-frequency
moment (see text).
Data shown in the upper frame of figure 1 dis-
plays the linewidths and amplitudes of a lorentzian
signal fitted to Ss(Q,ω). The best joint fit for
both the linewidth and amplitude to power laws
in DeffQm and [DeffQm]−1, respectively, yields a
subquadratic Q dependence ∆ωs(Q) = 2DeffQm
with a value for the apparent diffusion coefficient
Deff
s = 0.26 (3) meV Åm and an exponent m = 1.87
(28). A quadratic law yields a somewhat smaller
value for the diffusion coefficient Deff
s = 0.24 (9)
meV Å2 and a poorer fit. The value for the dif-
fusion coefficient comes significantly short of the
macroscopic value of 0.327 meV Å2. Such retarda-
tion of diffusive processes at microscopic scales is
usually attributed to a strong coupling with longi-
tudinal collective modes [13,14]. Put into different
words, under the present conditions mass diffusion
only takes place after a relatively large lapse of time
where liquid cages become incapable of maintaining
collective oscillatory motions. Spectral peak inten-
sities also follow such a sub-Fickian behavior and
are well accounted for by introducing a further fit-
ting parameter to serve as an overall normalization
constant.
The reduction in value of the self-diffusion
coefficient with respect to that obtained in the
macroscopic limit serves to evaluate the impor-
tance of the coupling to collective and slowly
varying dynamical modes. These ideas can be
put into numbers by first evaluating the dif-
fusion coefficient for a hard-sphere fluid DE
within the Enskog prescription in terms of the
temperature, particle mass and packing fraction
as [14],
M̃MC(iω = 0) =
kBT/M
Deff
s
[
1 −
Deff
s
DE
]
, (2)
and dividing it by the value for Deff
s given
by the quadratic fit. Equation (2) yields the
mode-coupling contribution to the Laplace trans-
98
Microscopic dynamics of liquids by scattering techniques
form of the memory-function associated with the normalized particle velocity autocorrela-
tion function 〈v(t) · v(0)〉/〈v2〉 at zero frequency. It yields a value of 11.2 ps−1 which
comes to be about twice the value reported for molten Na at an equivalent tempera-
ture [14].
Data for the coherent line width ∆ωc(Q) also shown in figure 1 displays a narrow minimum at
Qp and its overall shape displays an oscillation along the curve giving the Q dependence of the
incoherent linewidths. Within the region where equation (1) holds, that is about Qp and somewhat
beyond, the linewidth of the coherent component can be reproduced in terms of a single parameter
which is set to the value of the experimentally determined self-diffusion coefficient, since a value
for the particle diameter of about 4.7 Å was determined from the analysis of a previous inelastic
experiment [10].
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
<
v(
t)
.v
(0
)>
/<
v2 >
t (ps)
10
20
30
40
50
ττττ di
f (
ps
)
1
2
3
4
ττττ st
r (
ps
)
0.2
0.3
0.4
0 0.5 1 1.5 2 2.5
ττττ ef
f (
ps
)
Q (Å-1)
Figure 2. The upper frame displays the nor-
malized particle VACF as calculated from
the simulation results. Frames below depict
the characteristic times associated with self-
diffusion processes τdiff , the inverse of the co-
herent quasielastic linewidth τstr as calculated
from the current set of experiments, and the re-
laxation time associated with the propagation of
longitudinal acoustic excitations τeff , which was
derived in previous studies [10].
The picture that emerges from the data
just commented on portrays an intimate rela-
tionship of the microscopic motions appearing
in Sc(Q,ω) to those observable in the single-
particle quantity Ss(Q,ω). Our data also show
that diffusion-like processes either seen through
the motions of a tagged particle in Ss(Q,ω)
or given in terms of dynamic correlations be-
tween diffusing particles by ∆ωc(Q) take place
within significantly different times scales. To
illustrate such differences we plot in figure 2
the wave vector dependence of the inverse of
the incoherent linewidth τdif(Q) = 2π/∆ωs as
derived from the fits to the experimental val-
ues, the inverse of the coherent quasielastic
linewidth τstr(Q) = 2π/∆ωc as measured in
the present set of experiments, and the relax-
ation time τeff associated to the propagation of
longitudinal acoustic excitations, as studied in
previous works [10]. Hence, a neat separation
in timescales between the processes involving
particle vibratory oscillations as sampled by
τeff and mass transport phenomena explored
through τdif(Q) and τstr(Q) is inferred.
