Spatio-temporal anomalous diffusion in heterogeneous media by nuclear magnetic resonance

Abstract

For the first time, the diffusion phase diagram in highly confined colloidal systems, predicted by Continuous Time Random Walk (CTRW), is experimentally obtained. Temporal and spatial fractional exponents, α and µ, introduced within the framework of CTRW, are simultaneously measured by Pulse Field Gradient Nuclear Magnetic Resonance technique in samples of microbeads dispersed in water. We find that α depends on the disorder degree of the system. Conversely, µ depends on both bead sizes and magnetic susceptibility differences within samples. Our findings fully match the CTRW predictions.PACS numbers: 82.56. Lz, 87.64.kj, 61.05.Qr, Anomalous diffusion (AD), that predicts the meansquare displacement (MSD) of a diffusing particle to grow nonlinearly in time, [x(t) − x(0)] 2 ∝ t ν (with ν = 1), is a property of many complex systems and its related phenomena have been observed in various physical fields [5,8,14,16,17]. As shown by Metzler and coworkers [10], the features of AD can be described using fractional calculus. According to these Authors, molecular AD in media can be described by defining the motion propagator (MP) as the solution of fractional diffusion equations, which arise from the continuous time random walk (CTRW) model. These equations involve two fractional exponents, α and µ, which are, the orders of the time and space fractional derivatives, respectively [15]. The theoretical framework of CTRW is well established and has been corroborated by huge amounts of Monte Carlo simulations (see for example [20]) together with several experimental evidences, mainly obtained by using fluorescent spectroscopy [13,19]. However, to our knowledge, an experimental α vs µ phase diagram, showing the competition between superdiffusion and subdiffusion of diffusing molecules in heterogeneous media, has never been carried out. In this Letter, we experimentally challenge the Metzler et al. prediction of the CTRW theory [10] by measuring the fractional exponents α and µ by means of Nuclear Magnetic Resonance (NMR) methods, based on Pulse Field Gradient (PFG) technique [18], and by providing their interplay for the first time.In the last few decades, NMR water diffusion measurements have been a topic of extensive research, having broad applications in biophysics [2] and medicine [7]. Translational self-diffusion in liquid systems can be measured using PFG techniques, by applying magnetic gradient pulses (named diffusion gradients) to the system, in addition to the static magnetic field of the instrument itself. The signal attenuation, that depends on both dif- * Corresponding author: silvia.capuani@roma1.infn.it fusion gradient strength and diffusion time, is simply the Fourier Transform (FT) of the MP. When MP is Gaussian, NMR signal attenuation follows a mono-exponential Stejskal-Tanner decay [18]. Conversely, when the motion is described by a non-Gaussian propagator, the signal attenuation deviates from a mono-exponential decay [9,12]. In the present study, we have measured the fractional exponent α collecting the PFG ...

Spatio-temporal anomalous diffusion imaging: results in controlled phantoms and in excised human meningiomas

“…This is due to the high sensitivity of Mγ to Δχ which is about one order of magnitude higher in TiO 2 beads in water sample than that in polystyrene beads in water sample. According to our previous results [13,14], data reported in Fig. 3A) confirms that Mγ contrast depends on both the water dynamics and the Δχ.…”

“…Recently, we introduced a novel method [13] to obtain microstructural information from heterogeneous media by means of diffusion methods based on non-Gaussian diffusion theory. The approach utilizes the theory of the Continuous Time Random Walk (CTRW) [10] as an effective approach to describe the features of non normal, or anomalous diffusion [9,10].…”

“…We measured both the spatial AD index γ and the temporal AD index α, by using spectroscopic pulse gradient stimulated echo (PGSTE) methods [13] showing that γ quantifies pseudo-superdiffusion processes [13,14] while α quantifies subdiffusion processes [13]. Moreover, γ depends on local magnetic susceptibility (Δχ) [13,14] or on effective internal gradients [15][16][17] and it is equivalent to the stretching parameter of the AD stretched exponential imaging method, recently developed to investigate brain tissue [18][19][20]. Conversely, α provides information on the disorder degree of the system and does not depend on Δχ.…”

“…By taking into account the theory of CTRW [10,13], MP is defined as the solution of fractional diffusion equations. These equations introduce two parameters, the fractional exponent in time α and the fractional exponents in space γ, which are the time and space derivatives fractional orders.…”

“…1); capillary number 2 is filled with a mixture of 6 μm and 10 μm beads suspended in water. As a consequence it represents a more disordered system [13]; capillary number 3 is filled with TiO 2 5 μm beads monodispersed in water; capillary number 4 is filled with TiO 2 0.05 μm beads monodispersed in water; capillaries 5 and 6 are filled with 6 μm polystyrene beads monodispersed in water. These six microcapillaries were immersed in a 8 mm NMR tube ( Fig.…”