The Influence of the Magnetic Impurity Content on the Pore Size Distribution Determination via the DDIF Technique

The pore size distribution of porous media can be determined in a completely non-invasive manner using a new nuclear magnetic resonance (NMR) technique which monitors the magnetization decay due to diffusion in internal fields (DDIF). However, using of the DDIF technique is restricted to the low-phase encoding limit when only the relaxation mode and the first-order diffusion mode are excited. In the present work the fulfillment of such a limit is verified for a progressive increase of the magnetic impurity content of the porous media. If the higher order diffusion modes are excited they lead both to a stronger attenuation of the echo signal and to the appearance of ripples in the DDIF spectra which cannot be related to a pore size distribution. The samples used in this study are porous ceramics prepared using the replication technique and the magnetic impurity is iron (III) oxide which is introduced in an increasing concentration inside the porous matrix. All NMR experiments were done on water filling such porous ceramics using a low-field instrument operating at a proton resonance frequency of 20 MHz. The average pore dimension obtained with the DDIF technique in the weak encoding limit indicates a satisfactory agreement with that observed in optical microscopy images.     

Translational free random walk of spins in the presence of a parabolic magnetic field

The free random walk approach has been used to analyze the attenuation of the NMR signal due to spin dephasing in the presence of a constant and pulsed parabolic magnetic field. The spin echo sequence was chosen to examine the attenuation of the NMR signal resulting from self-diffusion. In the framework of the gaussian approach, the long-time limit calculations predict more pronounced diffusion weighting for the parabolic field than for linear magnetic field. Analytical results were obtained and compared with those from other approaches based on a variety of different of approximations.     

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.     

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.     

Low-frequency velocity correlation spectrum of fluid in a porous media by modulated gradient spin echo.

In addition to the fast correlation for local stochastic motion, the molecular velocity correlation function in a fluid enclosed within the pore boundaries features a slow long time-tail decay. Here we present its study by the NMR modulated gradient spin-echo method (MGSE) [1] on a system of water trapped in the space between the closely packed polystyrene beads. With MGSE pulse sequence, a repetitive train of RF pulses with interspersed gradient pulses periodically modulates the spin phase. It gives the spin echo attenuation proportional to a value of the molecular velocity correlation spectrum at the modulation frequency. Covering the frequency range between Hz and MHz, it is a complement to the quasi-elastic neutron scattering, and so a suitable technique for the investigation of low frequency molecular dynamics in fluids. In our experiment, it enables to extract the low frequency correlation spectrum of water molecules confined in porous media. The function exhibits a negative long time-tail characteristic (a low frequency decay of the spectrum), which can be interpreted as a molecular back scattering on boundaries. The results can be well fitted with the spectrum calculated from the solution of the Langevin equation for restricted diffusion (which exhibits an exponential decay) [2] as well as with the spectrum obtained when simulating the hydrodynamics of molecular motion constrained by capillary walls (which gives an algebraic decay) [3]. Despite much work on theories and simulation, which predict slow negative long time tail of molecular velocity correlation dynamics in confined fluids, the obtained velocity correlation spectrum is the first experimental evidence to confirm these effects. The obtained dependence of spin echo attenuation on time, gradient strength and modulation frequency is also the first experimental verification of the recently developed approach to the spin echo in porous media, that uses the spin phase average with the cumulant expansion to get the attenuation as a discord of spin spatial coherence [4].     

Restricted self-diffusion of water in a highly concentrated w/o emulsion studied using modulated gradient spin-echo NMR

Restricted diffusion of water in a highly concentrated w/o emulsion was studied using pulsed field gradient spin echo techniques. The standard two-pulse version of this technique, suitable for analysis in the time domain, fails to investigate the short time-scale for diffusion inside a single emulsion droplet with radius 0.7 microm. With a pulse-train technique, originally introduced by Callaghan and Stepisnik, shorter time-scales are accessible. The latter approach is analyzed in the frequency domain and yields frequency dependent diffusion coefficients. Predictions for the outcome of the experiment were calculated in the time domain using the Gaussian phase distribution and the pore hopping formalism expressions for the echo attenuation. The results of these calculations were transformed to the frequency domain via a numerical inverse integral transform in order to compare with the experimental results.     

