Strange kinetics, porous media, and NMR

Complex dynamics of partially freezable confined water revealed by combined experimental and computational studies

We investigate water dynamics in mesoporous silica across partial crystallization by combining broadband dielectric spectroscopy (BDS), nuclear magnetic resonance (NMR), and molecular dynamics simulations (MDS). Exploiting the fact that not only BDS but also NMR field-cycling relaxometry and stimulated-echo experiments provide access to dynamical susceptibilities in broad frequency and temperature ranges, we study both the fully liquid state above the melting point Tm and the dynamics of coexisting water and ice phases below this temperature. It is found that partial crystallization leads to a change in the temperature dependence of rotational correlation times τ, which occurs in addition to previously reported dynamical crossovers of confined water and depends on the pore diameter. Furthermore, we observe that dynamical susceptibilities of water are strongly asymmetric in the fully liquid state, whereas they are much broader and nearly symmetric in the partially frozen state. Finally, water in the nonfreezable interfacial layer below Tm does not exhibit a much debated dynamical crossover at ∼220 K. We argue that its dynamics is governed by a static energy landscape, which results from the interaction with the bordering silica and ice surfaces and features a Gaussian-like barrier distribution. Consistently, our MDS analysis of the motional mechanism reveals a hopping motion of water in thin interfacial layers. The rotational correlation times of the confined ice phases follow Arrhenius laws. While the values of τ depend on the pore diameter, freezable water in various types of confinements and mixtures shows similar activation energies of Ea ≈ 0.43 eV.

Ultrasonic study of water adsorbed in nanoporous glasses

Thermodynamic properties of fluids confined in nanopores differ from those observed in the bulk. To investigate the effect of nanoconfinement on water compressibility, we perform water sorption experiments on two nanoporous glass samples while concomitantly measuring the speed of longitudinal and shear ultrasonic waves in these samples. These measurements yield the longitudinal and shear moduli of the water-laden nanoporous glass as a function of relative humidity that we utilize in the Gassmann theory to infer the bulk modulus of the confined water. This analysis shows that the bulk modulus (inverse of compressibility) of confined water is noticeably higher than that of the bulk water at the same temperature. Moreover, the modulus exhibits a linear dependence on the Laplace pressure. The results for water, which is a polar fluid, agree with previous experimental and numerical data reported for nonpolar fluids. This similarity suggests that irrespective of intermolecular forces, confined fluids are stiffer than bulk fluids. Accounting for fluid stiffening in nanopores may be important for accurate interpretation of wave propagation measurements in fluid-filled nanoporous media, including in petrophysics, catalysis, and other applications, such as in porous materials characterization.

Assessing the Validity of NMR Relaxation Rates Obtained from Coarse-Grained Simulations of PEG-Water Mixtures

NMR relaxometry is a powerful and well-established experimental approach for characterizing dynamic processes in soft matter systems. All-atom (AA) resolved simulations are typically employed to gain further microscopic insights while reproducing the relaxation rates R₁. However, such approaches are limited to time and length scales that prevent to model systems such as long polymer chains or hydrogels. Coarse graining (CG) can overcome this barrier at the cost of losing atomistic details that impede the calculation of NMR relaxation rates. Here, we address this issue by performing a systematic characterization of dipolar relaxation rates R₁ on a PEG-H₂O mixture at two different levels of details: AA and CG. Remarkably, we show that NMR relaxation rates R₁ obtained at the CG level obey the same trends when compared to AA calculations but with a systematic offset. This offset is due to, on the one hand, the lack of an intramonomer component and, on the other hand, the inexact positioning of the spin carriers. We show that the offset can be corrected for quantitatively by reconstructing a posteriori the atomistic details for the CG trajectories.

NMR Investigation of Water in Salt Crusts: Insights from Experiments and Molecular Simulations

The evaporation of water from bare soil is often accompanied by the formation of a layer of crystallized salt, a process that must be understood in order to address the issue of soil salinization. Here, we use nuclear magnetic relaxation dispersion measurements to better understand the dynamic properties of water within two types of salt crusts: sodium chloride (NaCl) and sodium sulfate (Na₂SO₄). Our experimental results display a stronger dispersion of the relaxation time T₁ with frequency for the case of sodium sulfate as compared to sodium chloride salt crusts. To gain insight into these results, we perform molecular dynamics simulations of salt solutions confined within slit nanopores made of either NaCl or Na₂SO₄. We find a strong dependence of the value of the relaxation time T₁ on pore size and salt concentration. Our simulations reveal the complex interplay between the adsorption of ions at the solid surface, the structure of water near the interface, and the dispersion of T₁ at low frequency, which we attribute to adsorption-desorption events.

Dynamical Susceptibilities of Confined Water from Room Temperature to the Glass Transition

We confine water to narrow silica pores, where crystallization is suppressed, and determine the dynamical susceptibilities of the liquid from room temperature down to the glass transition by combining broadband dielectric spectroscopy (BDS) with ¹H and ²H nuclear magnetic resonance (NMR), in particular, by establishing NMR field-cycling relaxometry. For the correlation times, derivative analysis reveals Vogel-Fulcher-Tammann and Arrhenius regimes at T ≥ 215 K and T ≤ 160 K, respectively, which are separated by a broad crossover region. The continuous transition in the temperature dependence is accompanied by a gradual change from asymmetric high-temperature shapes of the dynamical susceptibilities to symmetric low-temperature ones and by a steady decrease of the dielectric relaxation strength. In the Arrhenius regime (E_a = 0.48 eV) at T ≤ 160 K, 2D ²H NMR spectra reveal quasi-isotropic water reorientation. We rationalize these results in terms of a crossover to an interface-affected, noncooperative relaxation involving both rotational and translational motions.

