Finite difference diagonalization to simulate nuclear magnetic resonance diffusion experiments in porous media

Grombacher, Denys; Nordin, Matias

doi:10.1002/cmr.a.21349

Cited by 3 publications

(3 citation statements)

References 40 publications

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“…Meantime, the eigenvalues of the stencil Laplacian (i.e. the free Hamiltonian) are in excellent agreement with analytical results for the discrete Laplacian [145] (also see [143]).…”

Section: A1 Testingsupporting

confidence: 73%

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Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations

Shamailov¹,

Brown²,

Haase³

et al. 2020

Preprint

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Motivated by rapid experimental progress in ultra-cold atomic systems, we aim to provide a simple, intuitive description of Anderson localisation that allows for a direct quantitative comparison to experimental data, as well as yielding novel insights. To this end, we advance, employ and validate a recently-discovered theory -Localisation Landscape Theory (LLT) -which has unparalleled strengths and advantages, both computational and conceptual, over alternative methods. We focus on two-dimensional systems with point-like random scatterers, although an analogous study in other dimensions and with other types of disorder would proceed similarly. We begin by showing that exact eigenstates cannot be efficiently used to extract the localisation length. We then provide a comprehensive review of known LLT, and show that the effective potential of LLT can, to some degree, replace the real potential in the Hamiltonian. Next, we use LLT to compute the localisation length and test our method against exact diagonalisation. Furthermore, we propose a transmission experiment that optimally detects Anderson localisation and link the simulated observations of such an experiment to the predictions of LLT. In addition, we study the dimensional crossover from one to two dimensions, providing a new explanation to the established trends. The prediction of a mobility edge coming from LLT is tested by direct Schrödinger time evolution and is found to be unphysical. Moreover, we investigate expanding wavepackets, and find interesting differences between wavepackets that are initiated within and outside the disorder. We explain these differences using LLT combined with multidimensional tunnelling. Then, we utilise LLT to uncover a connection between the Anderson model for discrete disordered lattices and continuous two-dimensional disordered systems, which provides powerful new insights. From here, we demonstrate that localisation can be distinguished from other effects by a comparison to dynamics in an ordered potential with all other properties unchanged. Finally, we thoroughly investigate the effect of acceleration and repulsive interparticle interactions, as relevant for current experiments.

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“…Meantime, the eigenvalues of the stencil Laplacian (i.e. the free Hamiltonian) are in excellent agreement with analytical results for the discrete Laplacian [145] (also see [143]).…”

Section: A1 Testingsupporting

confidence: 73%

“…Moreover, the eigenvalues and eigenfunctions of the free Hamiltonian (Laplacian term only, no external potential) are well known. We ensured that both the spectrum and eigenstates of the continuous Laplace operator are reproduced correctly by our code (see, e.g., [143]).…”

Section: A1 Testingmentioning

confidence: 99%

Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations

Shamailov¹,

Brown²,

Haase³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Simulations were run over a range of pore sizes and χ values for the solid phase. Specifics about the algorithm used to simulate the diffusion-editing sequence are given by Grombacher and Nordin (2015). Parameters for the simulated pulse sequence were selected to match those used in our laboratory study.…”

Section: Watermentioning

confidence: 99%

Investigating the effect of internal gradients on static gradient nuclear magnetic resonance diffusion measurements

2017

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Nuclear magnetic resonance (NMR) methods can be used to measure the diffusion coefficient D of fluids. In porous materials, diffusion of the pore fluid is restricted by pore boundaries, such that D may be smaller than the diffusion coefficient of the bulk fluid. This reduction in D provides information about the geometry of the pore space. Significant overestimates of D can, however, occur due to internal gradients caused by magnetic susceptibility contrasts between the pore fluid and the solid phase. We have investigated the way in which internal gradients can impact the measured diffusion coefficient and obscure the link to pore geometry in unconsolidated sediments. We focus on measurements of D obtained with a static gradient diffusion-editing sequence, which can be used with NMR logging tools to measure D in subsurface sediments. Laboratory measurements of D measured with a diffusion-editing D-T 2 sequence indicate significant impacts from internal gradients, including D values several orders of magnitude larger than D of bulk water. The log-mean D values were found to be highly correlated with estimated internal gradient magnitudes and indicate no clear relationship to pore size. Samples with heterogeneous magnetic susceptibility of the solid phase indicate D distributions with multiple peaks, reflecting the nonuniform distribution of internal gradients in the sediment. We found evidence of high internal gradients impacting a majority of our samples, resulting in increased D values that do not reflect pore size even in samples with low magnetic susceptibility.

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Quantification of Articular Cartilage Microstructure by the Analysis of the Diffusion Tensor

Tourell

Powell

Momot

2016

Biophysics and Biochemistry of Cartilage by NMR and MRI

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In this chapter, we present approaches to the numerical simulation of the diffusion of water molecules in fibre networks that serve as models of articular cartilage. The simulations are intended as a tool for the translation of experimental diffusion magnetic resonance imaging (MRI) data into quantitative microstructural and compositional characteristics of articular cartilage. The chapter begins with a brief introduction to diffusion nuclear magnetic resonance and diffusion imaging, focusing on diffusion tensor imaging. It discusses the current limitations of diffusion MRI in quantifying articular cartilage microstructure beyond the predominant direction of collagen fibre alignment. We then detail the construction of aligned and partially aligned networks of fibres that can serve as models of articular cartilage. We discuss the methods for the simulation of the diffusion of tracer molecules through the model networks (especially Langevin dynamics and Monte Carlo techniques), and reconstruction of the diffusion tensor from the simulated molecular trajectories. The aim of these simulations is to quantitatively link the eigenvalues and the fractional anisotropy of cartilage diffusion tensor to collagen fibre volume fraction and the degree of collagen fibre alignment. The global aim of this work is to move diffusion tensor imaging of articular cartilage beyond determination of the predominant direction of fibre alignment, and towards quantification of the fibre orientation distribution.

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Finite difference diagonalization to simulate nuclear magnetic resonance diffusion experiments in porous media

Cited by 3 publications

References 40 publications

Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations

Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations

Investigating the effect of internal gradients on static gradient nuclear magnetic resonance diffusion measurements

Quantification of Articular Cartilage Microstructure by the Analysis of the Diffusion Tensor

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