NMR Measurement of the Magnetic Field Correlation Function in Porous Media

Abstract

The structure factor provides a fundamental characterization of porous and granular materials as it is the key for solid crystals via measurements of x-ray and neutron scattering. Here, we demonstrate that the structure factor of the granular and porous media can be approximated by the pair correlation function of the inhomogeneous internal magnetic field, which arises from the susceptibility difference between the pore filling liquid and the solid matrix. In-depth understanding of the internal field is likely to contribute to further development of techniques to study porous and granular media. DOI: 10.1103/PhysRevLett.100.025501 PACS numbers: 76.60.Jx, 61.05.Qr, 61.43.Gt, 75.75.+a Sand piles, rocks, and colloids share a common structure element that they are examples of materials formed by aggregation of granular particles in another medium. Their physical properties are critically dependent on the grain-grain interactions and the structure of the grain arrangement [1][2][3]. Details of the packing are important for the presence of heterogeneous force causing jamming [4,5], unique acoustic properties [6], and clustering [1]. It was found that the spatial correlation functions of the electromagnetic fields in such porous media can be useful to predict fluid flow and structure factor. For example, twodimensional images have been used to construct the spatial correlation functions of porous media and to infer material properties [7][8][9][10], and it was shown that correlation length of electric field is similar to the hydraulic radius [11]. In biomedical imaging, magnetic field correlation was examined for a potential quantitative assessment of iron in brain [12]. It was shown that the magnetic field correlation function is closely related to the structure factor of the pore space [13,14]. Thus it is desirable to devise an experimental method to directly measure the correlation function of the electromagnetic field in such media.In this Letter, we present a NMR method to measure the spatial magnetic correlation function in model porous materials. When a granular sample is placed in a uniform magnetic field, the solid grain and the interstitial material are magnetized differently due to a susceptibility contrast, producing an inhomogeneous magnetic field in the pore space -often called internal field. We use nuclear spins to probe the magnetic field difference between two positions by the following method. First, we employ pulsed field gradient (PFG) NMR to select spins by their translational diffusion displacement. For this experiment, the effect of the internal field is nullified. Then, we perform a similar experiment but with the internal field effect. The field difference between the two positions causes a signal decay that is related to the magnetic field correlation at that displacement. We show that the method is robust for different grain sizes, diffusion, encoding times, and displacement resolution and reliably obtains surface-to-volume ratio (SVR).Conventional translation diffusion measurements ofte...

Magnetic Resonance Characterization of Porous Media Using Diffusion through Internal Magnetic Fields

“…It was recently shown that a pair of symmetric and asymmetric stimulated echo PFG-NMR experiments can be used to directly measure the correlation function of the internal magnetic field [28]. First, we used PFG-NMR to select spins by their translational diffusion displacements without effects from the internal field.…”

“…Cho et al . demonstrated that a pulsed-field gradient (PFG) method involving both applied and internal field gradients, can be used to obtain spatial magnetic correlation functions that are in good agreement with theoretical simulations [28]. For biological applications, such as brain imaging, Jensen et al .…”

“…Finally, a pulsed-field gradient-NMR (PFG-NMR) method that utilizes an asymmetric, stimulated echo sequence to extract the magnetic field correlation function will be discussed, describing the connections between the statistical properties of the internal field and a porous material’s structural 2-point correlation function. Representative examples in a system of randomly packed beads will be described using a pair of symmetric and asymmetric stimulated echoes [28]. Finally, these examples will be discussed in the broader context of microstructurally sensitive MR techniques, and future applications will be considered.…”

Generalized Moment Analysis of Magnetic Field Correlations for Accumulations of Spherical and Cylindrical Magnetic Perturbers

“…(A) Monoexponential approximations K i (t) of the correlation function K(t) (solid line) as obtained from Equation (B3) with moments given in Schulten et al [46]. The long time approximation K L (t) (dashed line) is given by Equation (20) where the first eigenvalue κ 1 is determined by Equation (17) (η = 0.001 in A,B). (B) Biexponential approximations for long times K (1,3) (t) [dashed line obtained from Equation (39)] and for short times K (2,2) (t) [dotted line obtained from Equation (37)] of the correlation function K(t) [solid line obtained from Equation (14)].…”

“…When spin dephasing around the local magnetic field inhomogeneity is Gaussian, a relation between magnetization time evolution and the frequency correlation function can be established [17] and it could be shown that information about correlation functions provide a more direct and measurable link to the local magnetic field inhomogeneity than relaxation rates [18]. Magnetic field correlation imaging techniques have already been utilized for MR measurements of iron accumulation in brain parenchyma [19] and for the MRanalysis of porous media [20]. In addition, since Carr-PurcellMeiboom-Gill (CPMG) sequence relaxation rates are connected to the correlation function [16], a better knowledge of the correlation function may aid in determining microstrucutral parameters that quantify local capillary size, density, and oxygen extraction fraction in muscle tissue [21,22].…”