Molecular velocity auto-correlation of simple liquids observed by NMR MGSE method

The velocity auto-correlation spectra of simple liquids obtained by the NMR method of modulated gradient spin echo show features in the low frequency range up to a few kHz, which can be explained reasonably well by a $t^{-3/2}$ long time tail decay only for non-polar liquid toluene, while the spectra of polar liquids, such as ethanol, water and glycerol, are more congruent with the model of diffusion of particles temporarily trapped in potential wells created by their neighbors. As the method provides the spectrum averaged over ensemble of particle trajectories, the initial non-exponential decay of spin echoes is attributed to a spatial heterogeneity of molecular motion in a bulk of liquid, reflected in distribution of the echo decays for short trajectories. While at longer time intervals, and thus with longer trajectories, heterogeneity is averaged out, giving rise to a spectrum which is explained as a combination of molecular self-diffusion and eddy diffusion within the vortexes of hydrodynamic fluctuations.

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 p-1 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, D = kB T 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] 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 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][22][23][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 where the dynamics of a single spin operator I xi in the magnetic field, and the molecular motion govern the evolution of the density matrix By additional factoring of the spin evolution operator U is (t), we detach the spin interaction with the main magnetic field U io (t), the interaction with RF field U iRF (t), and the interaction with gradient field G, [18,27] governed by the operator Here T is the time ordering operator, r i (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, has a peak at ω m = π/T m , where T m is the time interval between the π-RF pulses. The resulting spin echo attenuation [28]: p-3 is proportional to VAS at the modulation frequency 1/T m . By changing T m we can sample D z (ω), from which we can obtain the mean-squared displacement of the spin bearing particles using the relation [z(t) − z(0)] 2 = 4 π ∞ 0 D z (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 4 o C to 40 o 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 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: 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. ζ 1 and ζ 2 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 Only three parameters characterize the spectrum: the characteristic time of molecular coupling to attractive centers τ c = ζ1ζ2 (ζ1+ζ2)κ , the high-frequency plateau is the diffusion rate resulting from short almost unconstrained molecular jumps D(∞) = f (t) 2 τc ζ 2 1 , and the diffusion rate at long displacements D(0) = ( f (t) 2 + g(t) 2 )τc (ζ1+ζ2) 2 , 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 4 o 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 20 o 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 4 o C it is the combination of one and four hydrogen-bonds in almost equal proportion. VAS at 40 o C (not shown here) is almost identical to that at 20 o C, but with the plateau lifted to 2.9 × 10 −9 m 2 /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 p-5 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. * * *