液体NMR中平板间多量子相干受限扩散行为的有限差分模拟

将平板间单量子相干受限扩散理论表述推广到多量子相干,并结合积算符矩阵、Bloch方程和有限差分方法进行模拟. 通过模拟找出平板间受限扩散信号衰减随平板间距变化的规律,并与实际体系比较. 结果表明：平板间n量子相干的自旋回波信号衰减曲线与单量子类似,且其产生同样衍射图样所需的脉冲梯度场强度仅为单量子的1/|n|,可用于测量微小的平板间距. 本文的模拟方法可进一步推广到复杂体系的研究.     

Tortuosity measurement and the effects of finite pulse widths on xenon gas diffusion NMR studies of porous media.

We have extended the utility of NMR as a technique to probe porous media structure over length scales of approximately 100-2000 microm by using the spin 1/2 noble gas 129Xe imbibed into the system's pore space. Such length scales are much greater than can be probed with NMR diffusion studies of water-saturated porous media. We utilized Pulsed Gradient Spin Echo NMR measurements of the time-dependent diffusion coefficient, D(t), of the xenon gas filling the pore space to study further the measurements of both the pore surface-area-to-volume ratio, S/V(p), and the tortuosity (pore connectivity) of the medium. In uniform-size glass bead packs, we observed D(t) decreasing with increasing t, reaching an observed asymptote of approximately 0.62-0.65D(0), that could be measured over diffusion distances extending over multiple bead diameters. Measurements of D(t)/D(0) at differing gas pressures showed this tortuosity limit was not affected by changing the characteristic diffusion length of the spins during the diffusion encoding gradient pulse. This was not the case at the short time limit, where D(t)/D(0) was noticeably affected by the gas pressure in the sample. Increasing the gas pressure, and hence reducing D(0) and the diffusion during the gradient pulse served to reduce the previously observed deviation of D(t)/D(0) from the S/V(p) relation. The Pade approximation is used to interpolate between the long and short time limits in D(t). While the short time D(t) points lay above the interpolation line in the case of small beads, due to diffusion during the gradient pulse on the order of the pore size, it was also noted that the experimental D(t) data fell below the Pade line in the case of large beads, most likely due to finite size effects.     

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.     

Homogeneous length scale of shear-induced multilamellar vesicles studied by diffusion NMR

A recently developed protocol for pulsed gradient spin echo (PGSE) NMR is applied for the size determination of multilamellar vesicles (MLVs). By monitoring the self-diffusion behavior of water, the technique yields an estimate of the homogeneous length scale λ(hom), i.e. the maximum length scale at which there is local structural heterogeneity in a globally homogeneous material. A cross-over between local non-Gaussian to global Gaussian diffusion is observed by varying the experimentally defined length- and time-scales. Occasional observation of a weak Bragg peak in the PGSE signal attenuation curves permits the direct estimation of the MLV radius in favorable cases, thus yielding the constant of proportionality between λ(hom) and radius. The microstructural origin of the Bragg peak is verified through Brownian dynamics simulations and a theoretical analysis based on the center-of-mass diffusion propagator. λ(hom) is decreasing with increasing shear rate in agreement with theoretical expectations and results from (2)H NMR lineshape analysis.     

Computer simulation of the spin-echo spatial distribution in the case of restricted self-diffusion

This article concerns the question of a proper stochastic treatment of the spin-echo self-diffusion attenuation of confined particles that arises when short gradient pulse approximation fails. Diffusion is numerically simulated as a succession of random steps when motion is restricted between two perfectly reflecting parallel planes. With the magnetic field gradient perpendicular to the plane boundaries, the spatial distribution of the spin-echo signal is calculated from the simulated trajectories. The diffusion propagator approach (Callaghan, "Principles of Nuclear Magnetic Resonance Microscopy," Oxford Univ. Press, Oxford, 1991), which is just the same as the evaluation of the spin-echo attenuation by the method of cumulant expansion in the Gaussian approximation, with Einstein's approximation of the velocity correlation function (VCF) (delta function), agrees with the results of simulation only for the particle displacements that are much smaller than the size of the confinement. A strong deviation from the results of the simulation appears when the bouncing rate from the boundaries increases at intermediate and long gradient sequences. A better fit, at least for intermediate particle displacements, was obtained by replacing the VCF with the Oppenheim--Mazur solution of the Langevin equation (Oppenheim and Mazur, Physica 30, 1833--1845, 1964), which is modified in a way to allow for spatial dependence of particle displacements. Clearly, interplay of the correlation dynamics and the boundary conditions is taking place for large diffusion displacements. However, the deviation at long times demonstrates a deficiency of the Gaussian approximation for the spin echo of diffusion inside entirely closed pores. Here, the cumulants higher than the second one might not be negligible. The results are compared with the experiments on the edge enhancement by magnetic resonance imaging of a pore.     

