Dispersion measurements using time-of-flight remote detection MRI.

Remote detection nuclear magnetic resonance and magnetic resonance imaging can be used to study fluid flow and dispersion in a porous medium from a purely Eulerian point of view (i.e., in a laboratory frame of reference). Information about fluid displacement is obtained on a macroscopic scale in a long-time regime, while local velocity distributions are averaged out. It is shown how these experiments can be described using the common flow propagator formalism and how experimental data can be analyzed to obtain effective porosity, flow velocity inside the porous medium, fluid dispersion and flow tracing of fluid.

Low-frequency dielectric response of charged oblate spheroidal particles immersed in an electrolyte.

We study the low-frequency polarization response of a surface-charged oblate spheroidal particle immersed in an electrolyte solution. Because the charged spheroid attracts counterions which form the electric double layer around the particle, using usual boundary conditions at the interface between the particle and electrolyte can be quite complicated and challenging. Hence, we generalize Fixman's boundary conditions, originally derived for spherical particles, to the case of the charged oblate spheroid. Given two different counterion distributions in the thin electric double-layer limit, we obtain analytic expressions for the polarization coefficients to the first nontrivial order in frequency. We find that the polarization response normal to the symmetry axis depends on the total amount of charge carried by the oblate spheroid while that parallel to the symmetry axis is suppressed when there is less charge on the edge of the spheroid. We further study the overall dielectric response for a dilute suspension of charged spheroids. We find that the dielectric enhancement at low frequency, which is driven by the presence of a large ζ potential (surface charge), is suppressed by high ion concentrations in the electrolyte and depends on the size of the suspended particles. In addition, spheroids with higher aspect ratios will also lead to a stronger dielectric enhancement due to the combination of the electric double layer and textural effects. The characteristic frequency associated with the dielectric enhancement scales inversely with the square of the particle size, the major radius of the spheroid, and it has a weak dependence on the shape of spheroids.

Low magnetic fields for flow propagators in permeable rocks.

Pulsed field gradient NMR flow propagators for water flow in Bentheimer sandstone are measured at low fields (1H resonance 2 MHz), using both unipolar and bipolar variants of the pulsed gradient method. We compare with propagators measured at high fields (1H resonance 85 MHz). We show that (i) measured flow propagators appear to be equivalent, in this rock, and (ii) the lower signal to noise ratio at low fields is not a serious limitation. By comparing different pulse sequences, we study the effects of the internal gradients on the propagator measurement at 2 MHz, which for certain rocks may persist even at low fields.

Modeling diffusion in white matter in the brain: a composite porous medium.

We model diffusion in white matter fascicles as a problem of diffusion in an array of identical thick-walled cylindrical tubes immersed in an outer medium and arranged periodically in a regular lattice. The diffusing molecules have different diffusion coefficients and concentrations (or densities) within the tubes' inner core, membrane, myelin sheath, and within the outer medium. For an impermeable myelin sheath, diffusing molecules within the inner core are completely restricted, while molecules in the outer medium are hindered due to the tortuosity of the array of impenetrable tubes.

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.

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.

A new probe of pore geometry: diffusive MASS NMR.

NMR methods are widely used to probe the structure and fluid dynamics of porous materials, as they are uniquely suited to these studies since NMR records the correlation of changing local magnetic fields over a time scale of ns to seconds. The local magnetic fields are established by local variations in the bulk magnetic susceptibility of the sample (and so are directly tied to the sample's local structure). The fluctuation in field that a spin sees is due to molecular transport (including molecular diffusion) through these local fields, and so reports on the length scales of structures and impediments to transport. We have developed a new set of methods DIFFUSIVE-MASS to provide a means of systematically varying the effective time scale of the measurement and thus the effective length scale. This new handle permits a detailed, microscopic picture of the structure and dynamics. Diffusive MASS NMR methods will permit a systematic set of methods and analysis for characterizing the chemistry, structure and fluid dynamics of the mobile phase in porous materials. The approach will be applicable to any diamagnetic material. In particular, the industry of oil discovery depends on understanding heterogeneous porous media.

Influence of local geometry and transition to dispersive regime by mechanical mixing in porous media.

We study dispersion without diffusion via mechanical mixing, starting from the early ballistic regime where the mean-square displacement (MSd)<(xi-)2)> proportional to t(2) grows as the elapsed time squared, to the long-time, decoherent, asymptotic regime, where, MSd grows linearly in time. We find in the early time regime, the propagator has a fingerprint of local geometry. The quadratic t(2) term in the MSd may persist if a mechanical parameter aleph, long-range velocity deviation, which characterizes the deviation of the long-range mean velocity seen by each particle from the long-range mean velocity of the entire ensemble, is nonzero. The time dependence of dispersion coefficient, even in absence of diffusion, shows a behavior similar to that found in Telegraph equation for diffusion with drift.