Cadmium and lead ions adsorption on magnetite, silica, alumina, and cellulosic materials

The adsorption of small particles on the surface of an adsorbent depends on interfacial dynamics and associated parameters, including the adsorbate reactivity, adsorbent surface activity, and matrix porosity and tortuosity. Herein, the effect of the surfaces of magnetite, silica/alumina, and silica-cellulose matrix on cadmium adsorption is termed using spectroscopic methods. Atomic absorption spectroscopy was used to determine the adsorption of metal ions in the solid-liquid interfaces by the batch method with different pH, metal concentrations, and contact times. Cadmium (II) were well adsorbed on the magnetite-inorganic surface (around 90% adsorption) rather than other types of semi-organic surfaces, silica, silica-alumina and other cellulosic materials (less than 60% adsorption for Cadmium (II) and 80% of Lead (II) ions). The presence of lead (II) changed the cadmium adsorption behaviour, indicating that adsorption-desorption was a physical interaction on different surfaces. Most absorptions are pH-dependent, stable for Cadmium ions and vary for Lead ions. Moreover, the adsorption analysis using Langmuir and Freundlich isotherms showed no significant characteristics of chemical interaction of the ions with the surfaces as indicated by low R2 values (both around 0.5) for magnetite materials higher for cellulose materials of Langmuir and Freundlich isotherms. This study is beneficial for various fields, such as material science and environmental chemistry, which will play an essential role in the future.

Modeling Molecular Interactions with Wetting and Non-Wetting Rock Surfaces by Combining Electron Paramagnetic Resonance and NMR Relaxometry

Three types of natural rocks─Bentheimer and Berea sandstones, as well as Liège Chalk─have been aged by immersion in a bitumen solution for extended periods of time in two steps, changing the surface conditions from water-wet to oil-wet. NMR relaxation dispersion measurements were carried out on water and oil constituents, with saturated and aromatic molecules considered individually. In order to separate the different relaxation mechanisms discussed in the literature, ¹H and ¹⁹F relaxation times were compared to ²H for fully deuterated liquids: while ²H relaxes predominantly by quadrupolar coupling, which is an intramolecular process, the remaining nuclei relax by dipolar coupling, which potentially consists of intra- and intermolecular contributions. The wettability change becomes evident in an increase of relaxation rates for oil and a corresponding decrease for water. However, this expected behavior dominates only for the spin-lattice relaxation rate R₁ at very low field strengths and for the spin-spin relaxation rate R₂, while high-field longitudinal relaxation shows a much weaker or even reverse trend. This is attributed in part to a change of radical concentration on the pore surface upon coverage of the native rock surface by bitumen as well as by the change of surface chemistry and roughness. EPR and DNP measurements quantify the change of volume vs surface radical concentration in the rocks, and an improved understanding of the role of relaxation via paramagnetic centers is obtained. By means of comparing different fluids and nuclei in combination with a defined wettability change of natural rocks, a refined model for molecular dynamics in conjunction with NMR relaxation dispersion is proposed.

Low-field and variable-field NMR relaxation studies of H2O and D2O molecular dynamics in articular cartilage

Osteoarthritis (OA) as the main degenerative disease of articular cartilage in joints is accompanied by structural and compositional changes in the tissue. Degeneration is a consequence of a reduction of the amount of macromolecules, the so-called proteoglycans, and of a corresponding increase in water content, both leading to structural weakening of cartilage. NMR investigations of cartilage generally address only the relaxation properties of water. In this study, two-dimensional (T1-T2) measurements of bovine articular cartilage samples were carried out for different stages of hydration, complemented by molecular exchange with D2O and treatment by trypsin which simulates degeneration by OA. Two signal components were identified in all measurements, characterized by very different T2 which suggests liquid-like and solid-like dynamics. These measurements allow the quantification of separate hydrogen components and their assignment to defined physical pools which had been discussed repeatedly in the literature, i.e. bulk-like water and a combination of protein hydrogens and strongly bound water. The first determination of 2H relaxation dispersion in comparison to 1H dispersion suggests intramolecular interactions as the dominating source for the pronounced magnetic field dependence of the longitudinal relaxation time T1.

