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Register a new userDescribe NMR relaxation by effective diffusion equation
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Guoxing Lin
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This paper proposes an effective diffusion equation method to analyze nuclear magnetic resonance (NMR) relaxation. NMR relaxation is a spin system recovery process, where the evolution of the spin system is affected by the random field due to Hamiltonians, such as dipolar couplings. The evolution of magnetization can be treated as a random walk in phase space described either by a normal or fractional phase diffusion equation. Based on these phase diffusion equations, the NMR relaxation rates and equations can be obtained, exemplified in the analysis of relaxations affected by an arbitrary random field, and by dipolar coupling for both like and unlike spins. The obtained theoretical results are consistent with the reported results in the literature. Additionally, the anomalous relaxation expression obtained from the Mittag-Leffler function based time correlation function can successfully fit the previously reported 13C T1 NMR experimental data of polyisobutylene (PIB) in the blend of PIB and head-to-head poly(propylene) (hhPP). Furthermore, the proposed phase diffusion approach provides an intuitive way to interpret NMR relaxation, particularly for the fractional NMR relaxation, which is still a challenge to explain by the available theoretical methods. The paper provides additional insights into NMR and magnetic resonance imaging (MRI) relaxation experiments.
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This paper employs the general time-space fractional diffusion equation to derive correlation time function for analyzing nuclear magnetic resonance (NMR) relaxation. Both the anomalous rotational and translational diffusion are treated. NMR relaxation time affected by various Hamilton interactions such as dipolar or quadrupolar couplings can be calculated from the Mittag-Leffler type time correlation and their corresponded spectral density functions obtained. Additionally, to verify the results, the theoretical expressions are applied to fit reported experimental data of NMR quadrupolar coupling relaxation of head-to-head poly(propylene) (hhPP) in a polymer blend. The fitting is excellent and more convenient than the fitting utilizing the traditional modified Kohlrausch-Williams-Watts (KWW) formalism. Further, it is found that the temperature dependence behavior of the segmental dynamics in anomalous diffusion may obey a different Vogel-Tamman-Fulcher (VTF) expression. The paper proposes new, general formalisms for analyzing various NMR relaxation experiments in macromolecular systems.
Molecular dynamics (MD) simulations are used to investigate $^1$H nuclear magnetic resonance (NMR) relaxation and diffusion of bulk $n$-C$_5$H$_{12}$ to $n$-C$_{17}$H$_{36}$ hydrocarbons and bulk water. The MD simulations of the $^1$H NMR relaxation times $T_{1,2}$ in the fast motion regime where $T_1 = T_2$ agree with measured (de-oxygenated) $T_2$ data at ambient conditions, without any adjustable parameters in the interpretation of the simulation data. Likewise, the translational diffusion $D_T$ coefficients calculated using simulation configurations are well-correlated with measured diffusion data at ambient conditions. The agreement between the predicted and experimentally measured NMR relaxation times and diffusion coefficient also validate the forcefields used in the simulation. The molecular simulations naturally separate intramolecular from intermolecular dipole-dipole interactions helping bring new insight into the two NMR relaxation mechanisms as a function of molecular chain-length (i.e. carbon number). Comparison of the MD simulation results of the two relaxation mechanisms with traditional hard-sphere models used in interpreting NMR data reveals important limitations in the latter. With increasing chain length, there is substantial deviation in the molecular size inferred on the basis of the radius of gyration from simulation and the fitted hard-sphere radii required to rationalize the relaxation times. This deviation is characteristic of the local nature of the NMR measurement, one that is well-captured by molecular simulations.
A modified-Bloch equation based on the fractal derivative is proposed to analyze pulsed field gradient (PFG) anomalous diffusion. Anomalous diffusion exists in many systems such as in polymer or biological systems. PFG anomalous diffusion could be analyzed based on the fractal derivative or the fractional derivative. Compared to the fractional derivative, the fractal derivative is simpler, and it is faster in numerical evaluations. In this paper, the fractal derivative is employed to build the modified-Bloch equation that is a fundamental method to describe the spin magnetization evolution affected by fractional diffusion, Larmor precession, and relaxation. An equivalent form of the fractal derivative is proposed to convert the fractional diffusion equation, which can then be combined with the precession and relaxation equations to get the modified-Bloch equation. This modified-Bloch equation yields a general PFG signal attenuation expression that includes the finite gradient pulse width (FGPW) effect, namely, the signal attenuation during field gradient pulse. The FGPW effect needs to be considered in most clinical MRI applications, and including FGPW effect allows the detecting of slower diffusion that is often encountered in polymer systems. Additionally, the spin-spin relaxation effect can be analyzed, which provides a broad view of the dynamic process in materials. The modified-Bloch equation based on the fractal derivative could provide a fundamental theoretical model for PFG anomalous diffusion.
It is well known that water inside hydrophobic nano-channels diffuses faster than bulk water. Recent theoretical studies have shown that this enhancement depends on the size of the hydrophobic nanochannels. However, experimental evidence of this dependence is lacking. Here, by combining two-dimensional Nuclear Magnetic Resonance (NMR) diffusion-relaxation D-T2eff spectroscopy in the stray field of a superconducting magnet, and Molecular Dynamics (MD) simulations, we analyze the size dependence of water dynamics inside carbon nanotubes (CNTs) of different diameters (1.1 nm to 6.0 nm), in the temperature range of 265K to 305K. Depending on the CNTs diameter, the nanotube water is shown to resolve in two or more tubular components acquiring different self-diffusion coefficients. Most notable, a favourable CNTs diameter range 3.0-4.5 nm is experimentally verified for the first time, in which water molecule dynamics at the centre of the CNTs exhibit distinctly non-Arrhenius behaviour, characterized by ultrafast diffusion and extraordinary fragility, a result of significant importance in the efforts to understand water behaviour in hydrophobic nanochannels.
A general expression is derived for the dipolar NMR spin-lattice relaxation rate $1/T_1$ of a system exhibiting Brownian dynamics in a discrete and finite configuration space. It is shown that this approach can be particularly useful to model the proton relaxation rate in molecular rotors.
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