Three-dimensional water diffusion in impermeable cylindrical tubes: theory versus experiments

Abstract

Characterizing diffusion of gases and liquids within pores is important in understanding numerous transport processes and affects a wide range of practical applications. Previous measurements of the pulsed gradient stimulated echo (PGSTE) signal attenuation, E(q), of water within nerves and impermeable cylindrical microcapillary tubes showed it to be exquisitely sensitive to the orientation of the applied wave vector, q, with respect to the tube axis in the high-q regime. Here, we provide a simple three-dimensional model to explain this angular dependence by decomposing the average propagator, which describes the net displacement of water molecules, into components parallel and perpendicular to the tube wall, in which axial diffusion is free and radial diffusion is restricted. The model faithfully predicts the experimental data, not only the observed diffraction peaks in E(q) when the diffusion gradients are approximately normal to the tube wall, but their sudden disappearance when the gradient orientation possesses a small axial component. The model also successfully predicts the dependence of E(q) on gradient pulse duration and on gradient strength as well as tube inner diameter. To account for the deviation from the narrow pulse approximation in the PGSTE sequence, we use Callaghan's matrix operator framework, which this study validates experimentally for the first time. We also show how to combine average propagators derived for classical one-dimensional and two-dimensional models of restricted diffusion (e.g. between plates, within cylinders) to construct composite three-dimensional models of diffusion in complex media containing pores (e.g. rectangular prisms and/or capped cylinders) having a distribution of orientations, sizes, and aspect ratios. This three-dimensional modeling framework should aid in describing diffusion in numerous biological systems and in a myriad of materials sciences applications.

Figure 1

(a) PGSTE sequence with gradient strength, G, pulse duration, δ, gradient pulse separation, Δ, and echo time, TE. (b) Cartoon of a particle undergoing diffusion in a domain that is restricted along one direction and free along the other. Here random displacements are uncoupled in these two orthogonal directions. A random walk is depicted for three subsequent time points, t₁, t₂, and t₃. (c) A pack of impermeable tubes oriented parallel to the z axis in the laboratory coordinate system. Shown are the tube axis, the components of the q vector, q, parallel and perpendicular to the tube axis, q_// and q_⊥, respectively, the tube radius, a, and the azimuthal angle, θ.

Figure 2

PGSTE signal attenuation, E_⊥(q_⊥, Δ), as a function of q value, q_⊥, in the narrow and fat pulse regimens. (a) Experimental data juxtaposed with simulations of signal attenuations for a 9 μm ID pack of tubes. Simulations using the narrow and fat pulse (δ = 3 ms) are shown. (b) Same experimental data now plotted against simulation results for different tube IDs. (c) Signal attenuation vs q data for a 20 μm ID pack. The fat pulse simulations are in almost perfect agreement with experimental data.

Figure 3

PGSTE E_⊥(q_⊥, Δ) vs q_⊥ data for various pulse durations, δ. (a) Experimental data obtained for the nominal 9 μm ID pack of tubes along with simulations performed by setting the ID to 8.2 μm. (b, c) Experimental data and simulation results for 20 μm ID tubes. The data from 20 μm ID tubes are separated into two panels for clarity.

Figure 4

E(q, Δ) vs q experimental PGSTE data for different gradient orientations, α. Superimposed are simulated data obtained using the matrix operator framework. A tube ID of 8.2 μm is assumed in (a); 20 μm in (b). Excellent agreement is seen between the experimental data and the model.

Figure 5

Simulations examining the effect of differences in the variance of the diameter distribution of a cylindrical pack of tubes on the predicted E_⊥(q_⊥, Δ) vs q_⊥ curves. Increasing the variance of an assumed Gaussian distribution of tube diameters affects both the q values at which minima are predicted and the sharpness and depth of these minima

Figure 6

Superposition of the 3D model of diffusion in tubes to describe the behavior of E_⊥(q_⊥, Δ) in a pack of cylinders with different degrees of splay of their axial director or orientation vector. Increasing the variance of an assumed Gaussian distribution of splay angles around a mean angle of zero decreases the sharpness and depth of the diffraction minima observed in the E_⊥(q_⊥, Δ) profile.