Fast-field-cycling ultralow-field nuclear magnetic relaxation dispersion

Optically pumped magnetometers (OPMs) based on alkali-atom vapors are ultra-sensitive devices for dc and low-frequency ac magnetic measurements. Here, in combination with fast-field-cycling hardware and high-resolution spectroscopic detection, we demonstrate applicability of OPMs in quantifying nuclear magnetic relaxation phenomena. Relaxation rate dispersion across the nT to mT field range enables quantitative investigation of extremely slow molecular motion correlations in the liquid state, with time constants > 1 ms, and insight into the corresponding relaxation mechanisms. The 10-20 fT/\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\rm{H}}}{\rm{z}}$$\end{document}Hz sensitivity of an OPM between 10 Hz and 5.5 kHz ¹H Larmor frequency suffices to detect magnetic resonance signals from ~ 0.1 mL liquid volumes imbibed in simple mesoporous materials, or inside metal tubing, following nuclear spin prepolarization adjacent to the OPM. High-resolution spectroscopic detection can resolve inter-nucleus spin-spin couplings, further widening the scope of application to chemical systems. Expected limits of the technique regarding measurement of relaxation rates above 100 s⁻¹ are discussed.

Nuclear spin polarization and relaxation can be studied using nuclear magnetic resonance (NMR). Here the authors demonstrate a combination of fast-field cycling and optical magnetometry techniques, to realize a NMR sensor that operates in the region of very low frequency and high relaxation rate.

Slices of the Anomalous Phase Cube Depict Regions of Sub- and Super-Diffusion in the Fractional Diffusion Equation

Fractional-order time and space derivatives are one way to augment the classical diffusion equation so that it accounts for the non-Gaussian processes often observed in heterogeneous materials. Two-dimensional phase diagrams—plots whose axes represent the fractional derivative order—typically display: (i) points corresponding to distinct diffusion propagators (Gaussian, Cauchy), (ii) lines along which specific stochastic models apply (Lévy process, subordinated Brownian motion), and (iii) regions of super- and sub-diffusion where the mean squared displacement grows faster or slower than a linear function of diffusion time (i.e., anomalous diffusion). Three-dimensional phase cubes are a convenient way to classify models of anomalous diffusion (continuous time random walk, fractional motion, fractal derivative). Specifically, each type of fractional derivative when combined with an assumed power law behavior in the diffusion coefficient renders a characteristic picture of the underlying particle motion. The corresponding phase diagrams, like pages in a sketch book, provide a portfolio of representations of anomalous diffusion. The anomalous diffusion phase cube employs lines of super-diffusion (Lévy process), sub-diffusion (subordinated Brownian motion), and quasi-Gaussian behavior to stitch together equivalent regions.

Advances in the Interpretation of Frequency-Dependent Nuclear Magnetic Resonance Measurements from Porous Material

Fast-field-cycling nuclear magnetic resonance (FFC-NMR) is a powerful technique for non-destructively probing the properties of fluids contained within the pores of porous materials. FFC-NMR measures the spin-lattice relaxation rate R 1 ( f ) as a function of NMR frequency f over the kHz to MHz range. The shape and magnitude of the R 1 ( f ) dispersion curve is exquisitely sensitive to the relative motion of pairs of spins over time scales of picoseconds to microseconds. To extract information on the nano-scale dynamics of spins, it is necessary to identify a model that describes the relative motion of pairs of spins, to translate the model dynamics to a prediction of R 1 ( f ) and then to fit to the experimental dispersion. The principles underpinning one such model, the 3 τ model, are described here. We present a new fitting package using the 3 τ model, called 3TM, that allows users to achieve excellent fits to experimental relaxation rates over the full frequency range to yield five material properties and much additional derived information. 3TM is demonstrated on historic data for mortar and plaster paste samples.

Neutron spin echo monitoring of segmental-like diffusion of water confined in the cores of carbon nanotubes

Following the stream of increasing scientific interest in condensed-matter systems under ultra-hydrophobic confinement, the present work reports the first incoherent neutron spin echo assessment of the dynamics of water axially confined inside single-wall carbon nanotubes of diameter d∼ 1.4 nm. At the time scale of nanoseconds, two water populations are retrieved, whose relative proportion matches the one expected for a concentric shell + chain arrangement with cylindrical symmetry. The time dependence of the mean square displacement related to the external component is found to be subdiffusive, with peculiar resemblance to segmental diffusion typical of entangled polymeric systems.

Microstructure evolution during nano-emulsification by NMR and microscopy

HYPOTHESIS

Microstructure evolution in emulsions as a function of composition is of great interest but fundamentals have not yet been fully elucidated. Here, pulsed-field gradient (PFG) NMR diffusion measurements have been combined with confocal laser scanning microscopy (CLSM) to assess evolution of dynamics and microstructure during nano-emulsification.

EXPERIMENTS

Diffusion coefficients of emulsions made of water, mineral oil and surfactants (Span 20 and Tween 80) were measured as a function of water composition and compared with the morphological features of the emulsions obtained by CLSM.

FINDINGS

In the absence of water, two phases are visible from CLSM, and two diffusion components are observed with PFG NMR, a major fast component attributed to a continuous oil phase containing the more hydrophobic surfactant Span 20 with traces of Tween 80, and a minor slow component attributed to a dispersed phase of the more hydrophilic surfactant Tween 80 with traces of mineral oil and Span 20. At the inversion point (25 wt% water) the two-component diffusion behavior of the oil-rich phase is drastically reversed in terms of populations, with the slow diffusion process becoming dominant. This suggests a significant structuring of the oil-rich phase in the presence of surfactants enhanced by water, which can be explained by the formation of aggregates in the oil phase as reverse micelles or of a lamellar structure, and ties in well with the rheological measurements.