The graphs drawn in figure 2 together with
the plot showing the normalized velocity time
correlation function (VACF) 〈v(t) · v(0)〉/〈v2〉
calculated from a simulation [10] show that mi-
croscopic vibratory motions take place at times
shorter than some 0.5 ps where the VACF ex-
hibits a strong oscillatory structure including
a strong back-scattering (negative) region be-
tween 0.15 ps and 0.45 ps indicative of back-
flow effects. Such time range compares well to
the average time between collisions tcoll = 0.17
ps estimated from the thermal velocity vth =
4.52 Å ps−1 as well as from the mean free
path given by l = (
√
(2)πσ2n)−1 ≈ 0.8 Å . Relaxation phenomena sensed by quantities such as
τstr and τdif set in at times where all the oscillatory structure in the VACF has died away. Corre-
lated particle diffusion, that is, the translational motions of particles within the microscopic liquid
structures or cages sets in right after the complete decay of a density oscillation (sound mode).
τstr displays a strong momentum-transfer dependence with a narrow maximum at the peak of the
99
F.J.Bermejo, F.Fernandez-Alonso, C.Cabrillo
structure factor S(Q) and therefore suggests that these phenomena are linked to “structural re-
laxation” effects. Notice, however, that long-range particle diffusion leading to liquid-flow effects
takes place at significant longer times, as comparison of graphs for τdiff and τstr vividly exemplifies.
4. Molecular and complex liquids
Here we will briefly comment on two paradigmatic cases where the presence of multiple timescales
involving diffusive and finite-frequency motions has been firmly established by experimental means.
4.1. Diatomics: Fully separated rotations and translations
The molecular hydrogens, H2, D2 and HD, constitute a particularly useful benchmark for
the investigation of microscopic liquid dynamics. Their rather particular characteristics are best
assessed by comparison of two measures of quantum behavior for translational and rotational
motions, namely, λT = ~(Mεσ)−1/2, usually known as the de Boer parameter and λR = (2B/εφ)1/2
which is its rotational analogue. Here ε = 36.7 K and σ =2.96 Å are the potential parameters for
a Lennard-Jones potential, M stands for the molecular mass, B = ~
2/2I is the rotational constant
given in terms of the moment of inertia, and εφ sets an energy scale for the rotational motions.
While λT = 0.274, 0.224 and 0.193 provides a measure of quantum effects for translational motions
and compares to 0.479 for He or 0.0355 for N2, λR = 4.78 for H2 and 3.09 for D2 provides a measure
of the quantum nature of rotational motions and are to be compared to a meagre 0.46 for N2.
0
20
40
60
80
100
120
140
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
ττττ di
f (
ps
)
Q (Å-1)
0
1
2
3
4
ττττ st
r (
ps
)
0
0.5
1
0 0.5 1 1.5 2 2.5 3
ττττ ef
f (
ps
)
Q (Å-1)
Figure 3. The upper frame displays the charac-
teristic times associated with self-diffusion pro-
cesses τdif for molecular H2, the inverse of the
coherent quasielastic linewidth τstr as calculated
from the current set of experiments and the time
associated with the propagation of longitudinal
acoustic excitations τeff , which was derived in
previous studies [29].
The large value of the rotational constant
for hydrogen B = 85.25 K (≈ 7.35 meV) [15]
together with a consideration of the rotational
spacing given by rigid-rotor energy eigenvalues
EJ = BJ(J + 1) means that spectral transi-
tions from the para−H2 ground state to the
first rotationally excited state will take place
at ca. 15 meV, and therefore it is well sepa-
rated from quasielastic and inelastic parts of
the spectrum originating from centre-of-mass
translations. A characteristic timescale for ro-
tational motions is given in terms of the inverse
of the rotational frequency and amounts to ca.
