Theory and Application of NMR Diffusion Studies

Abstract

Theoretical Aspects Concepts of diffusion Self-diffusion is the random translational motion of molecules driven by their internal kinetic energy [1]. Translational diffusion and rotational diffusion can be distinguished. Diffusion is related to molecular size, as becomes apparent from the Stokes-Einstein equation: D = k B T / f (1) where D is the diffusion coefficient, k B is the Boltzmann constant, T is the temperature, and f is the friction coefficient. If the solute is considered to be a spherical particle with an effective hydrodynamic radius (i.e. Stokes radius) r S in a solution of viscosity η, then the friction coefficient is given by: f = 6πηr S (2) Pulse Sequences for PFG NMR Diffusion Measurements Using the pulsed-field gradient (PFG) method, motion is measured by evaluating the attenuation of a spin echo signal [2]. The attenuation is achieved by the dephasing of nuclear spins due to the combination of the translational motion and the imposition of gradient pulses. In contrast to relaxation methods, no assumptions concerning the relaxation mechanism(s) are necessary. The PFG NMR sequence (Figure 1) is the simplest for measuring diffusion [2]. During application of the gradient, which is along the direction of the static, spectrometer field, B 0 , the effective magnetic field for each spin is dependent on its position. Therefore, the precession frequency is also position dependent which leads to the development of position dependent phase angles. The 180 • pulse changes the direction of the precession. Hence, the second gradient of equal magnitude will cancel the effects of the first and re-focus all spins, provided that no change of position, with respect to the direction of the gradient, has occurred. If there is a change of position, the refocusing will not be complete. This results in a remaining dephasing which is proportional to the displacement during the period between the two gradients. Since diffusion is a random motion, there is a distribution of gradient-induced phase angles. These random phase shifts are averaged over the ensemble of spins contributing to the observed NMR signal. Hence, this signal is not phase shifted but attenuated, with the degree of attenuation depending on the displacement. In Ref. [1], this phenomenon is explained in more detail. It is shown in Ref. [1] that the signal intensity S(2τ) after the total echo time 2τ is given by: S(2τ) = S(0) exp − 2τ T 2 exp −γ 2 g 2 Dδ 2 − δ 3 = S(2τ) g=0 exp −γ 2 g 2 Dδ 2 − δ 3 (3) where S(0) is the signal intensity immediately after the 90 • pulse, T 2 is the spin-spin relaxation time of the species, γ is the gyromagnetic ratio of the observed nucleus, g is the strength of the applied gradient, and δ and are the length of the rectangular gradient pulses and the separation between them, respectively. Typically, δ is in the range of 0-10 ms, the diffusion time is in the range of milliseconds to seconds, and g is up to 20 T/m [1]. To determine diffusion coefficients, a series of experiments is performed in which either g, δ, or is varied while keeping τ constant to achieve identical attenua-tion due to relaxation. Normally, the gradient strength g is incremented in subsequent experiments. Non-linear regression of the experimental data can be used for the determination of D. Nowadays, the BPPLED pulse sequence (see Figure 2) is most often used for measuring diffusion since it allows eddy currents to decay and uses bipolar gradients which enables double effective strength as well as compensation for imperfections. This sequence is not affected by spin-spin coupling since it is based on the stimulated echo sequence. C 135 Graham A. Webb (ed.), Modern Magnetic Resonance, 135-143. 2008 Springer.