NMR study of microscopic mechanism of water molecules diffusion in natrolite

The mineral natrolite is a good object for studying the features of the molecular mobility of water molecules in spatially restricted structures by NMR techniques. The microscopic mechanisms of water molecule mobility in channels of monocrystal natural zeolite natrolite (Na₁₆Al₁₆Si₂₄O₈₀·16H₂O) have been investigated by broad-line (CW) NMR method. The CW NMR method, which accurately reproduces the shape of a Pake doublets of water molecules, has made it possible to trace diffusion of individual water molecules between specific positions in the crystal lattice as a function of temperature, which is important in understanding diffusion mechanisms in narrow channels of natrolite, where a diameter of the channel is smaller than the diameter of the water molecule. It has been shown that at temperatures higher than 250 K the regular diffusion of water molecules along the Schottky defects located along channels parallel to the c-crystal axis is a main kind of water molecules mobility. At temperature higher than 350 K the diffusion of water molecules in transversal channels of natrolite was observed also. Using the structural data the possible pathways of water molecules diffusion in natrolite channels have been discussed. The relationship between diffusion of water molecules and the dynamics of the zeolite framework and sodium cations is discussed on the basis of the assumption non-Markovian characte of diffusion in natrolite.

Physical characterization using diffusion NMR spectroscopy

NMR diffusion measurements (or dNMR) provide a powerful tool for analysis of solution organization and microgeometry of the environment by probing random molecular motion. Being a very versatile method, dNMR can be applied to a large variety of samples and systems. Here, a brief introduction into dNMR and a summary of recent advances in the field are presented. The research topics include restricted diffusion, anisotropic diffusion, polymer dynamics, solution structuring and dNMR method development. The dNMR studied systems include plants, cells (cell models), liquid crystals, polymer solutions, ionic liquids, supercooled solutions, untreated water, amino acid solutions and more. It is demonstrated how a variety of dNMR methods can be applied to a system to extract the data on particular structures present among, formed by or surrounding the diffusing particles. It is also demonstrated how dNMR methods can be developed to allow probing larger geometries, low sample concentrations and faster processes. Copyright © 2016 John Wiley & Sons, Ltd.

Explicit calculation of nuclear-magnetic-resonance relaxation rates in small pores to elucidate molecular-scale fluid dynamics

Nuclear-magnetic-resonance (NMR) spin-lattice (T_{1}^{-1}) and spin-spin (T_{2}^{-1}) relaxation rate measurements can act as effective nondestructive probes of the nanoscale dynamics of ^{1}H spins in porous media. In particular, fast-field-cycling T_{1}^{-1} dispersion measurements contain information on the dynamics of diffusing spins over time scales spanning many orders of magnitude. Previously published experimental T_{1}^{-1} dispersions from a plaster paste, synthetic saponite, mortar, and oil-bearing shale are reanalyzed using a model and associated theory which describe the relaxation rate contributions due to the interaction between spin ensembles in quasi-two-dimensional pores. Application of the model yields physically meaningful diffusion correlation times for all systems. In particular, the surface diffusion correlation time and the surface desorption time take similar values for each system, suggesting that surface mobility and desorption are linked processes. The bulk fluid diffusion correlation time is found to be two to five times the value for the pure liquid at room temperature for each system. Reanalysis of the oil-bearing shale yields diffusion time constants for both the oil and water constituents. The shale is found to be oil wetting and the water T_{1}^{-1} dispersion is found to be associated with aqueous Mn^{2+} paramagnetic impurities in the bulk water. These results escalate the NMR T_{1}^{-1} dispersion measurement technique as the primary probe of molecular-scale dynamics in porous media yielding diffusion parameters and a wealth of information on pore morphology.

Nuclear-magnetic-resonance relaxation due to the translational diffusion of fluid confined to quasi-two-dimensional pores

Nuclear-magnetic-resonance (NMR) relaxation experimentation is an effective technique for nondestructively probing the dynamics of proton-bearing fluids in porous media. The frequency-dependent relaxation rate T_{1}^{-1} can yield a wealth of information on the fluid dynamics within the pore provided data can be fit to a suitable spin diffusion model. A spin diffusion model yields the dipolar correlation function G(t) describing the relative translational motion of pairs of ^{1}H spins which then can be Fourier transformed to yield T_{1}^{-1}. G(t) for spins confined to a quasi-two-dimensional (Q2D) pore of thickness h is determined using theoretical and Monte Carlo techniques. G(t) shows a transition from three- to two-dimensional motion with the transition time proportional to h^{2}. T_{1}^{-1} is found to be independent of frequency over the range 0.01-100 MHz provided h≳5 nm and increases with decreasing frequency and decreasing h for pores of thickness h<3 nm. T_{1}^{-1} increases linearly with the bulk water diffusion correlation time τ_{b} allowing a simple and direct estimate of the bulk water diffusion coefficient from the high-frequency limit of T_{1}^{-1} dispersion measurements in systems where the influence of paramagnetic impurities is negligible. Monte Carlo simulations of hydrated Q2D pores are executed for a range of surface-to-bulk desorption rates for a thin pore. G(t) is found to decorrelate when spins move from the surface to the bulk, display three-dimensional properties at intermediate times, and finally show a bulk-mediated surface diffusion (Lévy) mechanism at longer times. The results may be used to interpret NMR relaxation rates in hydrated porous systems in which the paramagnetic impurity density is negligible.

