Non-Brownian molecular self-diffusion in bulk water
arXiv:1010.1175v1 [cond-mat.soft] 6 Oct 2010
J. Stepišnik1,2 , I. Serša2 and A. Mohorič1
1
University of Ljubljana, Faculty of Mathematics and Physics, Department of Physics Jadranska 19, 1000 Ljubljana, Slovenia
2
Josef Stefan Institute - Jamova 39, 1000 Ljubljana, Slovenia
PACS
PACS
PACS
PACS
PACS
PACS
05.40.Jc
76.60.Lz
66.10.cg
82.30.Rs
33.15.Fm
61.20.Ja
–
–
–
–
–
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Brownian motion
Spin echoes
Mass diffusion, including self-diffusion, mutual diffusion, tracer diffusion, etc.
Hydrogen bonding, hydrophilic effects
Bond strengths, dissociation energies
Computer simulation of liquid structure
Abstract – The paper presents the velocity autocorrelation spectrum of bulk water measured by
a new technique of NMR modulated gradient spin echo method. This technique is unprecedented
for the spectrum measurement in the frequency interval between a few Hz to about 100 kHz with
respect to directness and clarity of results and shows that a simple model of Brownian self-diffusion
is not applicable to describe the diffusion dynamics of water molecules. The observed temperature
dependant spectra of water show the existence of a slow chain-like dynamics in water, which we
explain by coupling of diffusing molecule to broken bonds in the hydrogen bond network.
Water properties are essential to many physical, chemical, biological, and geological
processes. Although a great deal is known about thermodynamic properties of bulk water,
the complex forces that govern its molecular arrangements and dynamics obscure a detailed
picture of instantaneous local structure of water and its evolution. Understanding these
phenomena has been one of the most important scientific challenges of the past 100 years.
In water, intermolecular hydrogen bonds lead to a highly associative structure that has
attracted the interest of scientists because it exists in the temperature range associated
with the existence of life. According to current knowledge, the structure of liquid water
is a disordered network of molecules connected by hydrogen bonds, which behaves on time
scales of 10−12 to 10−9 sec as a gel. The structure fluctuates and reorganizes, due to
rotations and other thermal motion, causing the unceasing breaking of individual hydrogen
bonds and their reformation into new configurations, whose extent and influence depend on
temperature and pressure. It is thought that the hydrogen-bond network does not remain
intact for long enough to be detected directly as an observable entity in ordinary bulk water
at normal pressures.
Insights into structural properties of water have come from x-ray and neutron-scattering
experiments, which lack dynamical information; insights into its dynamics have been gained
from ultrafast time-resolved experiments, which have lacked structural details [1, 2]. The
most detailed understanding of water properties derives from molecular dynamics simulations, which provide an atom-by-atom perspective on how hydrogen bonding changes with
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J. Stepišnik et al.
time. Although substantial efforts have been made to develop models for molecular dynamics simulations [3] that show some degree of success in reproducing the experimentally
observed radial distribution function [4], the models have difficulty addressing the unusual
nature of long-range ordering that strongly affects the properties of water and dynamics of
ordering has never been properly tested against the experiment.
The traditional experimental techniques are yielding diminishing returns, owing to the
collective efforts of many generations of researchers. Instead, new results should be expected
with techniques unrelated to these mainstream research efforts. For example, the recent
dielectric measurements show an anomalously slow relaxation process in water [5], which
is believed to be a kind of collective motion of hydrogen-bonded structures. It gave an
impetus to apply other nontraditional techniques for the study of long range dynamics in
water, among which the measurement of translation dynamics by a new NMR technique
can provide valuable information.
Many of the equilibrium and dynamic properties of liquid water show anomalous behavior
that is changing with temperature and pressure. Example is translational self-diffusion,
which is a key property for a long range molecular dynamics. Its measurements by tracer
methods [6], NMR techniques [7, 8] and indirectly by non-elastic neutron scattering [9]
reveal anomalously strong temperature dependence of self-diffusion coefficient that can be
attributed to the intermolecular hydrogen bonding in liquid water [10, 11]. However, the
diffusion coefficients calculated for many theoretical models of liquid water give mostly
larger values than experimentally observed [12].
