Pabitra N. Sen

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 systems 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.

SPIN ECHOES OF NUCLEAR MAGNETIZATION DIFFUSING IN A CONSTANT MAGNETIC FIELD GRADIENT AND IN A RESTRICTED GEOMETRY

We study the influence of restriction on Carr–Purcell–Meiboom–Gill spin echoes response of magnetization of spins diffusing in a bounded region in the presence of a constant magnetic field gradient. Depending on three main length scales: LS pore size, LG dephasing length and LD diffusion length during half-echo time, three main regimes of decay have been identified: free, localization and motionally averaging regime. In localization regime, the decay exponent depends on a fractional power (2/3) of the gradient, denoting a strong breakdown of the second cumulant or the Gaussian phase approximation (GPA). In the other two regimes, the exponent depends on the gradient squared, and the GPA holds. We find that the transition from the localization to the motionally averaging regime happens when the magnetic field gradients approach special values, corresponding to branch points of the eigenvalues. Transition from one regime to another as a function of echo number for a certain range of parameters is discussed. ...

Nuclear magnetic resonance spin echoes for restricted diffusion in an inhomogeneous field: Methods and asymptotic regimes

We develop systematic formulations for calculating the magnetization of spins diffusing in a bounded region in the presence of the surface relaxation and magnetic field inhomogeneity and compute explicitly the relaxation exponent for the Carr–Purcell–Meiboom–Gill spin echoes. The results depend on the echo number n, and three dimensionless parameters: Lρ/LS, D0=(LD/LS)2, the dimensionless diffusion constant, and γ=LD2LS/LG3=Δωτ, the dimensionless gyromagnetic ratio, where the restriction is characterized by a size LS, the magnetic field inhomogeneity by a dephasing length, LG, the diffusion length during half-echo time by LD, and a length Lρ characterizes the surface relaxation. Here Δω is the line broadening and 2τ is the echo period. Depending on the length scales, three main regimes of decay have been identified: short-time, localization, and motionally averaging regimes (MAv). The short-time and the MAv regimes are described well by the cumulant expansion in terms of powers of the “small” parameter ...

Correlation functions for inhomogeneous magnetic field in random media with application to a dense random pack of spheres.

In porous media subject to applied magnetic field, the internal field arises out of susceptibility contrast of the constituents. We have examined the spatial inhomogeneity of the internal fields in a random pack of spheres using numerical computation. We find that the pair-correlation function of the internal field (K2) is a close approximation to the structure factor of the material, thus K2 can be used to characterize pore geometry. The magnetic length scale LambdaM exhibited in K2 is shown to be related to the fluid transport in the medium.

Time dependent diffusion coefficient in a disordered medium

We present a closed form expression for the diffusion coefficient in a dilute suspension of reflecting spheres, which is valid for all times. Our result is in agreement with general long and short time asymptotic results, which have been used in the interpretation of nuclear magnetic resonance experiments.

Steady-state free precession experiments and exact treatment of diffusion in a uniform gradient

We derive an analytic solution for the magnetization of spins diffusing in a constant gradient field while applying a long stream of rf pulses, which is known as the steady-state free precession (SSFP) sequence. We calculate the diffusion-dependent amplitude of the free induction decay (FID) and higher order echoes for pulses with arbitrary flip angle α and pulse spacing TR. Stopped-SSFP experiments were performed in a permanent gradient field and the amplitudes of the first three higher order echoes were measured for a range of values of α and TR. Theoretical results are in excellent agreement with experimental results, using no adjustable parameters. We identify various diffusion regimes in a rather large parameter space of pulsing and relaxation times, diffusion coefficient, and flip angle and discuss the interplay of the relevant time scales present in the problem. This “phase diagram” provides a road map for designing experiments which enhance or suppress the sensitivity to diffusion. We delineate th...

Time-dependent diffusion coefficient as a probe of the permeability of the pore wall

The time dependence of the mean-square displacement (or equivalently of the diffusion coefficient) in the presence of a permeable barrier can be used as a probe of the surface-to-volume ratio and permeability of a membrane. An exact, universal, short-time asymptotics in a pack of cells, assuming that the surfaces are locally smooth, shows that the effects of nonzero permeability appear as a correction to the diffusion coefficient that is linear in time, whereas the surface-to-volume ratio enters as a square root in time. With κ as the permeability of the membrane, we find, for the particles released inside the cells, DR,eff(t)=DR[1−(SR/VR){4DRt/(9π)−κtDL(DL+DR)/(6DR)}]+⋯ . Here DR and DL are free (i.e., bulk) diffusion coefficients inside and outside of the cell, respectively, and SR/VR is the total internal surface divided by the total internal cell volume. The other terms linear in t that add to the right side of above equation are DR(SR/VR)[(1/6)ρt−(1/12)DRt〈(1/R1+1/R2)〉R], where ρ is a surface relaxat...

Characterization of coupled pore systems from the diffusion eigenspectrum

Complex structures often consist of many interconnected or “coupled” simpler regions. The problem frequently arises of determining the geometry of these individual subregions within the larger structure. We consider a simple model to argue that the high eigenmodes of the diffusion equation can be used to probe their geometry. We find that for a wide range of coupling, certain high eigenmodes preferentially remain within a particular subregion, thereby allowing the association of the corresponding eigenvalue with that subregion. We discuss an application of these results to the characterization of internal structure of porous media.

Restricted diffusion in grossly inhomogeneous fields.

We analyze the effects of geometrical restriction on the nuclear magnetization of spins diffusing in grossly inhomogeneous fields where radio-frequency (RF) pulses are weak relative to the total field inhomogeneity, making the rotation angle space-dependent and thus exciting multiple coherence pathways. We show how to separate the effects of restricted diffusion from the effects of the pulses in the case when the change in the field experienced by a diffusing spin in the course of the experiment is small compared to the RF magnitude. We then derive explicit formulas for the contribution of individual coherence pathways to the total magnetization in arbitrary pulse sequences. We find that, for long diffusion times, restriction can dramatically alter the spectrum and the shape of a particular echo, while for short times, the correction will be proportional to the pore space surface-to-volume ratio. We demonstrate these results on the example of the early echoes of the Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence.

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

We develop a general framework for analyzing the effects of restricted geometries and inhomogeneous (nonuniform-gradient) magnetic fields on the relaxation of nuclear magnetization. The formalism naturally separates the effects of radio-frequency pulses by introducing the field scattering kernel F(t)≡〈[B(t)−B(0)]2〉 which captures all the interactions of the diffusing spins with the inhomogeneous field and with the confining walls. F(t) is the fundamental building block in the computation of the magnetization in any sequence of pulses. We use it to derive explicit formulas for the attenuation of the echoes of a general coherence pathway and thus arbitrary pulse trains. The short-time and long-time results, proved rigorously, are model-independent and hold for arbitrary geometries, both closed, such as a single cell or pore, and open, such as a connected porous medium. In open geometries, we compute the magnetization for all times, using a model form of the time-dependent diffusion coefficient. We apply our...