Thermal wave: from nonlocal continuum to molecular dynamics

To accommodate effects of thermomass and size-dependency of thermophysical properties on heat transport and to remove the theoretical singularity of temperature gradients across the thermal wavefront NL FTPL heat conduction, corroborated with MD simulation, is introduced.

Tissue microstructure features derived from anomalous diffusion measurements in magnetic resonance imaging

OBJECTIVES

Tissue microstructure features, namely axon radius and volume fraction, provide important information on the function of white matter pathways. These parameters vary on the scale much smaller than imaging voxels (microscale) yet influence the magnetic resonance imaging diffusion signal at the image voxel scale (macroscale) in an anomalous manner. Researchers have already mapped anomalous diffusion parameters from magnetic resonance imaging data, but macroscopic variations have not been related to microscale influences. With the aid of a tissue model, we aimed to connect anomalous diffusion parameters to axon radius and volume fraction using diffusion-weighted magnetic resonance imaging measurements.

EXPERIMENTAL DESIGN

An ex vivo human brain experiment was performed to directly validate axon radius and volume fraction measurements in the human brain. These findings were validated using electron microscopy. Additionally, we performed an in vivo study on nine healthy participants to map axon radius and volume fraction along different regions of the corpus callosum projecting into various cortical areas identified using tractography.

PRINCIPAL OBSERVATIONS

We found a clear relationship between anomalous diffusion parameters and axon radius and volume fraction. We were also able to map accurately the trend in axon radius along the corpus callosum, and in vivo findings resembled the low-high-low-high behaviour in axon radius demonstrated previously.

CONCLUSIONS

Axon radius and volume fraction measurements can potentially be used in brain connectivity studies and to understand the implications of white matter structure in brain diseases and disorders. Hum Brain Mapp 38:1068-1081, 2017. © 2016 Wiley Periodicals, Inc.

Direct correlation of diffusion and pore size distributions with low field NMR

The time-dependent diffusion coefficient (D) is a powerful tool to probe microstructure in porous media, and can be obtained by the NMR method. In a real porous sample, molecular diffusion is very complex. Here we present a new method which directly measures the relationship between effective diffusion coefficients and pore size distributions without knowing surface relaxivity. This method is used to extract structural information and explore the relationship between D and a in porous media having broad pore size distributions. The diffusion information is encoded by the Pulsed Field Gradient (PFG) method and the pore size distributions are acquired by the Decay due to Diffusion in the Internal Field (DDIF) method. Two model samples were measured to verify this method. Restricted diffusion was analyzed, and shows that most fluid molecules experience pore wall. The D(a) curves obtained from correlation maps were fitted to the Padé approximant equation and a good agreement was found between the fitting lines and the measured data. Then a sandstone sample with unknown structure was measured. The state of confined fluids was analyzed and structural information, such as pore size distributions, were extracted. The D - T1 correlation maps were also obtained using the same method, which yielded surface relaxivities for different samples. All the experiments were conducted on 2MHz NMR equipment to obtain accurate diffusion information, where internal gradients can be neglected. This method is expected to have useful applications in the oil industry, particularly for NMR logging in the future.

Confined Water as Model of Supercooled Water

Water in confined geometries has obvious relevance in biology, geology, and other areas where the material properties are strongly dependent on the amount and behavior of water in these types of materials. Another reason to restrict the size of water domains by different types of geometrical confinements has been the possibility to study the structural and dynamical behavior of water in the deeply supercooled regime (e.g., 150-230 K at ambient pressure), where bulk water immediately crystallizes to ice. In this paper we give a short review of studies with this particular goal. However, from these studies it is also clear that the interpretations of the experimental data are far from evident. Therefore, we present three main interpretations to explain the experimental data, and we discuss their advantages and disadvantages. Unfortunately, none of the proposed scenarios is able to predict all the observations for supercooled and glassy bulk water, indicating that either the structural and dynamical alterations of confined water are too severe to make predictions for bulk water or the differences in how the studied water has been prepared (applied cooling rate, resulting density of the water, etc.) are too large for direct and quantitative comparisons.

Superstatistics model for T₂ distribution in NMR experiments on porous media

We propose analytical functions for T2 distribution to describe transverse relaxation in high- and low-fields NMR experiments on porous media. The method is based on a superstatistics theory, and allows to find the mean and standard deviation of T2, directly from measurements. It is an alternative to multiexponential models for data decay inversion in NMR experiments. We exemplify the method with q-exponential functions and χ(2)-distributions to describe, respectively, data decay and T2 distribution on high-field experiments of fully water saturated glass microspheres bed packs, sedimentary rocks from outcrop and noisy low-field experiment on rocks. The method is general and can also be applied to biological systems.