0.3 ps. Due to the quantum nature of such para
−→ ortho-H2 transition (necessarily involving
a nuclear-spin flip), spectral broadenings of this
rotational transition are only caused by inter-
actions with neighboring ortho molecules, and
are estimated to be of the order of several tens
of µeV. We note in passing that a separation
between translations and internal rotations in
H2 is likely to break down when it is confined
in strongly interacting substrates, as recently
shown in neutron scattering studies of H2 ad-
sorbed in carbon nanohorns [26].
Under equilibrium (SVP) conditions, liquid
hydrogen is primarily in its rotational ground
state as para-H2(J = 0) and the lowest fre-
quency motions correspond to diffusive mass
transport. A recent experiment [26] has mea-
sured in detail such translational spectra for a
75:25 ortho:para H2 mixture, which was cho-
sen to isolate the single-particle spectrum since
100
Microscopic dynamics of liquids by scattering techniques
para-H2 is a mostly coherent scatterer of cold neutrons. Such measurements provide an experimen-
tal estimate of τdif in much the same way as derived for molten K. Here, due to the large differences
in neutron cross-sections between the lowerst-lying ortho- and para-H2 species, our measurements
provide direct access to self-diffusion of the ortho nuclear-spin isomer. The results are depicted in
the upper frame of figure 3 and show that at low wave vectors, the characteristic times scales are
above 100 ps. Also shown in figure 3 are data pertaining to the inverse of the coherent quasielas-
tic linewidth τstr which, for reasons given earlier, we associate with structural relaxation effects.
Such data were derived from coherent quasielastic scattering of para-H2 in a different experiment.
Remarkably, τstr displays a narrow maximum at wave vectors where S(Q) shows its maximum.
The spectrum of finite-frequency excitations for a liquid of particles interacting via a spherically
symmetric interaction potential such as para-H2(J=0) only contains acoustic phonons [29]. Thus,
characteristic times are to be compared with diffusive processes here defined as the inverse of the
phonon linewidth, that is, an estimate of excitation lifetimes. The data plotted as τeff in figure 3
are thus analogous to the same quantities derived for molten K and plotted in figure 2. They were
derived from previous experimental works devoted to the spectrum of collective excitations [29].
A comparison between data shown in figure 2 and those in figure 3 reveals a substantial simi-
litude in terms of the temporal scales involved in diffusion-like and propagating excitations. The
similitude is most striking if allowance is made for the rather different systems involved, namely, a
molten metal at T = 370 K and an insulating material at T = 14 K, the latter being characterized
by strong quantum effects.
4.2. Diatomics: Experimental separation of timescales
Here we wish to illustrate the emergence of two additional timescales in the stochastic dynam-
ics of molecular liquids. The first and the lowest in frequency of the two timescales arises from
small particle reorientations about the molecular centre of mass taking place at an average zero
frequency. Such motions, usually lumped together as rotational diffusion, are common to all molec-
ular liquids and their spectra is not always easy to isolate from contributions to the total spectral
intensity arising from translational and collective dynamics. The second timescale concerns collec-
tive dynamics at high-frequencies due to concerted, out-of-phase motions of neighboring molecules,
which are reminiscent of optic-like excitations occurring in the parent crystals.
Our system of interest here is hydrogen fluoride (HF), which is known to have the strongest as
well as the simplest hydrogen bond. These features make liquid HF a model system to which the
behavior of more complex liquids such as water can be referred to.
Recent experiments [27] have provided accurate data for the low-frequency diffusional dynamics
of this liquid. Data plotted in figure 4 correspond to the inverses of the translational and rotational
contributions to the quasielastic spectrum. The large differences in characteristic timescales for
either motion permitted a reliable separation of these two modes in the total dynamic structure
factor, accessible via inelastic neutron scattering.
Two features worth remarking relate to the rather disparate timescales in which molecular
translation and rotational diffusion take place, a difference of one-to-two orders of magnitude,
and also, as pointed out in [27], the striking differences in the thermal activation characteristics
of both sets of motions. While τdif is seen to be coupled to the shear viscosity and follows a
modified Stokes-Einstein law, τrot is effectively decoupled from such a quantity and shows a very
mild dependence with temperature. Put into different words, while translational diffusion displays
features common to molten alkali-metals such as K or liquid H2, rotational motions take place on a
far more rapid timescale and are weakly hindered by the potential fields of neighboring molecules.
