Two-Dimensional Correlation of Isotropic and Directional Diffusion Using NMR

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Two-Dimensional Correlation of Isotropic and Directional Diffusion Using NMR

João P. de Almeida Martins and Daniel Topgaard

Phys. Rev. Lett. 116, 087601 – Published 23 February 2016

Abstract

Diffusion nuclear magnetic resonance (NMR) is a powerful technique for studying porous media, but yields ambiguous results when the sample comprises multiple regions with different pore sizes, shapes, and orientations. Inspired by solid-state NMR techniques for correlating isotropic and anisotropic chemical shifts, we propose a diffusion NMR method to resolve said ambiguity. Numerical data inversion relies on sparse representation of the data in a basis of radial and axial diffusivities. Experiments are performed on a composite sample with a cell suspension and a liquid crystal.

Received 23 September 2015

DOI:https://doi.org/10.1103/PhysRevLett.116.087601

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Diffusion

Nuclear magnetic resonance

Condensed Matter, Materials & Applied PhysicsInterdisciplinary Physics

Authors & Affiliations

João P. de Almeida Martins^* and Daniel Topgaard

Division of Physical Chemistry, Department of Chemistry, Lund University, 221 00 Lund, Sweden

^*Corresponding author. joao.martins@fkem1.lu.se

Article Text

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References

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Issue

Vol. 116, Iss. 8 — 26 February 2016

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Images

Figure 1
NMR pulse sequence for correlating isotropic and directional diffusion. The series of $90 °_{x}$ (thin vertical line) and $180 °_{y}$ (thick vertical lines) radiofrequency (rf) pulses gives rise to a signal (decaying sinusoidal), the amplitude of which is encoded for diffusion using three sets of ramped gradient pulses as indicated with the numbered and colored braces. The dashed box shows a magnification of the first bipolar gradient pulse. The gradient amplitude $G$ and the time intervals $δ$ , $Δ$ , $ϵ$ , and $τ$ enter the calculation of the trace of the $b$ tensor in Eq. (9). The bottom right panel illustrates the unit vectors ( $n_{1}$ , $n_{2}$ , $n_{3}$ ) of the three sets of gradient pulses. The vectors form the angle $ζ$ with the $z$ axis and are located with threefold symmetry on a right circular cone with apex at the origin. (Adapted with permission from Ref. [59]. Copyright 2015 by Elsevier.)
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Figure 2
Data sampling strategy for 2D correlation of isotropic and directional diffusion using the pulse sequence in Fig. 1. (a) 2D space spanned by the linear $b_{L}$ and spherical $b_{S}$ components of the diffusion-encoding tensor $b$ as defined in Eq. (3). Isotropic (blue) and directional (red) encoding correspond to the colored perpendicular lines in the ( $b_{L}$ , $b_{S}$ ) space. (b) 2D mesh of gradient amplitudes $G$ and angles $ζ$ calculated from the square grid in the ( $b_{L}$ , $b_{S}$ ) space by inverting Eqs. (9) and (10).
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Figure 3
Schematics of geometry and microstructure of a composite yeast cell suspension and liquid crystal sample giving three distinct water components. (a) Assembled sample with coaxial glass tubes located in a 10 mm rf coil. (b) Cutaway view revealing the geometries of the solutions in the inner and outer tubes with outer diameters of 5 and 10 mm, respectively. (c) Magnification of a section of a single microdomain in the polydomain liquid crystal in the inner tube. (d) Micrometer-scale structure of the yeast cell suspension in the 10 mm tube. The spherical shells symbolize the cell membranes that separate the intra- and extracellular water.
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Figure 4
2D correlation of isotropic and directional water diffusion for the three-component sample in Fig. 3. (a) Water signal amplitude $S (b_{L}, b_{S}) / S_{0}$ sampled on a geometrically spaced rectangular grid in the 2D ( $b_{L}$ , $b_{S}$ ) space (circles, experimental; lines, calculated from the sparse basis analysis). The traces corresponding to purely isotropic and directional encoding are highlighted with the colors blue and red, respectively. (b) 2D directional and isotropic diffusion correlation spectrum $P (D_{z z}, D_{iso})$ obtained by 2D inverse Laplace transformation. (c) 2D correlation spectrum of radial and axial diffusivities $P (D_{⊥}, D_{∥})$ estimated by numerical inversion of Eq. (7). The result of a three-component fit of Eq. (7) is indicated with green crosses and percentages for the relative weights of the components. (d) 2D spectrum $P (D_{z z}, D_{iso})$ obtained from $P (D_{⊥}, D_{∥})$ in panel (c) by transformation according to Eq. (6). In (b)–(d), the contour lines are linearly spaced from 5% to 90% of the maximum value and the dashed lines indicate the diagonals. The traces above and to the left of each contour plot represent the 1D projections of the 2D distributions onto the respective axis. Note that the three components are widely separated in the sparse basis in panel (c).
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