Nuclear magnetic resonance relaxometry of water in two and quasi-two dimensions

Molecular dynamics (MD) and Monte Carlo (MC) methods are used to determine the spin-pair correlation function G(*)(t) for the diffusion of bulk water in three dimensions (3D) and pore water in two dimensions (2D) and quasi-two dimensions (Q2D). The correlation function is required for the determination of the nuclear magnetic resonance spin-lattice and spin-spin relaxation times T(1) and T(2). It is shown that the analytic form of the powder-average correlation function, introduced by Sholl [Sholl, J. Phys. C: Solid State Phys. 7, 3378 (1974)] for the diffusion of spins on a 3D lattice, is of general validity. An analytic expression for G(*)(t) for a uniform spin fluid is derived in 2D. An analytic expression for the long-time behavior of G(*)(t) is derived for spins diffusing on 3D, 2D, and Q2D lattices. An analytic correction term, which accounts for spin pairs outside the scope of the numerical simulations, is derived for 3D and 2D and shown to improve the accuracy of the simulations. The contributions to T(1) due to translational and rotational motion obtained from the MD simulation of bulk water at 300 K are 7.4 s and 10±1 s, respectively, at 150 MHz, leading to an overall time of 4.3±0.4 s compared to the experimental value of 3.8 s. In Q2D systems, in which water is confined by alpha-quartz surfaces to thicknesses of 1-5 nm, T(1) for both translational and rotational relaxation is reduced due to the orientation and adsorption of spins at the surfaces. A method of parametrizing the MC lattice-diffusion simulations in 3D, 2D, and Q2D systems is presented. MC results for G(*)(t) for 3D and 2D systems are found to be consistent with an analytic uniform fluid model for t~/>40 ps. The value of TT(1) for translational diffusion obtained from the MC simulation of bulk water is found to be 4.8 s at 15 MHz. G(*)(t) obtained from MC simulations of Q2D systems, where water is confined by hard walls, is found to execute a distinct transition from 3D to 2D behavior. The T(1) is found to be similar to the 3D bulk water result at all pore thicknesses.

Chromatographic NMR with size exclusion chromatography stationary phases

Chromatographic NMR describes the use of stationary phases or solvent additives, such as polymers, to modify the diffusion properties of analyte molecules and thereby improve the observed resolution in the diffusion domain. This paper demonstrates similar ideas using size exclusion chromatographic media and characterises the changes in the observed diffusion coefficient using a series of polymer molecular weight reference standards of known polydispersity. The results are interpreted in terms of a simple description of the size exclusion phenomena.

Nuclear magnetic resonance studies on the rotational and translational motions of ionic liquids composed of 1-ethyl-3-methylimidazolium cation and bis(trifluoromethanesulfonyl)amide and bis(fluorosulfonyl)amide anions and their binary systems including lithium salts

Room temperature ionic liquids (ILs) are stable liquids composed of anions and cations. 1-ethyl-3-methyl-imidazolium (EMIm, EMI) is a popular and important cation that produces thermally stable ILs with various anions. In this study two amide-type anions, bis(trifluoro-methanesulfonyl)amide [N(SO(2)CF(3))(2), TFSA, TFSI, NTf(2), or Tf(2)N] and bis(fluorosulfonyl)amide [(N(SO(2)F)(2), FSA, or FSI] were investigated by multinuclear NMR spectroscopy. In addition to EMIm-TFSA and EMIm-FSA, lithium-salt-doped binary systems were prepared (EMIm-TFSA-Li and EMIm-FSA-Li). The spin-lattice relaxation times (T(1)) were measured by (1)H, (19)F, and (7)Li NMR spectroscopy and the correlation times of (1)H NMR, τ(c)(EMIm) (8 × 10(-10) to 3 × 10(-11) s) for the librational molecular motion of EMIm and those of (7)Li NMR, τ(c)(Li) (5 × 10(-9) to 2 × 10(-10) s) for a lithium jump were evaluated in the temperature range between 253 and 353 K. We found that the bulk viscosity (η) versus τ(c)(EMIm) and cation diffusion coefficient D(EMIm) versus the rate 1/τ(c)(EMIm) have good relationships. Similarly, linear relations were obtained for the η versus τ(c)(Li) and the lithium diffusion coefficient D(Li) versus the rate 1∕τ(c)(Li). The mean one-jump distances of Li were calculated from τ(c)(Li) and D(Li). The experimental values for the diffusion coefficients, ionic conductivity, viscosity, and density in our previous paper were analyzed by the Stokes-Einstein, Nernst-Einstein, and Stokes-Einstein-Debye equations for the neat and binary ILs to clarify the physicochemical properties and mobility of individual ions. The deviations from the classical equations are discussed.

Analytic treatment of nuclear spin-lattice relaxation for diffusion in a cone model

We consider nuclear spin-lattice relaxation rate resulted from a diffusion equation for rotational wobbling in a cone. We show that the widespread point of view that there are no analytical expressions for correlation functions for wobbling in a cone model is invalid and prove that nuclear spin-lattice relaxation in this model is exactly tractable and amenable to full analytical description. The mechanism of relaxation is assumed to be due to dipole-dipole interaction of nuclear spins and is treated within the framework of the standard Bloemberger, Purcell, Pound-Solomon scheme. We consider the general case of arbitrary orientation of the cone axis relative the magnetic field. The BPP-Solomon scheme is shown to remain valid for systems with the distribution of the cone axes depending only on the tilt relative the magnetic field but otherwise being isotropic. We consider the case of random isotropic orientation of cone axes relative the magnetic field taking place in powders. Also we consider the cases of their predominant orientation along or opposite the magnetic field and that of their predominant orientation transverse to the magnetic field which may be relevant for, e.g., liquid crystals. Besides we treat in details the model case of the cone axis directed along the magnetic field. The latter provides direct comparison of the limiting case of our formulas with the textbook formulas for free isotropic rotational diffusion. The dependence of the spin-lattice relaxation rate on the cone half-width yields results similar to those predicted by the model-free approach.