Novel NMR techniques for porous media research

Diffusion in porous media has been used as a probe of pore geometry in various NMR techniques. We will examine the effect of time-dependent diffusion in CPMG by showing that the diffusion time in CPMG is approximately the echo time, even in grossly inhomogeneous magnetic fields. Extension of the diffusion time in modified CPMG sequences is discussed. Diffusion in the susceptibility-contrast induced internal field is discussed as a means to probe pore size and pore shape. Finally, we present the general concept of two-dimensional relaxation-type experiments for study of molecules, fluids, materials and their dynamics that are characterized by spin relaxation and diffusion.     

Diffusion and relaxation effects in general stray field NMR experiments

We analyze the evolution of magnetization following any series of radiofrequency pulses in strongly inhomogeneous fields, with particular attention to diffusion and relaxation effects. When the inhomogeneity of the static magnetic field approaches or exceeds the strength of the RF field, the magnetization has contributions from different coherence pathways. The diffusion or relaxation induced decay of the signal amplitude is in general nonexponential, even if the sample has single relaxation times T(1), T(2) and a single diffusion coefficient D. In addition, the shape of the echo depends on diffusion and relaxation. It is possible to separate contributions from different coherence pathways by phase cycling of the RF pulses. The general analysis is tested on stray field measurements using two different pulse sequences. We find excellent agreement between measurements and calculations. The inversion recovery sequence is used to study the relaxation effects. We demonstrate two different approaches of data analysis to extract the relaxation time T(1). Finite pulse width effects on the timing of the echo formation are also studied. Diffusion effects are analyzed using the Carr--Purcell--Meiboom--Gill sequence. In a stray field of a constant gradient g, we find that unrestricted diffusion leads to nonexponential signal decay versus echo number N, but within experimental error the diffusion attenuation is still only a function of g(2)Dt(3)(E)N, where t(E) is the echo spacing.     

1H and 2H NMR studies of benzene confined in porous solids: melting point depression and pore size distribution

The pore size distributions of four controlled pore glasses and three silica gels with nominal diameters in the range 4-24 nm were determined by measuring the 1H and 2H NMR signals from the non-frozen fraction of confined benzene and perdeuterated benzene as a function of temperature, in steps of ca. 0.1-1 K. The liquid and solid components of the adsorbate were distinguished, on the basis of the spin-spin relaxation time T2, by employing a spin-echo sequence. The experimental intensity curves of the liquid component are well represented by a sum of two error functions. The mean melting point depression of benzene and perdeuterated benzene confined in the four controlled pore glasses, with pore radius R, follows the simplified Gibbs-Thompson equation DeltaT=kp/R with a kp value of 44 K nm. As expected, the kp value mainly determines the position of the pore size distribution curve, i.e., the mean pore radius, while the transition width determines the shape of the pore size distribution curve. The excellent agreement between the results from the 1H and 2H measurements shows that the effect of the background absorption from protons in physisorbed water and silanol groups is negligible under the experimental conditions used. The overall pore size distributions determined by NMR are in reasonable agreement with the results specified by the manufacturer, or measured by us using the N2 sorption technique. The NMR method, which is complementary to the conventional gas sorption method, is particularly appropriate for studying pore sizes in the mesoporous range.     