In this context, it should be noted that HF displays the largest known rotational constant for a
diatom (B ∼ 2.6 meV) after H2 [15] as well as dynamic and kinematic viscosities in the liquid that
are 4–5 times smaller than those of water at ambient temperature. Also, quantum delocalization of
the proton has been shown to play an important role in describing the properties of liquid HF [16].
Even though this feature has been largely neglected in the past, it is now within the reach of
ab-initio molecular dynamics methodologies.
101
F.J.Bermejo, F.Fernandez-Alonso, C.Cabrillo
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
ττττ ro
t
(p
s)
Q (Å
_1
)
0
1
2
3
4
0 1 2 3 4
ττττ st
r (
ps
)
Q (Å-1)
50
100
150
200
250
ττττ di
f (
ps
)
Figure 4. The upper frame displays the char-
acteristic times associated with self-diffusion
processes τdif for hydrogen fluoride. The
times associated with molecular reorientation
processes are shown in the middle frame as
τrot. The inverse of the coherent quasielastic
linewidth τstr obtained from experiments using
DF, a mostly coherent scatterer, is shown in
the lower frame [30].
Recent experiments on liquid DF at T =
205 K, that is, a few degrees above melting did
not reveal the presence of well-defined excita-
tions at finite frequencies. The spectrum was
describable in terms of a lorentzian component
in addition to those attributable to the transla-
tional and rotational motions described above.
The inverse of the spectral linewidths for such a
lorentzian component is also shown in figure 4
as τstr. With these data at hand, a comparison
between figure 4 and figures 2 and 3 is in order,
as both τdiff and τstr display common features
in all three systems. The absolute timescales
are not too dissimilar and appear within a rel-
atively narrow interval. In fact, the main dif-
ferences between such quantities for the three
liquids are of a quantitative rather than a quali-
tative nature.
Interest in HF/DF also stems from reports
from molecular dynamics simulations [17]. In
these works, detailed analysis of the frequency
spectrum for single-particle and collective mo-
tions shows the presence of well-defined finite-
frequency features not accountable in terms
of acoustic-like excitations. Recent neutron ex-
periments employing relatively high incident
energies [28] have confirmed the presence of
such finite-frequency excitations in liquid DF at
T = 205 K. These modes appear at around 28
meV and 80 meV, and have been assigned to an
optic-like collective mode and to rapid molecu-
lar reorientational librations, respectively. Such
relatively fast phenomena thus set the shortest
times here considered to τvib ∼ 0.05 − 0.15 ps.
5. Complex liquids: Dynamics under highly viscous regimes
Our last case to consider pertains the most extreme example of disparate timescales, namely,
highly viscous liquids close to the liquid→glass transition temperature Tg. The latter is currently
viewed as a purely dynamical phenomenon characterized by metastable translational arrest, that
is, a thermodynamic anomaly not related to any underlying phase transition [19]. Kinetic theories
portray such a transition in terms of the onset of stochastic mass transport and reorientational
motions signalling liquid-flow taking place beyond a characteristic temperature usually referred to
as Tc located a few tens of Kelvin above the calorimetric glass-transition temperature Tg. This
theory constitutes the only predictive tool presently at our disposal which is capable of describing
glassy phenomena in simple liquids [19], molecular liquids [22], and molecular crystals [23].
Due to the huge viscosities characteristic of a deeply supercooled liquid (SCL), one expects the
above referred motions to be strongly coupled to others of vibrational nature that persist even
within the glass-phase [20,21]. The presence of such strongly coupled motions poses a number of
difficulties to isolate by experimental means phenomena related to the glass transition from others
pertaining to the material characteristics which may exhibit distinct temperature dependences
from that followed by translational mass-diffusion and may, therefore, hide any sign of criticality
at Tc, as has now been recognized [22].