Water behavior in mesoporous materials as studied by NMR relaxometry

Water in mesoporous materials possessing a two-dimensional hexagonal structure has been studied by the variation of its NMR longitudinal relaxation time T(1) as a function of the static magnetic field value, or equivalently of the NMR measurement frequency. This technique, dubbed relaxometry, has been applied from 5 kHz (measurement frequency) up to 400 MHz with various instruments including a variable-field spectrometer operating between 8 and 90 MHz. Moreover, the range 0-5 kHz could be investigated by transverse relaxation, T(2) denoting the corresponding relaxation time, and relaxation in the rotating frame, T(1ρ) denoting the corresponding relaxation time. Measurements of proton relaxation rates (inverse of relaxation times) have been performed with H(2)O and HOD (residual protons of heavy water) at water volumes of 80%, 60%, and 40% relative to the porous volume. Comparison between H(2)O and HOD shows clearly that, above 1 MHz where both sets of data are superposed, relaxation is purely intermolecular and due to paramagnetic relaxation (dipolar interactions of water protons with unpaired electrons of paramagnetic entities). Below 1 MHz, it is possible to subtract the intermolecular contribution (given by HOD data) from H(2)O data so that one is left with intramolecular relaxation which is solely due to water reorientational motions. The analysis of these low-frequency data (in terms of Lorentzian functions) reveals two types of water within the pores: one interacting strongly with the surface and the other corresponding to a second layer. High-frequency data, which arise from paramagnetic relaxation, exhibit again two types of water. Due to their correlation times, one type is assigned to relatively free water within the pores while the other type corresponds to bulk (interparticular) water. Their proportions, given as a function of the volume fraction, are consistent with the above assignments.

Anomalous NMR relaxation in cartilage matrix components and native cartilage: fractional-order models

We present a fractional-order extension of the Bloch equations to describe anomalous NMR relaxation phenomena (T(1) and T(2)). The model has solutions in the form of Mittag-Leffler and stretched exponential functions that generalize conventional exponential relaxation. Such functions have been shown by others to be useful for describing dielectric and viscoelastic relaxation in complex, heterogeneous materials. Here, we apply these fractional-order T(1) and T(2) relaxation models to experiments performed at 9.4 and 11.7 Tesla on type I collagen gels, chondroitin sulfate mixtures, and to bovine nasal cartilage (BNC), a largely isotropic and homogeneous form of cartilage. The results show that the fractional-order analysis captures important features of NMR relaxation that are typically described by multi-exponential decay models. We find that the T(2) relaxation of BNC can be described in a unique way by a single fractional-order parameter (α), in contrast to the lack of uniqueness of multi-exponential fits in the realistic setting of a finite signal-to-noise ratio. No anomalous behavior of T(1) was observed in BNC. In the single-component gels, for T(2) measurements, increasing the concentration of the largest components of cartilage matrix, collagen and chondroitin sulfate, results in a decrease in α, reflecting a more restricted aqueous environment. The quality of the curve fits obtained using Mittag-Leffler and stretched exponential functions are in some cases superior to those obtained using mono- and bi-exponential models. In both gels and BNC, α appears to account for micro-structural complexity in the setting of an altered distribution of relaxation times. This work suggests the utility of fractional-order models to describe T(2) NMR relaxation processes in biological tissues.

Non-Gaussian diffusion imaging: a brief practical review

The departure from purely mono-exponential decay of the signal, as observed from brain tissue following a diffusion-sensitized sequence, has prompted the search for alternative models to characterize these unconventional water diffusion dynamics. Several approaches have been proposed in the last few years. While multi-exponential models have been applied to characterize brain tissue, several unresolved controversies about the interpretations of the results have motivated the search for alternative models that do not rely on the Gaussian diffusion hypothesis. In this brief review, diffusional kurtosis imaging (DKI) and anomalous diffusion imaging (ADI) techniques are addressed and compared with diffusion tensor imaging. Theoretical and experimental issues are briefly described to allow readers to understand similarities, differences and limitations of these two non-Gaussian models. However, since the ultimate goal is to improve specificity, sensitivity and spatial localization of diffusion MRI for the detection of brain diseases, special attention will be paid on the clinical feasibility of the proposed techniques as well as on the context of brain pathology investigations.