NMR pulsed-gradient-spin-echo NMR method (PGSE) plays a prominent role in the
measurements of the molecular self-diffusion. PGSE determines the molecular displacement
in a time interval between two pulses of applied magnetic field gradient via the spin echo
attenuation [13]. However, the results of measurements in water at different temperatures
and pressures differ among authors [6, 8, 14], on a scale exceeding the experimental uncertainty. Presumably the uncertainties in the gradient calibration and convection flows due to
temperature variation in the sample are the cause. The measurements with the new NMR
technique presented here show that the scattering might be a result of the time-dependent
self-diffusion coefficient, i.e. non-Brownian diffusion of water molecules [15] that has been
overlooked in the PGSE experiments. Nevertheless, PGSE measurements brought forth the
poor validity of the hydrodynamic model of water [8]. Namely, the obtained self-diffusion
coefficient D expressed in terms of the Stokes-Einstein-Debye formula of particle diffusion,
kB T
D = 6πRη
, shows a dramatic change in the effective radius R of the diffusing species in the
temperature interval from 300 to about 600 K. This necessities a new molecular description
of self-diffusion in water that takes intermolecular hydrogen bonding into account.
A step in this direction comes from a new viewpoint on the labelling of moving spinbearing molecules by a sequence of magnetic field gradient and radio-frequency (RF) pulses
in NMR. It turns out as a natural relation of the accumulated time dependent spin phase
to the velocity autocorrelation spectrum (VAS) of molecular translation motion [16], which
is defined as [17]
Z
1 ∞
h∆vz (t) ∆vz (0)ie−iωt dt ,
(1)
Dz (ω) =
π −∞
where ∆vz (t) = vz (t) − hvz (t)i is the component of the velocity fluctuation in the direction
of the applied gradient (z-axis in our case). Repetitive sequence of RF and magnetic field
gradient pulses results in a periodic spin phase modulation. The frequency spectrum of
this modulation has a narrow peak, which can be used to sample VAS by changing the
modulation frequency [18]. This technique of modulated gradient spin echo (MGSE) acts as
a filter with an adjustable detection window into the frequency domain of spin translation
dynamics because it samples only molecular motion that is in pace with the spin phase
modulation. The method is unique in a clear discrimination of fast diffusion-like motion
from slow collective one, e.g. flow or convection in fluids. Therefore, it does not require a
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Non-Brownian molecular self-diffusion in bulk water
special temperature stabilization to prevent convection flows within the sample, which is a
big obstacle in the PGSE measurement. VAS of ordinary liquids decreases at frequencies
higher than the reciprocal correlation time of velocity fluctuations being in the pico-second
range. Its constant value at low frequencies determines the Brownian character of molecular
self-diffusion [19, 20]. MGSE measurements of liquid reveal whether character of molecular
self-diffusion is Brownian or not and cast a new light on the structural dynamics.
The pulsed-gradient version of the MGSE method has already been used to study flow
and self-diffusion in porous media [21–24], but the inductance of gradient coils limits the
switching time of gradient pulses to the modulation frequencies below 1 kHz. For this
reason, the technique does not attract a lot of attention. For instance, in porous media, it
cannot explore the characteristic of VAS typical for restricted diffusion in pores smaller than
1 µm. Recently, we demonstrated [25] that a standard Carr-Purcell-Meiboom-Gill sequence
of π-RF pulses [26] applied simultaneously with a constant magnetic field gradient acts as
a MGSE method. This technique overcomes the high-frequency deficiency of the pulsed
MGSE technique, because the absence of gradient-switching induced eddies enables VAS
measurement up to the frequencies of about 100 kHz. In the NMR receiving coil aligned
along x-axis, spins induce signal
E(τ ) ∼
d X
T rρi (τ )Ixi ,
dt i
(2)
where the dynamics of a single spin operator Ixi in the magnetic field, and the molecular
motion govern the evolution of the density matrix
+
+
ρi (τ ) = Ui (t)ρo Ui+ (t) = Uim (t)Uis (t)ρo Uis
(t)Uim
(t).
