Effects of curved midline and varying width on the description of the effective diffusivity of Brownian particles

Diffusion coefficients and MSD measurements on curved membranes and porous media

We study some geometric aspects that influence the transport properties of particles that diffuse on curved surfaces. We compare different approaches to surface diffusion based on the Laplace-Beltrami operator adapted to predict concentration along entire membranes, confined subdomains along surfaces, or within porous media. Our goal is to summarize, firstly, how diffusion in these systems results in different types of diffusion coefficients and mean square displacement measurements, and secondly, how these two factors are affected by the concavity of the surface, the shape of the possible barriers or obstacles that form the available domains, the sinuosity, tortuosity, and constrictions of the trajectories and even how the observation plane affects the measurements of the diffusion. In addition to presenting a critical and organized comparison between different notions of MSD, in this review, we test the correspondence between theoretical predictions and numerical simulations by performing finite element simulations and illustrate some situations where diffusion theory can be applied. We briefly reviewed computational schemes for understanding surface diffusion and finally, discussed how this work contributes to understanding the role of surface diffusion transport properties in porous media and their relationship to other transport processes.

Surface diffusion in narrow channels on curved domains

We study the transport properties of diffusing particles restricted to confined regions on curved surfaces. We relate particle mobility to the curvature of the surface where they diffuse and the constraint due to confinement. Applying the Fick-Jacobs procedure to diffusion in curved manifolds shows that the local diffusion coefficient is related to average geometric quantities such as constriction and tortuosity. Macroscopic experiments can record such quantities through an average surface diffusion coefficient. We test the accuracy of our theoretical predictions of the effective diffusion coefficient through finite-element numerical solutions of the Laplace-Beltrami diffusion equation. We discuss how this work contributes to understanding the link between particle trajectories and the mean-square displacement.

Modeling pyramidal silicon nanopores with effective ion transport

While the electrical models of the membrane-based solid-state nanopores have been well established, silicon-based pyramidal nanopores cannot apply these models due to two distinctive features. One is its 35.3° half cone angle, which brings additional resistance to the moving ions inside the nanopore. The other is its rectangular entrance, which makes calculating the access conductance challenging. Here, we proposed and validated an effective transport model (ETM) for silicon-based pyramidal nanopores by introducing effective conductivity. The impact of half cone angle can be described equivalently using a reduced diffusion coefficient (effective diffusion coefficient). Because the decrease of diffusion coefficient results in a smaller conductivity, effective conductivity is used for the calculation of bulk conductance in ETM. In the classical model, intrinsic conductivity is used. We used the top-down fabrication method for generating the pyramidal silicon nanopores to test the proposed model. Compared with the large error (≥25% in most cases) when using the classical model, the error of ETM in predicting conductance is less than 15%. We also found that the ETM is applicable when the ratio of excess ion concentration and bulk ion concentration is smaller than 0.2. At last, it is proved that ETM can estimate the tip size of pyramidal silicon nanopore. We believe the ETM would provide an improved method for evaluating the pyramidal silicon nanopores.

Enhanced diffusion in soft-walled channels with a periodically varying curvature

The motion of particles along channels of finite width is known to be hindered by either the presence of energy barriers along the channel direction or by variations in the width of the channel in the transverse direction (rugged channel). Remarkably, when both features are present, they can interact to produce a counterintuitive result: adding energy barriers to a rugged channel can enhance the rate of diffusion along it. This is the result of competing energetic and entropic effects. Under the approximation of particles instantaneously in equilibrium in the transverse direction, one can tailor the energy barriers to the ruggedness to recover free diffusion. However, such fine-tuning and potentially restrictive approximations are not necessary to observe an enhanced rate of diffusion as we demonstrate by adding a range of (non-fine-tuned) energy barriers to a channel of sinusoidally varying curvature. Furthermore, this was observed to hold for systems with a finite characteristic timescale for motion in the transverse direction, thus, suggesting that the phenomenon lends itself to be exploited for practical applications.

Hydrodynamic separation of colloidal particles in tubes: Effective one-dimensional approach

We investigate diffusion of colloidal particles carried by flow in tubes of variable diameter and under the influence of an external field. We generalize the method mapping the three-dimensional confined diffusion onto an effective one-dimensional problem to the case of nonconservative forces and use this mapping for the problem in question. We show that in the presence of hydrodynamic drag, the lowest approximation (the Fick-Jacobs approximation) may be insufficient, and inclusion of at least the first-order correction is desirable to obtain more reliable results. As a practical application, we use the method for investigation of separation of colloidal particles carried by a fluid flow according to their size, using flotation and centrifugation.

Effect of particle size oscillations on drift and diffusion along a periodically corrugated channel

We study diffusive transport of a particle in a channel with periodically varying cross-section, occurring when the size of the particle periodically switches between two values. In such a situation, the entropy potential, which accounts for the area accessible for diffusion particle, varies both spatially (along the channel axis) and temporally. This underlies the complex interplay between different timescales of the system and leads to novel dynamic regimes. The most notable observations are: emergence of directed motion (in case of asymmetric channel) and resonant diffusion, both controlled by the switching frequency. Resonantlike behaviors of the drift velocity and the effective diffusion coefficient are shown and discussed. Based on heuristic arguments, an approximate analytical treatment of the transport process is proposed. As a comparison with the results obtained from Brownian dynamics simulations indicates, this approach provides a satisfactory way to handle the problem analytically.