Taylor dispersion with absorbing boundaries: a stochastic approach.

We solve the problem of Taylor dispersion in the presence of absorbing boundaries using an exact stochastic formulation. In addition to providing a clear stochastic picture of Taylor dispersion, our method leads to closed-form expressions for all the moments of the convective displacement of the dispersing particles in terms of the transverse diffusion eigenmodes. We also find that the cumulants grow asymptotically linearly with time, ensuring a Gaussian distribution in the long-time limit. As a demonstration of the technique, the first two longitudinal cumulants (yielding respectively the effective velocity and the Taylor diffusion constant) as well as the skewness (a measure of the deviation from normality) are calculated for fluid flow in the parallel plate geometry. We find that the effective velocity and the skewness are enhanced while Taylor dispersion is suppressed due to absorption at the boundary.

Time dependent diffusion in a disordered medium with partially absorbing walls: a perturbative approach.

We present an analytical study of the time dependent diffusion coefficient in a dilute suspension of spheres with partially absorbing boundary condition. Following Kirkpatrick [J. Chem. Phys. 76, 4255 (1982)] we obtain a perturbative expansion for the time dependent particle density using volume fraction f of spheres as an expansion parameter. The exact single particle t operator for partially absorbing boundary condition is used to obtain a closed form time dependent diffusion coefficient D(t) accurate to first order in the volume fraction f. Short and long time limits of D(t) are checked against the known short time results for partially or fully absorbing boundary conditions and long time results for reflecting boundary conditions. For fully absorbing boundary condition the long time diffusion coefficient is found to be D(t)=5a(2)/(12fD(0)t)+O((D(0)t/a(2))(-2)) to the first order of perturbation theory. Here f is small but nonzero, D(0) the diffusion coefficient in the absence of spheres, and a the radius of the spheres. The validity of this perturbative result is discussed.

Diffusivity of asphaltene molecules by fluorescence correlation spectroscopy.

Using fluorescence correlation spectroscopy (FCS) we measure the translational diffusion coefficient of asphaltene molecules in toluene at extremely low concentrations (0.03-3.0 mg/L): where aggregation does not occur. We find that the translational diffusion coefficient of asphaltene molecules in toluene is about 0.35 x 10(-5) cm(2)/s at room temperature. This diffusion coefficient corresponds to a hydrodynamic radius of approximately 1 nm. These data confirm previously estimated size from rotational diffusion studied using fluorescence depolarization. The implication of this concurrence is that asphaltene molecular structures are monomeric, not polymeric.

A model for diffusion in white matter in the brain.

Diffusion of molecules in brain and other tissues is important in a wide range of biological processes and measurements ranging from the delivery of drugs to diffusion-weighted magnetic resonance imaging. Diffusion tensor imaging is a powerful noninvasive method to characterize neuronal tissue in the human brain in vivo. As a first step toward understanding the relationship between the measured macroscopic apparent diffusion tensor and underlying microscopic compartmental geometry and physical properties, we treat a white matter fascicle as an array of identical thick-walled cylindrical tubes arranged periodically in a regular lattice and immersed in an outer medium. Both square and hexagonal arrays are considered. The diffusing molecules may have different diffusion coefficients and concentrations (or densities) in different domains, namely within the tubes' inner core, membrane, myelin sheath, and within the outer medium. Analytical results are used to explore the effects of a large range of microstructural and compositional parameters on the apparent diffusion tensor and the degree of diffusion anisotropy, allowing the characterization of diffusion in normal physiological conditions as well as changes occurring in development, disease, and aging. Implications for diffusion tensor imaging and for the possible in situ estimation of microstructural parameters from diffusion-weighted MR data are discussed in the context of this modeling framework.

Spatial and temporal coarse graining for dispersion in randomly packed spheres.

The propagator for molecular displacements P(zeta, t) and its first three cumulants were measured for Stokes flow in monodisperse bead packs with different sphere sizes d and molecular diffusion coefficients D(m). We systematically varied the normalized mean displacement /d and diffusion length L(D)=sqrt[2D(m)t]/d. The experimental results map onto each other with this scaling. For L(D)/d<0.2 the propagator remains non-Gaussian, and thus an advection diffusion equation is not obeyed, for mean displacements measured up to >10d. A Gaussian shape is approached for large mean displacements when L(D)>0.3d.

Nonresonant multiple spin echoes.

Nonresonant manipulation of nuclear spins can probe large volumes of sample situated in inhomogeneous fields outside a magnet, a geometry suitable for mobile sensors for the inspection of roads, buildings, and geological formations. However, the interference by Earth's magnetic field causes rapid decay of the signal within a few milliseconds for protons and is detrimental to this method. Here we describe a technique to suppress the effects of Earth's field by using adiabatic rotations and sudden switching of the applied fields. We observed hundreds of spin echo signals lasting for more than 600 milliseconds and accurately measured the relaxation times of a liquid sample.