Pulsed field gradient signal attenuation of restricted anomalous diffusions in plate, sphere, and cylinder with wall relaxation

Xinli Liao, Shaokuan Zheng, and Guoxing Lin

Phys. Rev. E 101, 012128 – Published 24 January 2020

Abstract

The effect of boundary relaxation on pulsed field gradient (PFG) anomalous restricted diffusion is investigated in this paper. The PFG signal attenuation expressions of anomalous diffusion in plate, sphere, and cylinder are derived based on fractional calculus. In addition, approximate expressions for boundary relaxation induced short time signal attenuation under zero gradient field and boundary relaxation affected short time apparent diffusion coefficients are given in this paper. Unlike the exponential signal attenuation in normal diffusion, the PFG signal attenuation in anomalous diffusion with boundary relaxation is either a Mittag-Leffler-function-based attenuation or a stretched-exponential-function-based attenuation. The stretched exponential attenuations of all three structures clearly show the diffractive pattern. In contrast, only in the plate structure does the Mittag-Leffler-function-based attenuation display an obvious diffractive pattern. Additionally, anomalous diffusion with smaller time derivative order α has a weaker diffractive pattern and less signal attenuation. Moreover, the results demonstrate that boundary relaxation induced signal attenuation is significantly affected by the anomalous diffusion when no gradient field is applied. Meanwhile, the boundary relaxation significantly affects PFG signal attenuation of anomalous diffusion in the following ways: The boundary relaxation results in reduced radius from the minimum of the diffractive patterns, and it results in an increased apparent diffusion coefficient and decreased surfaces to volume ratio in varying the diffusion time experiment; the boundary relaxation also substantially affects the apparent diffusion coefficient of sphere structure in the variation of gradient experiment.

Received 6 September 2019

DOI:https://doi.org/10.1103/PhysRevE.101.012128

Physics Subject Headings (PhySH)

Spin diffusion

Interdisciplinary Physics

Authors & Affiliations

Xinli Liao ¹, Shaokuan Zheng², and Guoxing Lin ^3,*

¹Chemistry Department, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, 361005, China
²Department of Radiology, UMASS Medical School, Worcester, Massachusetts 01655, USA
³Carlson School of Chemistry and Biochemistry, Clark University, Worcester, Massachusetts 01610, USA

^*Corresponding author: glin@clarku.edu

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Vol. 101, Iss. 1 — January 2020

