The narrow pulse approximation and long length scale determination in xenon gas diffusion NMR studies of model porous media

We report a systematic study of xenon gas diffusion NMR in simple model porous media, random packs of mono-sized glass beads, and focus on three specific areas peculiar to gas-phase diffusion. These topics are: (i) diffusion of spins on the order of the pore dimensions during the application of the diffusion encoding gradient pulses in a PGSE experiment (breakdown of the narrow pulse approximation and imperfect background gradient cancellation), (ii) the ability to derive long length scale structural information, and (iii) effects of finite sample size. We find that the time-dependent diffusion coefficient, D(t), of the imbibed xenon gas at short diffusion times in small beads is significantly affected by the gas pressure. In particular, as expected, we find smaller deviations between measured D(t) and theoretical predictions as the gas pressure is increased, resulting from reduced diffusion during the application of the gradient pulse. The deviations are then completely removed when water D(t) is observed in the same samples. The use of gas also allows us to probe D(t) over a wide range of length scales and observe the long time asymptotic limit which is proportional to the inverse tortuosity of the sample, as well as the diffusion distance where this limit takes effect (approximately 1-1.5 bead diameters). The Padé approximation can be used as a reference for expected xenon D(t) data between the short and the long time limits, allowing us to explore deviations from the expected behavior at intermediate times as a result of finite sample size effects. Finally, the application of the Padé interpolation between the long and the short time asymptotic limits yields a fitted length scale (the Padé length), which is found to be approximately 0.13b for all bead packs, where b is the bead diameter.     

NMR diffusometry and the short gradient pulse limit approximation

In NMR diffusometry, one often uses the short gradient pulse (SGP) limit approximation in the interpretation of data from systems with restricted diffusion. The SGP limit approximation means that the gradient pulse length, delta, is so short that the spins do not diffuse during the pulse duration, but this condition is rarely met. If the length scale of the pores corresponds to the molecular mean square displacement during the gradient pulse, the measured echo intensities become a function of the gradient pulse length. Here, we have studied highly concentrated emulsions to show how the length of the gradient pulse influences NMR diffusion experiments. We have focused on molecules confined to one pore and molecules that can migrate through the porous system. For the former the echo decays give smaller pores than the actual case and for the latter we show large changes in echo decay depending on the gradient pulse length, everything else being equal.     

Effects of finite-width pulses in the pulsed-field gradient measurement of the diffusion coefficient in connected porous media

We analytically compute the apparent diffusion coefficient D(app) for an open restricted geometry, such as an extended porous medium, for the case of a pulsed-field gradient (PFG) experiment with finite-width pulses. In the short- and long-time limits, we give explicit, model-independent expressions that correct for the finite duration of the pulses and can be used to extract the pore surface-to-volume (S/V) ratio as well as the tortuosity. For all times, we compute D(app) using a well-established model form of the actual time-dependent diffusion coefficient D(t) that can be obtained from an ideal narrow-pulse PFG. We compare D(app) and D(t) and find that, regardless of pulse widths and geometry-dependent parameters, the two quantities deviate by less than 20%. These results are in sharp contrast with the studies on closed geometries [J. Magn. Reson. A 117 (1995) 209], where the effects of finite gradient-pulse widths are large. The analytical results presented here can be easily adapted for different pulse protocols and time sequences.     

Measurements of restricted diffusion using an oscillating gradient spin-echo sequence

An oscillating gradient spin-echo (OGSE) pulse sequence was used to measure the apparent diffusion coefficient (D(app)) of water in the short diffusion time regime in the presence of restrictions. The diffusion coefficients of water in a simple water sample and a water and oil mixture were measured to be the same for different periods of the gradient oscillation, as expected when there are no restriction effects. The D(app) of water in the spaces between closely packed beads was also measured as a function of the gradient oscillation periods in the range 11 to 80 ms. The D(app) of water in restricted systems varies with the period of the gradient oscillation and the dispersion depends on the scale of the restriction. For a sample of packed beads of diameter 9.1 +/- 0.7 microm, the pore surface-to-volume ratio was estimated experimentally by this method to be 1.3 +/- 0.1 microm(-1), corresponding to a mean pore diameter of 6.4 +/- 0.7 microm. A Monte Carlo computer simulation of the NMR OGSE signal from the spins diffusing in a system of compartments was also implemented and the D(app) demonstrated similar behavior with gradient oscillation periods.     