102
Microscopic dynamics of liquids by scattering techniques
-24
-20
-16
-12
-8
-4
100 150 200 250 300
Lo
g.
c
ha
ra
ct
er
is
tic
ti
m
es
(
s)
T(K)
Figure 5. Dielectric relaxation times τα are
shown by open circles, those characteristic of the
high-frequency shoulder of ε′′(ω) are shown by
open lozenges and those for the β-type relax-
ation are shown by open triangles. Lines joining
the data are guides to the eye. Neutron data are
depicted by solid circles for τdif and full squares
and triangles for τr1 and τr2, respectively.
Neutron quasielastic scattering on the
molecular liquid 1-propanol (1Pr) within
its normal and deeply supercooled liquid
range [25] shows three different dynamic pro-
cesses attributed to mass-diffusion with char-
acteristic time τdif and two molecular reorien-
tational motions with times τr1 and τr2 at-
tributable to reorientations along the princi-
pal molecular axis (methyl-group motions take
place on much faster time scales). The tempera-
ture dependence of characteristic times for such
motions is compared in figure 5 to that followed
by the three main peaks (τα, secondary relax-
ation τ2nd and τβ) observed in the dielectric
permittivity spectrum ε(ω).
Data shown in figure 5 span some ten or-
ders of magnitude in dynamics. For temper-
atures well within the normal liquid range,
that is, above ∼ 170 K (melting temperature
Tm = 148 K), the dielectric τα and the neu-
tron τdif data merge with each other and fol-
low similar temperature dependences. As the
temperature is lowered towards melting and enters the SCL, dielectric and neutron data start to
diverge. While τα exhibits a temperature dependence like that of the shear viscosity within the
SCL range, all the three components of the neutron spectra show significantly milder temperature
dependences. Furthermore, data for both τα and τ2nd diverge as the temperature is lowered towards
Tg = 98 K, without showing any hint of criticality for Tc > Tg, whereas the neutron data follow a
rather mild trend down to the lowest temperatures investigated.
0.0
0.2
0.4
0.6
0.8
1.0
λλλλ T
F
(
T
)
/
λλλλ
T
F
(
T
=
0
K
)
T
g
γγγγ = 0.686, T
c
= 128(4) K
T
m
12
13
14
15
0 50 100 150 200
R
el
ax
. A
m
pl
. (
ar
b.
un
its
)
T (K)
Figure 6. The upper frame shows the tem-
perature dependence of the relaxation rate for
1PR under a transverse field of 20 G within
its glass, deeply supercooled, and normal liquid
ranges. The solid line shows the approximation
described in the text. The inset shows the esti-
mate for the critical temperature and γ factor
(see text).
Muon spin rotation (µSR) spectroscopy
provides access to a frequency region inter-
mediate between neutron and dielectric spec-
troscopy. µSR spectra yield relaxation or
damping rates that are the simple inverse of
the T2 relaxation times: λTF(T ) = T−1
2 (T ). By
heating from the glassy phase, these relaxation
rates are expected to decrease with increas-
ing temperature due to the onset of molecu-
lar motions: this is analogous to motional nar-
rowing in NMR. The results are shown in fig-
ure 6 where one sees a mild decrease in relax-
ation rate accompanied by a small increase in
signal amplitudes up to the calorimetric glass-
transition temperatures. There are no distinct
features at Tg, but a strong drop in the relax-
ation rates, accompanied by a marked increase
in signal amplitude, is seen at higher temper-
atures. Finally, a clear change of regime is at-
tained for temperatures around 1.3 times Tg,
where the relaxation rates follow a smooth, ap-
proximately linear, dependence with increasing
temperature.
To model the data shown in figure 6 re-
course is made to the predictions of Mode-
103
F.J.Bermejo, F.Fernandez-Alonso, C.Cabrillo
Coupling Theories (MCT) [19,22] concerning the Lamb-Mössbauer factor. On the grounds of such
an analogy, we describe our data using a term proportional to
√
|T − Tc| /Tc, where Tc stands for
a critical temperature, and the relaxation rate follows
λ(T ) = Aλglass(T )
√
Tred + ζ(Tred), T 6 Tc ,
λ(T ) = ζ(Tred), T > Tc (3)
with
Tred = |T − Tc| /Tc, ζ(Tred) = α0 + β0Tred. (4)
The high temperature data is modelled by a mild, linear temperature dependence characterized
by a slope easily calculated from the measured data once the value for Tc is set. Fitting the
relaxation rates and leaving Tc and r as adjustable parameters yields a value of Tc = 128(4) K,
that is some 30 K above Tg, in agreement with theoretical predictions.