Single-Shot Diffusion Measurement in Laser-Polarized Gas

A single-shot pulsed gradient stimulated echo sequence is introduced to address the challenges of diffusion measurements of laser polarized 3He and 129Xe gas. Laser polarization enhances the NMR sensitivity of these noble gases by >10(3), but creates an unstable, nonthermal polarization that is not readily renewable. A new method is presented which permits parallel acquisition of the several measurements required to determine a diffusive attenuation curve. The NMR characterization of a sample's diffusion behavior can be accomplished in a single measurement, using only a single polarization step. As a demonstration, the diffusion coefficient of a sample of laser-polarized 129Xe gas is measured via this method.     

Long-path measurements of ultrasonic attenuation and velocity for very dilute slurries and liquids and detection of contaminates

The objective was to use multiple paths through the slurry to determine the lowest concentration that provided accurate attenuation measurements and to measure the velocity of sound through an effective long path. Ultrasonic attenuation measurements were obtained for slurries of silica (10 microm diameter) in water for concentrations of 0.1%, 0.25%, 0.5%, 0.75% and 1% silica by weight. Attenuation measurements for concentrations less than 0.1% may prove useful for process control to detect contaminants. A long path is obtained due to multiple reflections occurring within the stainless steel (SS) vessel used; broad-band transducers are affixed on the outside of the thick-walled vessel. The signal in the receive transducer permits the measurement of the attenuation and also the velocity by measuring the time-of-flight. The FFT of the appropriate signal for each echo was obtained and compared with that for water to yield the attenuation as a function of frequency. The attenuation measurements are self-calibrating because they are not affected by changes in the pulser voltage. The data show the feasibility for measuring a concentration of 0.025 wt% silica, which is equivalent to 0.25 g of silica in 1 l of water. Therefore, such measurements can prove useful for detecting contaminants in liquid. The velocity of sound measurements for solutions of hydrogen peroxide in water were obtained and accurate to about 0.3m/s, or 0.02% uncertainty.     

A global inversion method for multi-dimensional NMR logging

We describe a general global inversion methodology of multi-dimensional NMR logging for pore fluid typing and quantification in petroleum exploration. Although higher dimensions are theoretically possible, for practical reasons, we limit our discussion of proton density distributions as a function of two (2D) or three (3D) independent variables. The 2D can be diffusion coefficient and T(2) relaxation time (D-T(2)), and the 3D can be diffusion coefficient, T(2), and T(1) relaxation times (D-T(2)-T(1)) of the saturating fluids in rocks. Using the contrast between the diffusion coefficients of fluids (oil and water), the oil and water phases within the rocks can be clearly identified. This 2D or 3D proton density distribution function can be obtained from either two-window or regular type multiple CPMG echo trains encoded with diffusion, T(1), and T(2) relaxation by varying echo spacing and wait time. From this 2D/3D proton density distribution function, not only the saturations of water and oil can be determined, the viscosity of the oil and the gas-oil ratio can also be estimated based on a previously experimentally determined D-T(2) relationship.     

Relation of self-diffusion and the topological structure of flexible polymers as studied by NMR

A general approach to deriving formulas for calculating the diffusion attenuation of spin echo decay signal is developed and applied to obtaining data on random processes in linear and crosslinked polymers with different topological structures. The analysis explains the experimentally observed anomalous diffusion in polymer networks.     

Pore-size determination of mesoporous materials by1H NMR spectroscopy

The pore-size distributions of a series of mesoporous silica materials were determined by measuring the¹H nuclear magnetic resonance (NMR) signal from the nonfrozen fraction of organic probe molecules as a function of temperature. The melting point distribution curves of confined benzene reveal 2–3 transition points. The high-temperature transition point, corresponding to the temperature at the first maximum of the melting point distribution curve, is interpreted as the average depressed melting point of the confined substance. However, the intensity data reveal that a measurable portion of the confined benzene apparently remains nonfrozen even 120 K below the bulk melting point in the 4–10 nm pore systems. The component at lowest temperature is largely attributed to the liquidlike molecules at the pore wall, while the component at the intermediate temperature might result from pockets in the solid matrix or even a bimodal pore-size distribution. The average pore-size distributions obtained by NMR agree fairly well with those obtained by N₂ sorption. However, NMR gives a more detailed picture of the distribution, revealing two or three well-defined peaks. The peak at the smallest pore size, however, reflects the surface layer rather than a pore-size distribution.