Combined effects of diffusion, nonuniform-gradient magnetic fields, and restriction on an arbitrary coherence pathway

A single-shot measurement of time-dependent diffusion over sub-millisecond timescales using static field gradient NMR

Time-dependent diffusion behavior is probed over sub-millisecond timescales in a single shot using a nuclear magnetic resonance static gradient time-incremented echo train acquisition (SG-TIETA) framework. The method extends the Carr-Purcell-Meiboom-Gill cycle under a static field gradient by discretely incrementing the π-pulse spacings to simultaneously avoid off-resonance effects and probe a range of timescales (50-500 µs). Pulse spacings are optimized based on a derived ruleset. The remaining effects of pulse inaccuracy are examined and found to be consistent across pure liquids of different diffusivities: water, decane, and octanol-1. A pulse accuracy correction is developed. Instantaneous diffusivity, Dinst(t), curves (i.e., half of the time derivative of the mean-squared displacement in the gradient direction) are recovered from pulse accuracy-corrected SG-TIETA decays using a model-free log-linear least squares inversion method validated by Monte Carlo simulations. A signal-averaged 1-min experiment is described. A flat Dinst(t) is measured on pure dodecamethylcyclohexasiloxane, whereas decreasing Dinst(t) is measured on yeast suspensions, consistent with the expected short-time Dinst(t) behavior for confining microstructural barriers on the order of micrometers.

Microstructural analysis of foam by use of NMR R2 dispersion

The spin-spin relaxation rate R2 (=1/T2) in hydrogel foams measured by use of a multiple spin echo sequence is found to be dependent on the echo time spacing. This property, referred to as R2-dispersion, originates to a large extent from molecular self-diffusion of water within internal field gradients that result from magnetic susceptibility differences between the gel and air phase. Another contribution to the R2 relaxation rate is surface relaxation. Numerical simulations are performed to investigate the relation between the foam microstructure (the mean air bubble radius and standard deviation of the air bubble radius) and foam composition properties (such as magnetic susceptibilities, diffusion coefficient and surface relaxivity) at one hand and the R2-dispersion at the other hand. The simulated R2-dispersions of gel foam are in agreement with the measured R2-dispersions. By correlating the R2-dispersion parameters and simulated microstructure properties a semi-empirical relationship is obtained that enables the mean air bubble size to be derived from measured R2-dispersion curves. The R2-derived mean air bubble size of a hydrogel foam is in agreement with the bubble size measured with X-ray micro-CT. This illustrates the feasibility of using 1H R2-dispersion measurements to determine the size of air bubbles in hydrogel foams and of alveoli in lung tissue.

Analytical solution for restricted diffusion in circular and spherical layers under inhomogeneous magnetic fields

We propose an analytical solution for restricted diffusion of spin-bearing particles in circular and spherical layers in inhomogeneous magnetic fields. More precisely, we derive exact and explicit formulas for the matrix representing an applied magnetic field in the Laplacian eigenbasis and governing the magnetization evolution. For thin layers, a significant difference between two geometrical length scales (thickness and overall size) allows for accurate perturbative calculations. In these two-scale geometries, apparent diffusion coefficient (ADC) as a function of diffusion time exhibits a new region with a reduced but constant value. The emergence of this intermediate diffusion regime, which is analogous to the tortuosity regime in porous media, is explained in terms of the underlying Laplace operator eigenvalues. In general, regions with constant ADCs would be reminiscent of multiscale geometries, and their observation can potentially be used in experiments to detect the length scales by varying diffusion time.

Spectral characterization of diffusion in porous media by the modulated gradient spin echo with CPMG sequence

Carr-Purcell-Meiboom-Gill train of radiofrequency pulses applied to spins in the constant magnetic field gradient is an efficient variant of the modulated magnetic field gradient spin echo method, which provides information about molecular diffusion in the frequency-domain instead in the time-domain as with the two-pulse gradient spin echo. The frequency range of novel technique is broad enough to sample the power spectrum of displacement fluctuation in water-saturated pulverized silica (SiO(2)) and provides comprehensive information about the molecular restricted motion as well as about the structure of medium.

Probing short length scales with restricted diffusion in a static gradient using the CPMG sequence

We experimentally verify a new method of extracting the surface-to-volume ratio (S/V) of porous media with diffusion NMR. In contrast to the widely used pulsed field gradient (PFG) technique, which employs the stimulated echo coherence pathway, we use here the direct Carr-Purcell-Meiboom-Gill (CPMG) path. Even for high echoes, which exhibit ample attenuation due to diffusion in the field gradient, the relevant ruler length for the direct pathway is fixed by the diffusion length during a single inter-pulse spacing. The direct path, therefore, is well suited for probing shorter length scales than is possible with the conventional approach. In our experiments in a low-field static-gradient system, the direct CPMG pathway was found to be sensitive to structure an order of magnitude smaller than accessible with the stimulated-echo pathway.

Short-time restricted diffusion in a static gradient and the attenuation of individual coherence pathways

We experimentally explore some of the implications of a recent theoretical study [J. Magn. Reson. 64 (2003) 145] for the measurement of restricted diffusion in connected porous media in a static gradient. In particular, we examine how restriction affects the short-time attenuation of different coherence pathways, all excited with the same sequence of slice-selective radiofrequency (RF) pulses, and how the various pathways make the transition to the long-time or tortuosity regime. We confirm that every pathway contains equivalent diffusional information and, for short times, yields the surface-to-volume ratio (S/V) of the confining space. We find also, in agreement with the theoretical predictions, that different pathways are controlled by different time scales and, thus, exhibit different sensitivity to restriction. This property might be exploited when designing optimal sequences to study restricted motion.

Effect of internal gradients in the nuclear magnetic resonance measurement of the surface-to-volume ratio

We consider a system of spins diffusing in a static inhomogeneous (nonuniform-gradient) magnetic field B in a restricted geometry and in the presence of surface relaxation. We show that the short-time diffusional decay of nuclear magnetization is controlled by the field scattering kernel F(t) identical with [B(t)-B(0)](2), which is a measure of the average field inhomogeneity sampled by the spins in time t and does not depend on the particular sequence of radio-frequency pulses used. Magnetization in arbitrary sequences can be straightforwardly computed by evaluating elementary integrals of F(t). Diffusion takes place while the field is on, so that the spins precess as they diffuse, in contrast to the simpler problem of purely classical diffusion considered in [P. P. Mitra, P. N. Sen, and L. M. Schwartz, Phys. Rev. B 47, 8565 (1993)] which is applicable only to the ideal pulsed-field gradient experiment. We compute the short-time asymptotic form of F(t) and find that it depends on the surface-to-volume ratio (S/V) of the pore space as well as on the average of the gradients over the bounding surface. In a system with nonuniform gradients that vary faster near the surface than in the bulk, as for internal susceptibility fields, this gradient surface average may be much larger than the gradients in the bulk, significantly enhancing the apparent S/V. We discuss the application of our results to the widely used Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence as well as proposing a modification of it, which we term "padded" CPMG, that may be preferable in systems with significant surface relaxation. We indicate how each sequence can be used to probe the internal fields.