Pulsed-gradient spin-echo monitoring of restricted diffusion in multilayered structures

A general mathematical basis is developed for computation of the pulsed-gradient spin-echo signal attenuated due to restricted diffusion in multilayered structures (e.g., multiple slabs, cylindrical or spherical shells). Individual layers are characterized by (different) diffusion coefficients and relaxation times, while boundaries between adjacent layers are characterized by (different) permeabilities. Arbitrary temporal profile of the applied magnetic field can be incorporated. The signal is represented in a compact matrix form and the explicit analytical formulas for the elements of the underlying matrices are derived. The implemented algorithm is faster and much more accurate than classical techniques such as Monte Carlo simulations or numerical resolutions of the Bloch-Torrey equation. The algorithm can be applied for studying restricted diffusion in biological systems which exhibit a multilayered structure such as composite tissues, axons and living cells.

Continuous-time random-walk model of transport in variably saturated heterogeneous porous media

We propose a unified physical framework for transport in variably saturated porous media. This approach allows fluid flow and solute migration to be treated as ensemble averages of fluid and solute particles, respectively. We consider the cases of homogeneous and heterogeneous porous materials. Within a fractal mobile-immobile continuous time random-walk framework, the heterogeneity will be characterized by algebraically decaying particle retention times. We derive the corresponding (nonlinear) continuum-limit partial differential equations and we compare their solutions to Monte Carlo simulation results. The proposed methodology is fairly general and can be used to track fluid and solutes particles trajectories for a variety of initial and boundary conditions.

Subdiffusion in a bounded domain with a partially absorbing-reflecting boundary

The exit time of a subdiffusive process from a bounded domain with a partially absorbing/reflecting boundary is considered. The short-time and long-time behaviors of the exit time probability density are investigated by using a spectral decomposition on the basis of the Laplace operator eigenfunctions. Rotation-invariant domains are analyzed in depth in order to illustrate the use of theoretical formulas and to compare them to numerical simulations. The asymptotic results obtained are relevant for describing subdiffusion inside a living cell with a semipermeable membrane, in a chemical reactor filled with catalytic grains of finite reactivity, or in mineral or biological samples which are probed by nuclear magnetic resonance measurements subject to surface relaxation.

Tagged-particle dynamics in a fluid adsorbed in a disordered porous solid: interplay between the diffusion-localization and liquid-glass transitions

A mode-coupling theory for the slow single-particle dynamics in fluids adsorbed in disordered porous media is derived, which complements previous work on the collective dynamics [V. Krakoviack, Phys. Rev. E 75, 031503 (2007)]. Its equations, such as the previous ones, reflect the interplay between confinement-induced relaxation phenomena and glassy dynamics through the presence of two contributions in the slow part of the memory kernel, which are linear and quadratic in the density correlation functions, respectively. From numerical solutions for two simple models with pure hard-core interactions, it is shown that two different scenarios result for the diffusion-localization transition depending on the strength of the confinement. For weak confinement, this transition is discontinuous and coincides with the ideal glass transition, such as in one-component bulk systems, while, for strong confinement, it is continuous and occurs before the collective dynamics gets nonergodic. In the latter case, the glass transition manifests itself as a secondary transition, which can be either continuous or discontinuous, in the already arrested single-particle dynamics. The main features of the anomalous dynamics found in the vicinity of all these transitions are reviewed and illustrated with detailed computations.

Fractional Fokker-Planck equation for Lévy flights in nonhomogeneous environments

The fractional Fokker-Planck equation, which contains a variable diffusion coefficient, is discussed and solved. It corresponds to the Lévy flights in a nonhomogeneous medium. For the case with the linear drift, the solution is stationary in the long-time limit and it represents the Lévy process with a simple scaling. The solution for the drift term in the form lambda sgn(x) possesses two different scales which correspond to the Lévy indexes micro and micro+1 (micro<1) . The former component of the solution prevails at large distances but it diminishes with time for a given x . The fractional moments, as a function of time, are calculated. They rise with time and the rate of this growth increases with lambda .

Self-organization, optical, and electrical properties of alpha-quinquethiophene-dinucleotide conjugates

The synthesis and properties of (5')TA(3')-t5 (8a) and (5')CG(3')-t5 (8b) conjugates, in which the self-complementary dinucleotides TA and CG are covalently bound to the central ring of alpha-quinquethiophene (t5), are described. According to molecular mechanics calculations, the preferred conformation of both 8a and 8b is that with the dinucleotide folded over the planar t5 backbone, with the nucleobases facing t5 at stacking distance. The calculations show that the aggregation process of 8a and 8b is driven by a mix of nucleobase-thiophene interactions, hydrogen bonding between nucleobases (non Watson-Crick (W&C) in TA, and W&C in CG), van der Waals, and electrostatic interactions. While 8b is scarcely soluble in any solvents, 8a is soluble in water, indicating that the aggregates of the former are more stable than those of the latter. Microfluidic-induced self-assembly studies of 8a showed the formation of lamellar, spherulitic, and dendritic supramolecular structures, depending on the concentration and solvent evaporation time. The self-assembled structures displayed micrometer dimensions in the xy plane of the substrate and nanometer dimensions in the z direction. Spatially resolved confocal microscopy and spectroscopy showed that the aggregates were characterized by intense fluorescence emission. Cast films of 8a from water solutions showed chirality transfer from the dinucleotide to t5. The hole mobility of the cast films of 8a was estimated using a two-electrode device under high vacuum and found to be up to two orders of magnitude greater than those previously measured for dinucleotide-quarterthiophene conjugates under the same experimental conditions.