Probing porous media with gas diffusion NMR.

We show that gas diffusion nuclear magnetic resonance (GD-NMR) provides a powerful technique for probing the structure of porous media. In random packs of glass beads, using both laser-polarized and thermally polarized xenon gas, we find that GD-NMR can accurately measure the pore space surface-area-to-volume ratio, S/V rho, and the tortuosity, alpha (the latter quantity being directly related to the system's transport properties). We also show that GD-NMR provides a good measure of the tortuosity of sandstone and complex carbonate rocks.     

Microstructural features assessment of different waterlogged wood species by NMR diffusion validated with complementary techniques

Wood is a hygroscopic, multi-scale and anisotropic natural material composed of pores with different size and differently oriented. In particular, archaeologically excavated wood generally is waterlogged wood with very high moisture content (400%–800%) that need to have a rapid investigation at the microstructural level to obtain the best treatment with preservative agents. Time-dependent diffusion coefficient D(t) quantified by Pulse Field Gradient (PFG) Nuclear Magnetic Resonance (NMR) techniques provides useful information about complex porous media, such as the tortuosity (τ) describing pore connectivity and fluid transport through media, the average-pore size, the anisotropic degree (a_n). However, diffusion NMR is intrinsically limited since it is an indirect measure of medium microstructure and relies on inferences from models and estimation of relevant diffusion parameters. Therefore, it is necessary to validate the information obtained from NMR diffusion parameters through complementary investigations. In this work, the structures of five waterlogged wood species were studied by PFG of absorbed water. D(t) and τ of water diffusing along and perpendicular to vessels/tracheids main axes together with relaxation times and a_n were quantified. From these parameters, the pore sizes distribution and the wood microstructure characterization were obtained. Results among wood species were compared, validated and integrated by micro-imaging NMR (μ-MRI), environmental-scanning electron-microscope (ESEM) images, wood dry density and imbibition times measurement of all woods. The work suggests that a_n vs τ rather than the estimated pore size diversifies and characterize the different wood species. As a consequence diffusion-anisotropy vs tortuosity could be an alternative method to characterize and differentiate wood species of waterlogged wood when high resolution images (μ-MRI and ESEM) are not available. Moreover, the combined use of D(t) and micro-MRI expands the scale of dimensions observable by NMR covering all the interesting length scales of wood.     

Random-walk technique for simulating NMR measurements and 2D NMR maps of porous media with relaxing and permeable boundaries

We revisit random-walk methods to simulate the NMR response of fluids in porous media. Simulations reproduce the effects of diffusion within external inhomogeneous background magnetic fields, imperfect and finite-duration B(1) pulses, T(1)/T(2) contrasts, and relaxing or permeable boundaries. The simulation approach consolidates existing NMR numerical methods used in biology and engineering into a single formulation that expands on the magnetic-dipole equivalent of spin packets. When fluids exhibit low T(1)/T(2) contrasts and when CPMG pulse sequences are used to acquire NMR measurements, we verify that classical NMR numerical models that neglect T(1) effects accurately reproduce surface magnetization decays of saturated granular porous media regardless of the diffusion/relaxation regime. Currently, analytical expressions exist only for the case of arbitrary pore shapes within the fast-diffusion limit. However, when fluids include several components or when magnetic fields are strongly inhomogeneous, we show that simulations results obtained using the complete set of Bloch's equations differ substantially from those of classical NMR models. In addition, our random-walk formulation accurately reproduces magnetization echoes stemming from coherent-pathway calculations. We show that the random-walk approach is especially suited to generate parametric multi-dimensional T(1)/T(2)/D NMR maps to improve the characterization of pore structures and saturating fluids.     

Pore diameter mapping using double pulsed-field gradient MRI and its validation using a novel glass capillary array phantom

Double pulsed-field gradient (d-PFG) MRI can provide quantitative maps of microstructural quantities and features within porous media and tissues. We propose and describe a novel MRI phantom, consisting of wafers of highly ordered glass capillary arrays (GCA), and its use to validate and calibrate a d-PFG MRI method to measure and map the local pore diameter. Specifically, we employ d-PFG Spin-Echo Filtered MRI in conjunction with a recently introduced theoretical framework, to estimate a mean pore diameter in each voxel within the imaging volume. This simulation scheme accounts for all diffusion and imaging gradients within the diffusion weighted MRI (DWI) sequence, and admits the violation of the short gradient pulse approximation. These diameter maps agree well with pore sizes measured using both optical microscopy and single PFG diffusion diffraction NMR spectroscopy using the same phantom. Pixel-by-pixel analysis shows that the local pore diameter can be mapped precisely and accurately within a specimen using d-PFG MRI.     