The result illustrates how the sought phenomena associated with the dynamic glass transition
are best seen if the single-particle dynamics is explored within mesoscopic time scales, which are
slower than 10−8 s but faster than some fraction of a millisecond. Motions within such limits are
free from contributions from internal and deformational modes that considerably complicate the
response of the SCL if explored using higher-frequency probes.
6. Concluding remarks
Our main aim in reporting on the four cases here considered was to illustrate the close interre-
lationships between the dynamics of a single-tagged molecule executing diffusion-like motions and
cooperative rearrangements involving groups of particles within the liquid having an average zero
frequency. From the data and arguments here given for τdif , we see that the Brownian diffusion
regime is only achieved after some hundreds of picoseconds and that within such a lapse of time,
many particle rearrangements with a characteristic time τstr will take place. In contrast, coherent
motions of groups of particles with well-defined frequencies will take place within intervals of time
spanning at most some tenths of a picosecond as clearly illustrated by data on τeff and τvib.
The data here reported show that the characteristic times under consideration can only be
related to relevant macroscopic transport coefficients such as that for the shear viscosity by means
of SE or SED approaches under rather stringent conditions. Mass-diffusion motions as sampled
by τdif are certainly coupled to the viscosity within the normal liquid range but, as the data for
our last case study show, diffusion within times far shorter than those characteristic of viscosity
relaxation, may also take place. Even within the normal liquid range, recent data for HF have shown
the emergence of fractional SE behavior [27], which marks a significant departure from Brownian
dynamics approaches. Finally, the data shown here concerning rotational motions indicate that
while SED behavior may constitute an apt prescription for the calculation of the rotational diffusion
coefficient for a large object such a macromolecule or a colloidal particle immersed within a liquid
composed of low-molecular-weight particles, its relevance to rationalize data concerning simple
molecular liquids, should be considered as a first-order, rather approximate approach at best.
Finally, the current results point towards the need of further developments of theoretical tools
capable of accommodating the current observations and thus bridging the gap between theory and
experiment. In fact, theories on collective dynamics of liquids still lag well behind experimental
observations and conversely, analysis of experimental data is many times carried out using ex-
pressions only valid within the hydrodynamic realm. Efforts carried out along the lines sketched
by treatments based upon the Generalized Collective Modes [31] approach appear as a promis-
ing alternative and their further advancement should be pursued in order to develop a consistent
framework useful in analyzing bare experimental data.
7. Acknowledgements
FFA gratefully acknowledges financial support from the UK Science and Technology Facilities
Council.
104
Microscopic dynamics of liquids by scattering techniques
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105
F.J.Bermejo, F.Fernandez-Alonso, C.Cabrillo
Рiзнi часовi масштаби в мiкроскопiчнiй динамiцi простих та
складних рiдин з дослiджень по радiацiйному розсiянню
Ф.Х.Бермехо1, Ф.Фернандез-Алонсо2,3, К.Кабрiйо 1
1 Iнститут структури матерiї, Наукова Рада Дослiджень та факультет науки та технологiї, Унiверситет
Паїс Васко, Бiльбао, Iспанiя
2 Лабораторiя Резерфорд Аплтон, Чiлтон, Великобританiя
3 Факультет фiзики i астрономiї, Лондонський унiверситетський коледж, Лондон, Великобританiя
Отримано 10 жовтня 2007 р.
Зроблено огляд експериментальних дослiджень часових масштабiв, що охоплюють колективну ди-
намiку простих (металiчних) та молекулярних рiдин, на основi експериментiв по розсiянню нейтронiв
та ротацiї спiна мюонiв.
Ключовi слова: динамiка, рiдини, стекла, розсiяння нейтронiв, мюонна спектроскопiя
PACS: 61.25.Mv, 66.10.-x, 66.30.Fq, 65.20.+v
106
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