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Images

Figure 1
The effect of different diffusion time on the PFG signal attenuation in the presence of boundary relaxation: (a) plate structure, where the distance between the two plate surfaces is $a$ ; (b) sphere structure of radius $a$ ; (c) cylinder structure of radius $a$ , where its symmetric axis is perpendicular to the gradient field direction. Three different diffusion delays are used, with $D_{2} Δ^{α} / a^{2}$ equaling 0.5, 1, and 2, respectively. The parameter for the boundary relaxation is $M a / D_{2} = 2$ , and $D_{1} Γ (1 + α) = D_{2}$ . The PFG signal attenuation with nonrelaxing (NR) boundary condition (namely, $M = 0$ ) at $D_{2} Δ^{α} / a^{2} = 2$ is also plotted for comparison. Both the Mittag-Leffler-function (MLF) -based attenuation and the stretched-exponential-function (SEF) -based attenuation are plotted.
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Figure 2
Boundary relaxation effects on the PFG signal attenuation of restricted anomalous diffusion in plate, sphere, and cylinder structures. Four different $M a / D_{2}$ values, 0, 0.5, 1, and 2, are used. $α = 0.75, D_{1} Γ (1 + α) = D_{2}$ , and $D_{2} Δ^{α} / a^{2}$ is fixed at 2 where $a$ is the radius of the sphere and cylinder, and the distance between the two surfaces in the plate. Both the Mittag-Leffler-function (MLF) -based attenuation and the stretched-exponential-function (SEF) -based attenuation are plotted.
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Figure 3
The short time NMR signal intensity is affected by the combined effect of boundary relaxation and restricted anomalous diffusion when no gradient field is applied, namely, $q = 0$ . These discrete data points (solid diamonds, spheres, triangles, and squares) in (a) plates, (b) spheres, and (c) cylinders are theoretical signal intensities $A_{q = 0} (Δ)$ calculated based on Eqs. (16), (22), and (27) respectively, while the curves in (a)–(c) display the fitting of these $A_{q = 0} (Δ)$ based on Eq. (30a). These fitting parameters, $C_{l m}$ and $C_{s m}$ , are listed in Table 1, and plotted as data points in (d). The curves in (d) represent the fittings of these $C_{l m}$ and $C_{s m}$ values base on Eqs. (30b) and (30c), respectively, and the obtained fitting parameters ( $c_{\infty m}$ and $ɛ_{l m}$ , $c_{0 m}$ and $ɛ_{s m}$ ) are listed in Table 1. In (a)–(c), four different $M a / D_{2}$ values, 0.25, 1, 2, and 50, are used, and all $D_{2} Δ^{α} / a^{2}$ values are smaller than 0.1. Other parameters used are $D_{2} = 7.0 \times 10^{- 10} m^{2} / s^{α}$ , $α = 0.75$ , and $a = 30 μ m$ which is the radius of the sphere and cylinder, and the distance between the two surfaces in the plate. Only the results from the fractional derivative model are plotted.
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Figure 4
Boundary relaxation effects on the apparent diffusion coefficient obtained under the variation of the diffusion time in typical structures: (a) plate, (b) sphere, and (c) cylinder. Four different $M a / D_{2}$ values, 0.5, 2, 10, and 100, are used. Other parameters used are $α = 0.75$ , $D_{2} = 7.0 \times 10^{- 10} m^{2} / s^{α}$ , and $q a = 0.25$ where $a = 30 μ m$ is the radius of the sphere and cylinder, and the distance between the two surfaces in the plate. The theoretical data points (solid spheres, triangles, diamonds, and squares) are obtained by Eq. (34), which is applied to fit the numerical signal attenuations predicted from Eqs. (15), (21), and (26). The solid gray line represents $M = 0$ described by Eq. (32), while the dotted gray curve represents $M = \infty$ described by Eq. (37). The curves in (a)–(c) display the fitting based on Eq. (38), and the fitting parameters $D_{20}$ , $C_{1 v t}$ , and $C_{m v t}$ are summarized on Table 2. Only the results from the fractional derivative model are plotted.
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Figure 5
Boundary relaxation effects on the apparent diffusion coefficient obtained under the variation of the gradient intensity. The typical signal attenuation versus the square of $2 π q$ is plotted in (a). The apparent diffusion coefficient versus diffusion time is shown for plates in (b), spheres in (c), and cylinders in (d). Four different $M a / D_{2}$ values, 0.5, 2, 10, and 100, are used. Other parameters used are $α = 0.75$ , $D_{2} = 7.0 \times 10^{- 10} m^{2} / s^{α}$ , and $a = 30 μ m$ that is the radius of the sphere and cylinder, and the distance between the two surfaces in the plate. All the data points in (b)–(d) (solid spheres, triangles, diamonds, and squares) are theoretical data obtained from using Eq. (33) to fit the numerical signal attenuations by Eqs. (15), (21). and (26), where the maximum gradient for each data set is chosen to have its signal intensity decrease 20% from the signal intensity of the corresponding smallest gradient used. All the maximum $D_{2} Δ^{α} / a^{2}$ values are equal to 0.1. Except for these gray lines that are the $D_{a p p v g, i} (t)$ of $M = 0$ calculated by Eq. (32), these curves in (b)–(d) display the fitting of these $D_{a p p v g, i} (t)$ based on Eq. (39); the fitting results are summarized in Table 3. Only the results from the fractional derivative model are plotted.
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Figure 6
The effect of time-fractional derivative order $α$ on PFG signal attenuation of fractional diffusion in a plate. $M a / D_{2} = 1$ and $D_{2} Δ^{α} / a^{2} = 0.5$ are used. Only the Mittag-Leffler-function-based attenuations are plotted.
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