Characterization of partially sintered ceramic powder compacts using fluorinated gas NMR imaging.

We use nuclear magnetic resonance (NMR) imaging of C2F6 gas to characterize porosity, mean pore size, and permeability of partially sintered ceramic (Y-TZP Yttria-stabilized tetragonal-zirconia polycrystal) samples. Conventional measurements of these parameters gave porosity values from 0.18 to 0.4, mean pore sizes from 10 nm to 40 nm, and permeability from 4 nm(2) to 25 nm(2). The NMR methods are based on relaxation time measurements (T(1)) and the time dependent diffusion coefficient D(Delta). The relaxation time of C2F6 gas is longer in pores than in bulk gas and it increases as the pore sizes decrease. NMR yielded accurate porosity values after correcting for surface adsorption effects. A model for T(1) dependence on pore size that accounts for collisions between gas molecules and walls as well as surface adsorption effects is proposed. The model fits the experimental data well. Finally, the long time limit of D(Delta)/D(o), where D(o) is the bulk gas diffusion coefficient is useful for measuring tortuosity, while the short time limit was not achieved experimentally and could not be used for calculating surface-area to volume (S/V) ratios.     

Determination of pore space shape and size in porous systems using NMR diffusometry. Beyond the short gradient pulse approximation

The influence of finite length gradient pulses on NMR diffusion experiments on liquids confined to diffuse between two parallel planes is investigated. It is experimentally verified that the pore size decreases when determined using finite gradient pulses if the results are analyzed within the short gradient pulse approximation. The results are analyzed using the matrix formulation. The observed minima in the echo decay profiles are considerably less sharp than theoretical analysis would indicate and we suggest that this is due to the presence of a distribution of pore sizes in the sample. In addition, effects due to the presence of background gradients are discussed. It is argued that effects due to the finite length gradient pulses are relatively minor and in realistic applications the effects due to inhomogeneities in pore sizes and effects due to background gradients will constitute more serious problems in pore size determinations by means of NMR diffusometry.     

Measuring pore connectivity by pulsed field gradient diffusion editing with hydrocarbon gases

Measurements of time-dependent diffusion are performed on a rock sample saturated first with water, then methane and finally ethane. The gases were selected because their increased diffusivities and relaxation times allow probing greater length scales than water and because of their practical relevance. The nuclear magnetic resonance measurements employed pulse field gradient diffusion editing pulse sequences, allowing analysis of D(t) as a function of relaxation time. Very different D(t) behaviors are observed for different relaxation times, including indications of connected pore networks at moderate relaxation times.     

Studies of porous media by thermally polarized gas NMR: current status

Three examples of thermally polarized gas NMR performed at New Mexico Resonance are presented to demonstrate its unique advantages in porous media studies. 1) In-vivo animal lung imaging by Kuethe et al., in which useful quality 3D images of rat lungs were obtained in 30 min. It is conjectured that comparable human lung images would take much less time to make, possibly by the ratio of body weights, a factor of several hundred. 2) The success of the lung imaging suggested other porous media as candidates for thermally polarized gas NMR. Caprihan and coworkers obtained excellent images from partially sintered ceramics and Vycor glass. Since then, Beyea has developed the technique of spatially resolved BET curves for ceramics and other nanoporous solids. In this way, surface area, pore size, and porosity, averaged over an image voxel, can be spatially resolved. This greatly aids in the characterization of such materials, especially with regards to spatial heterogeneities. 3) Finally, we describe Codd's propagator experiments on propane gas flowing through a packed bed of 300 microm beads. In order to increase signal-to-noise ratio, the flowing gas was pressurized to 170 kPa. Excellent quality propagators showing the discrete nature of the bead pack were obtained. This type of information is not available in comparable liquid studies because most spins will not diffuse far enough to sample the walls in the time available.     

