Revealing mesoscopic structural universality with diffusion

Abstract

Measuring molecular diffusion is widely used for characterizing materials and living organisms noninvasively. This characterization relies on relations between macroscopic diffusion metrics and structure at the mesoscopic scale commensurate with the diffusion length. Establishing such relations remains a fundamental challenge, hindering progress in materials science, porous media, and biomedical imaging. Here we show that the dynamical exponent in the time dependence of the diffusion coefficient distinguishes between the universality classes of the mesoscopic structural complexity. Our approach enables the interpretation of diffusion measurements by objectively selecting and modeling the most relevant structural features. As an example, the specific values of the dynamical exponent allow us to identify the relevant mesoscopic structure affecting MRI-measured water diffusion in muscles and in brain, and to elucidate the structural changes behind the decrease of diffusion coefficient in ischemic stroke.

Fig. 1.

Time-dependent diffusion distinguishes between structural universality classes in one dimension, represented here by the placement of identical permeable barriers with the same mean density. (A) Order (red), hyperuniform disorder (green), short-range disorder (blue), and strong disorder (magenta) are shown. (B) The barrier densities have qualitatively different large-scale fluctuations, reflected in the small-k behavior of their density correlator (see Examples of Structural Universality Classes). (C) Numerical results confirming the relation 1. The time-dependence 3 clearly distinguishes between the four arrangements, while the value is the same for all of them. The dashed lines are the exact power laws from Eqs. S14, S19, and S23, and the exponential decrease is from the exact solution, Eq. S25. Strong disorder occurs for ; here .

Fig. 2.

Structural universality classes in dimension d > 1. (A and B) The examples of analogs of the d = 1 classes, corresponding to Fig. 1 (blue and red). (C–E) The extended universality classes inherent to d > 1. (C) Random membranes, with representatives shown for d = 2 and d = 3, result in ϑ = 1/2 for any d. (D) Random rods, with a representative shown for d = 3, result in ϑ = 1 for any d. (E) Structure correlator Γ(k) ∼ k^p (numerically calculated and angular averaged, arbitrary units) for C (magenta in d = 2 and green in d = 3), and for D (gray), exhibits the negative structural exponent p = –d_s.

Fig. 3.

Time-dependent diffusion transverse to muscle fibers from ref. reveals extended structural disorder class of in , provided by the muscle fiber membrane (sarcolemma). (A) The longitudinal, , and the transverse, , diffusion tensor components for calf tongue genioglossus (TG) (blue circles) and heart (H) (red diamonds). Solid lines are the fit of to derived from Eq. S27 with . For fit results see Table S1. (B) Data for replotted as function of consistent with . Eq. 1 yields ; hence, (see Extended Disorder Provided by Muscle Fiber Walls and Fig. 2C). (C) Muscle slice across the fibers. (D) calculated from image intensity in C. Tight cell packing achieved by straight cell walls in C results in exponent of the extended disorder class of Fig. 2C, yielding .

Fig. 4.

Dispersive in cerebral gray matter consistent with diffusion along narrow neurites in the presence of short-range disorder , both in normal and globally ischemic rat brain. (A) Original data (20) for d-sin and cos gradient waveforms, fitted to Eq. 6, yields for normal and for postmortem brain. The role of the disorder (the slope) increases after global ischemia onset. (B) Varicose axons from rat hippocampus area CA1 (22) rationalizing the picture of an effectively 1D diffusion inside narrow randomly oriented and structurally disordered neurites.

Fig. 5.

Cumulative diffusion coefficient, Eq. 4, for the 1D example of Fig. 1. Dashed lines correspond to the asymptotic power law decrease of . For and (blue represents short-range disorder and magenta strong disorder), the power law in coincides with that in (in accord with Eq. 5) whereas for (red represents periodic and green hyperuniform), it is masked by the term. Taking the derivative reveals the values of ϑ (as shown in Fig. 1) but increases noise.