(3)
By additional factoring of the spin evolution operator Uis (t), we detach the spin interaction with the main magnetic field Uio (t), the interaction with RF field UiRF (t), and the
interaction with gradient field G, [18, 27] governed by the operator
Z τ
UiG (τ ) = T exp[iγG ·
ri (t){Izi cos πa(t) +
0
+
Ixi sin πa(t)}dt].
(4)
Here T is the time ordering operator, ri (t) is the fluctuating location of the i-th spin, and
the factor a(t) alternates between 0 and ±1 with the rate of spin phase inversion by π-RF
pulses [16]. The second term of the exponent includes the resonance offset effects, which
have to be reduced as much as possible. Proper phase cycling of π-RF pulses suppresses the
second term to an extent that the combination of RF pulses and constant magnetic field
gradient behaves as an effective gradient with the sign alternating with the rate of applied
π-RF pulses, when the amplitude of RF field is larger than the difference of the non-uniform
magnetic field across the sample. This condition implies that the gradient selected slice
must not be smaller than the sample. The gradient spectrum,
Z τ
G(ω, τ ) = G
cos(πa(t))eiωt dt ,
0
has a peak at ωm = π/Tm , where Tm is the time interval between the π-RF pulses. The
resulting spin echo attenuation [28]:
Z
Dz (ω) ∗
γ2 ∞
G (ω, τ ) dω
G(ω, τ )
β(τ ) =
2π −∞
ω2
=
8γ 2 G2 τ Dz (ωm )
2
π2
ωm
p-3
(5)
J. Stepišnik et al.
Fig. 1: Measured velocity correlation spectra of water at 21o C (blue) and at 4o C (red) are
fitted with the spectra calculated from the Langevin equations of harmonically coupled Brownian
particles. The spectrum of ethanol at 21o C (green) is given to emphasize the difference between
water and liquid with a weaker hydrogen bonding.
is proportional to VAS at the modulation frequency 1/Tm. By changing Tm we can sample Dz (ω), from which we can obtain the mean-squared
displacement of the spin bearing
R∞
particles using the relation h[z(t) − z(0)]2 i = π4 0 Dz (ω)(1 − cos(ωt))/ω 2 dω.
Our experiments were carried out on a TecMag NMR spectrometer with a 2.35 T horizontal bore superconductive magnet equipped with micro-imaging accessories and Maxwell
gradient coils. Samples were sealed in a small glass tube of 0.7 cm diameter and 2 cm
length. With this equipment, we have demonstrated the use of the new MGSE technique
by measuring VAS of restricted diffusion in porous medium with pores smaller than 0.1 µm
[25] and by the first observation of VAS of fluidized granular system [29].
In a study of self-diffusion in gel systems by this technique, water was meant to be used
as a calibrator at first. However, much to our surprise VAS of water exhibited a very distinct
low-frequency decrease in the range below 2 kHz as shown in fig. 1. Systematic error was
ruled out by identical measurements of other liquids: nitrobenzene, toluene and ethanol,
which all yielded a flat VAS in this low frequency range (fig. 1). Possibility of water heating
by a high duty cycle of RF pulse train was ruled out by performing experiments with the
same modulation frequencies but 40 ms and 400 ms long CPMG RF trains. Both measurements give identical frequency dependencies. The expected result of MGSE measurements
is also a strong temperature dependence of water VAS in the temperature interval of our
measurements, from 4o C to 40o C, which at low frequencies converge within experimental
error towards the results of Mills. Mills measurements by the diaphragm technique are
commonly considered as the most accurate measurements of long time water self-diffusion.
The observed VAS of water has features similar to those of restricted self-diffusion in
porous media [22, 25] or of polymer self-diffusion [30], which indicates interaction of the
diffusing molecule with surrounding and thus constraining its displacements. As shown
in the following, a simple model of a Brownian particle interacting with local attractive
centers [31] fits well the observed low frequency decay of VAS. Therefore, we assume that an
association of the diffusing molecule with incessantly breaking bonds of the hydrogen bond
network in water is responsible for its constrained motion. This also explains a stronger low
frequency decrease of VAS at lower temperature, fig. 1, as the structure of hydrogen bonds
becomes more stable.