Pore-Size Distributions and Tortuosity in Heterogeneous Porous Media

The diffusion coefficient, measured at long observation times by pulsed-held-gradient NMR, can in principle be used to estimate the tortuosity of a porous medium. This method is useful for glass-sphere packs, but we find that it does not generally work for porous sedimentary rock. Natural sedimentary rocks are characterized by complex microgeometries and broad distributions of pore sizes, which cannot be adequately sampled by diffusing molecules in experimentally accessible observation times. The time-dependent diffusion coefficient D(t) can be distinctly irregular for rocks with very large pores. In heterogeneous porous media, determination of pore-size distribution by relaxation-time measurements and tortuosity by PFG diffusion measurements are mutually exclusive.     

Pulsed-Field-Gradient Measurements of Time-Dependent Gas Diffusion

Pulsed-field-gradient NMR techniques are demonstrated for measurements of time-dependent gas diffusion. The standard PGSE technique and variants, applied to a free gas mixture of thermally polarized xenon and O₂, are found to provide a reproducible measure of the xenon diffusion coefficient (5.71 × 10⁻⁶m²s⁻¹for 1 atm of pure xenon), in excellent agreement with previous, non-NMR measurements. The utility of pulsed-field-gradient NMR techniques is demonstrated by the first measurement of time-dependent (i.e., restricted) gas diffusion inside a porous medium (a random pack of glass beads), with results that agree well with theory. Two modified NMR pulse sequences derived from the PGSE technique (named the Pulsed Gradient Echo, or PGE, and the Pulsed Gradient Multiple Spin Echo, or PGMSE) are also applied to measurements of time dependent diffusion of laser polarized xenon gas, with results in good agreement with previous measurements on thermally polarized gas. The PGMSE technique is found to be superior to the PGE method, and to standard PGSE techniques and variants, for efficiently measuring laser polarized noble gas diffusion over a wide range of diffusion times.     

Correlation time and diffusion coefficient imaging: application to a granular flow system

A parametric method for spatially resolved measurements for velocity autocorrelation functions, R(u)(tau) = , expressed as a sum of exponentials, is presented. The method is applied to a granular flow system of 2-mm oil-filled spheres rotated in a half-filled horizontal cylinder, which is an Ornstein-Uhlenbeck process with velocity autocorrelation function R(u)(tau) = e(- ||tau ||/tau(c)), where tau(c) is the correlation time and D = tau(c) is the diffusion coefficient. The pulsed-field-gradient NMR method consists of applying three different gradient pulse sequences of varying motion sensitivity to distinguish the range of correlation times present for particle motion. Time-dependent apparent diffusion coefficients are measured for these three sequences and tau(c) and D are then calculated from the apparent diffusion coefficient images. For the cylinder rotation rate of 2.3 rad/s, the axial diffusion coefficient at the top center of the free surface was 5.5 x 10(-6) m(2)/s, the correlation time was 3 ms, and the velocity fluctuation or granular temperature was 1.8 x 10(-3) m(2)/s(2). This method is also applicable to study transport in systems involving turbulence and porous media flows.     

Probing four orders of magnitude of the diffusion time in porous silica glass with unconventional NMR techniques

The combined use of two unconventional NMR diffusometry techniques permits measurements of the self-diffusion coefficient of fluids confined in porous media in the time range from 100 microseconds to seconds. The fringe field stimulated echo technique (FFStE) exploits the strong steady gradient in the fringe field of a superconducting magnet. Using a standard 9.4 T (400 MHz) wide-bore magnet, for example, the gradient is 22 T/m at 375 MHz proton resonance and reaches 60 T/m at 200 MHz. Extremely short diffusion times can be probed on this basis. The magnetization grid rotating frame imaging technique (MAGROFI) is based on gradients of the radio frequency (RF) field. The RF gradients not necessarily need be constant since the data are acquired with spatial resolution along the RF gradient direction. MAGROFI is also well suited for unilateral NMR applications where all fields are intrinsically inhomogeneous. The RF gradients reached depend largely on the RF coil diameter and geometry. Using a conic shape, a value of at least 0.3 T/m can be reached which is suitable for long-time diffusion measurements. Both techniques do not require any special hardware and can be implemented on standard high RF power NMR spectrometers. As an application, the influence of the tortuosity increasing with the diffusion time is examined in a saturated porous silica glass.     

Rapid surface-to-volume ratio and tortuosity measurement using Difftrain

Analysis of diffusion measurements as a function of observation time (Delta), to calculate surface-to-volume ratios (S/V) and tortuosities (kappa), is a useful tool in the characterisation of porous media using NMR. However, using conventional pulsed field gradient (PFG) measurements, this requires long total experiment times (typically hours). Here, we show how the rapid diffusion measurement pulse sequence, Difftrain, can be used to provide the required experimental data much more rapidly (typically within minutes) with a consequential reduction in total experiment time of typically over an order of magnitude. Several novel modifications to the Difftrain pulse sequence are also presented to tailor it to this particular application; these include a variable delay between echoes (to ensure optimal echo position with respect to Delta) and a variable tip angle for the refocusing pulse (to ensure optimal use of available signal). Difftrain is applied to measure both S/V and kappa for a model glass bead pack; excellent agreement is found with both a conventional PFG measurement and with a bulk gravimetric measurement of S/V.     

MAG-PGSTE: a new STE-based PGSE NMR sequence for the determination of diffusion in magnetically inhomogeneous samples

A new stimulated echo based pulsed gradient spin-echo sequence, MAG-PGSTE, has been developed for the determination of self-diffusion in magnetically inhomogeneous samples. The sequence was tested on two glass bead samples (i.e., 212-300 and <106 microm glass bead packs). The MAG-PGSTE sequence was compared to the MAGSTE (or MPFG) (P.Z. Sun, J.G. Seland, D. Cory, Background gradient suppression in pulsed gradient stimulated echo measurements, J. Magn. Reson. 161 (2003) 168-173; P.Z. Sun, S.A. Smith, J. Zhou, Analysis of the magic asymmetric gradient stimulated echo sequence with shaped gradients, J. Magn. Reson. 171 (2004) 324-329; P.Z. Sun, Improved diffusion measurement in heterogeneous systems using the magic asymmetric gradient stimulated echo (MAGSTE) technique, J. Magn. Reson. 187 (2007) 177-183; P. Galvosas, F. Stallmach, J. K?rger, Background gradient suppression in stimulated echo NMR diffusion studies using magic pulsed field gradient ratios, J. Magn. Reson. 166 (2004) 164-173, P. Galvosas, PFG NMR-Diffusionsuntersuchungen mit ultra-hohen gepulsten magnetischen Feldgradienten an mikropor?sen Materialien, Ph.D. Thesis, Universit?t Leipzig, 2003, P.Z. Sun, Nuclear Magnetic Resonance Microscopy and Diffusion, Ph.D. Thesis, Massachusetts Institute of Technology, 2003] sequence and Cotts 13-interval [R.M. Cotts, M.J.R. Hoch, T. Sun, J.T. Marker, Pulsed field gradient stimulated echo methods for improved NMR diffusion measurements in heterogeneous systems, J. Magn. Reson. 83 (1989) 252-266] sequence using both glass bead samples. The MAG-PGSTE and MAGSTE (or MPFG) sequences outperformed the Cotts 13-interval sequence in the measurement of diffusion coefficients; more interestingly, for the sample with higher background gradients (i.e., the <106 microm glass bead sample), the MAG-PGSTE sequence provided higher signal-to-noise ratios and thus better diffusion measurements than the MAGSTE and Cotts 13-interval sequences. In addition, the MAG-PGSTE sequence provided good characterization of the surface-to-volume ratio for the glass bead samples.     

Restricted diffusion and exchange of water in porous media: average structure determination and size distribution resolved from the effect of local field gradients on the proton NMR spectrum

NMR Pulsed field gradient measurements of the restrained diffusion of confined fluids constitute an efficient method to probe the local geometry in porous media. In most practical cases, the diffusion decay, when limited to its principal part, can be considered as Gaussian leading to an apparent diffusion coefficient. The evolution of the latter as a function of the diffusion interval yields average information on the surface/volume ratio of porosities and on the tortuosity of the network. In this paper, we investigate porous model systems of packed spheres (polystyrene and glass) with known mean diameter and polydispersity, and, in addition, a real porous polystyrene material. Applying an Inverse Laplace Transformation in the second dimension reveals an evolution of the apparent diffusion coefficient as a function of the resonance frequency. This evolution is related to a similar evolution of the transverse relaxation time T2. These results clearly show that each resonance frequency in the water proton spectrum corresponds to a particular magnetic environment produced by a given pore geometry in the porous media. This is due to the presence of local field gradients induced by magnetic susceptibility differences at the liquid/solid interface and to slow exchange rates between different pores as compared to the frequency differences in the spectrum. This interpretation is nicely confirmed by a series of two-dimensional exchange experiments.