This reasoning follows the lines of a widely accepted view of tetrahedral arrangements
of water molecules ordered by hydrogen bonds, where the mechanism of fast switching between the hydrogen-bonded partners remains unclear. Recent combined studies by 2D IR
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Non-Brownian molecular self-diffusion in bulk water
spectroscopy and molecular dynamics simulations [32, 33] revealed that a typical configuration has many instances of molecules with apparently broken hydrogen bonds. Simulation
studies [34] showed that although the majority of molecules in liquid water participate in
four hydrogen bonds, many of them participate also in two, three or five hydrogen bonds at
a given time. Recent observation of slow dynamics in the frequency range below 1 kHz by
the dielectric measurement of water [5] indicated a possibility that the distorted hydrogenbond structure behaves like a chain in polymers. Although translational molecular motion
itself does not lead to a significant bond breaking or bond restoring, the results of MGSE
measurement give a new insight into interactions of molecule diffusing within the hydrogen
bond network in water.
To interpret the distorted VAS in water, we use a simple model of Brownian particle
harmonically coupled to attractive centers [31, 35] i.e., to broken bonds of the hydrogen
bond network. We assume that instability of the network introduces the fluctuation of
the attractive centers, which is independent of the forces acting on the diffusing molecule.
As the relaxation times of the forces associated with the motion of libration are short in
comparison to the times of the translational motion, the Markovian approximation of the
stochastic process permits to describe the molecular displacements along the applied gradient
with coupled Langevin equations without the oscillating terms:
dz
+ κ(z − Z) = f (t)
dt
dZ
−ζ2
+ κ(Z − z) = g(t).
dt
− ζ1
(6)
Here κ is the coupling constant between the diffusing molecule, at the coordinate z, to
the attractive point in the hydrogen bond network located at Z. ζ1 and ζ2 are friction
coefficients while f (t) and g(t) are stochastic forces acting on the diffusing molecule and on
the attractive point respectively. Calculation gives the low frequency part of VAS as
D(ω) = D(0)
D(∞) 2 2
D(0) ω τc
1 + ω 2 τc2
1+
.
(7)
Only three parameters characterize the spectrum: the characteristic time of molecular cou1 ζ2
pling to attractive centers τc = (ζ1ζ+ζ
, the high-frequency plateau is the diffusion rate
2 )κ
resulting from short almost unconstrained molecular jumps D(∞) =
(hf (t)2 i+hg(t)2 i)τc
,
(ζ1 +ζ2 )2
hf (t)2 iτc
,
ζ12
and the dif-
fusion rate at long displacements D(0) =
is affected by the network interaction. For fixed centers, the value of D(0) is zero but, in our model, the random breaking
and restoring of bonds results in a finite value of D(0) and is a measure of network stability.
By assuming the probability that the diffusing molecule can form multiple bonds with
the free nodes on the network during its travel through the medium, we have obtained the
model spectrum that fits well to the experimental data as shown in fig. 1. At 4o C, the best
fit is obtained if the molecule is coupled 40% of the time with a force four times weaker then
the coupling force for the remaining 60% of the time. At 20o C the best fit is obtained if
the coupling is mostly of the weaker type. Fit with intermediate values of coupling deviates
from the experimental data. We attribute the four-to-one ratio of the forces to the molecular
association with one hydrogen bond for the weaker type and with four hydrogen bonds for
the stronger one. Thus, effective interaction of diffusing molecule at room temperature is
one-hydrogen bond, while at 4o C it is the combination of one and four hydrogen-bonds
in almost equal proportion. VAS at 40o C (not shown here) is almost identical to that
at 20o C, but with the plateau lifted to 2.9 × 10−9 m2 /s, what could mean that one-bond
interaction is dominant at higher temperatures, while the tetrahedral arrangements
prevail
√
near the freezing point. Plots of measured spectra as a function of ω show a linear
dependence at very low frequencies, which might indicate a chain-like motion similar to
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J. Stepišnik et al.
that in polymers [36], when only one hydrogen bond is involved. According to this model,
a larger difference D(∞) − D(0) at 4 o C compared to those at higher temperatures could
indicate a higher network stability at low temperatures.
Evidently, the results of MGSE measurement of the bulk water open some interesting
questions about the impact of the network-like structure on the molecular self-diffusion.
A simple theoretical explanation for VAS of water is provided, however, a more detailed
theoretical analysis still needs to be done.
∗∗∗
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