Journal of Chemistry

Volume 2015 (2015), Article ID 986402, 10 pages

http://dx.doi.org/10.1155/2015/986402

## Unbiased Diffusion through a Linear Porous Media with Periodic Entropy Barriers: A Tube Formed by Contacting Ellipses

Universidad Autónoma Metropolitana Iztapalapa, Avenue San Rafael Atlixco 186, 09340 México, DF, Mexico

Received 18 February 2015; Accepted 26 May 2015

Academic Editor: Demeter Tzeli

Copyright © 2015 Yoshua Chávez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work is devoted to the study of unbiased diffusion of point-like Brownian particles through channels with radial symmetry of varying cross-section and elliptic shape. The effective one-dimensional reduction is used with distinct forms of a position-dependent diffusion coefficient, , found in literature, to obtain expressions for (I) narrow escape times from a single open-ended tube, (II) its correspondent effective diffusion coefficient, both as functions of the eccentricity of the tube, *ε*, where *ε* = 0 returns the system to a spherical vesicle with two open opposite sides, and (III) finally, Lifson-Jackson formula that is used to compute expressions to assess the mean effective diffusion coefficient for a periodic elliptic channel formed by contacting ellipses, also as a function of the eccentricity. Mathematical expressions are presented and contrasted against computational simulations to validate them.

#### 1. Introduction

Among many systems, well spread in nature and technological applications, the unbiased diffusion of point-like particles confined within quasi-one-dimensional systems, such as pores and channels, is ubiquitous and has been subjected to increasing attention in recent years [1–6]. Diffusive transport in confined environments arises from different contexts of practical and theoretical interest. It is relevant to study in fields as nanotechnology, chemistry, and biology, having direct applications to channels such as pores in zeolites [7], carbon nanotubes [8], synthetic nanopores [9–11], artificial pores produced in solid thin films [12], channels in biological systems [13], and single-nanopore sensors, designed to detect, quantify, and characterize many different types of molecules, for example, single- and double-stranded DNA chains. Experimental techniques, such as high-resolution crystallography of bacterial porins and other large channels, have demonstrated that these can be envisaged as tubes with significantly varying cross-sections along their principal axis. In some of these channels, variations in cross-section areas exceed one order of magnitude [14, 15]. This leads to the so-called entropic-like walls and barriers in the theoretical description of transport through such structures.

When diffusion occurs in quasi-one-dimensional structures, it becomes intuitively appealing to introduce an effective one-dimensional description. In a three-dimensional tube of varying radius, , the -axis directed along the centerline of the tube, the one-dimensional concentration of point-like particles is related to their three-dimensional concentration by the expressionwhich averages the spatial concentration over the channel’s position-dependent cross-section, . Given the condition of uniform concentration in any cross-section, satisfies the Fick-Jacobs equationwhere is the particle’s bulk diffusion coefficient. Later, this result was generalized by Zwanzig [16], showing that the diffusion entering in (2) becomes position-dependent, , by means of introducing fluctuations in the concentration of particles in any cross-section, provided is a slowly varying function of ; that is, . Thus the generalized Fick-Jacobs equation takes the form Zwanzig also proposed an explicit form to :which has been extensively used throughout literature but in many cases poorly adjusted to simulated data. Later, Reguera and Rubí generalized Zwanzig’s result, and based on heuristic arguments they suggested that entering into (3) is given by [17]That has proved to be more reliable given several geometrical conditions [18–22], when used along with (3). Surprisingly, in the same literature, this approach has also shown to be valid, even for a less restrictive condition upon the radius’ rate of change: , thus giving the generalized Fick-Jacobs equation a considerably extended range of applicability. In the same spirit Kalinay and Percus [23, 24] developed a more general theory of reduction to the effective one-dimensional description for radially symmetrical two- (2D) and three-dimensional (3D) tubes, and Dagdug and Pineda [25] later extended these results to nonsymmetrical 2D systems.

A wide range of stochastic processes of practical interest underlies first-passage events, such as the first-passage time, namely, the probability that a diffusing particle or a random walk will first reach a specified site (or set of sites) at a specified time [26]. Indeed, chemical and biochemical reactions [27, 28], animals searching for food [29], and trafficking receptors on biological membranes [30] are often controlled by first-encounter events [31]. Among first-passage events, the narrow escape time (NET)—the mean time a Brownian particle spends before being trapped by an opening window in exiting the cavity for the first time—is a quantity is of particular importance. Its applications range from cellular biology to biochemical reactions in cellular microdomains, such as dendritic spines, synapses, and micro vesicles, among others [30, 32]. The NET is the limiting quantity and the first step in the mathematical modeling of such processes where particles must first exit their domains in order to live up to their biological function [31].

To determine which form of to enter in (3) for a given set of geometries and boundary conditions, we took advantage of the fact that the Mean First-Passage Time (MFPT)—defined as the time it takes a random walker to reach a specified place for the first time, averaged over all the trajectories or realizations of the random walk—, *τ*, is a quantity often obtained by means of computer simulations. The MFPT is found to satisfy a backward equation:where , is the Boltzmann constant, and is the absolute temperature, and is the initial position where any particle begins its random walk. The potential is defined as follows:Equation (7), called* entropic-like* potential, accounts for the change in cross-section area along the axial length of the tube, taken to be zero at (a reference position, placed somewhere in the system). Then (6) is solved for the appropriate boundary conditions, to obtain an algebraic expression that relates with and the system’s geometrical parameters.

Complex geometries, such as those found in nature [13] and manmade structures at the small scale [33, 34], hardly resemble the simple shapes used so far to model them. Nonetheless, particle transport through these very complex geometries still can be tackled as a problem of diffusion in quasi-one-dimensional structures [5]. Again, Zwanzig had worked, back in 1983, a solution to the study of diffusion of Brownian particles in two-dimensional channels made of periodic boundaries [35], using a sophisticated mathematical result, known as the Lifson-Jackson theorem [36]. More recently, the correspondent Lifson-Jackson formulahas been successfully used in several works on 2D and 3D systems [19, 20, 22, 37, 38], to obtain a mathematical expression for the effective one-dimensional diffusion coefficient averaged for periodic channels, which represents a good assessment of the effective diffusion coefficient behavior in real long, narrow channels (i.e., their length is significantly greater than their radii), with internal structure (also known as* corrugated* channels).

The present work is divided in three main parts:(I)Following the methodology developed in previous studies [20, 22], we treat the system shown in Figure 1, a single tube of uniform cross-section and elliptic shape (which we call* elliptic tube* hereafter) of length , where is given by , is the semilength of the minimum diameter (minor axis), gamma is the semi length of the maximum diameter (major axis) given by the expression , and epsilon is the eccentricity. The openings placed at opposite sides of the tube are of radius . In this system we solve (6) entering (7) and several expressions for the position dependent diffusion coefficient : (the* Fick-Jacobs approximation*), , and , given by (4) and (5), respectively, and subject to the appropriate boundary conditions (see Figure 3), thus obtaining algebraic expressions of as a function of geometrical parameters of the system (, , ). The resulting theoretical curves will be compared with data obtained by computer simulations.(II) We assess the effective diffusion coefficient for this tube, using the exact solution of (6) for a cylindrical tube (given , and ) to establish the relation , where we have assumed that has the same structure than the bulk diffusion . Then we compare the theoretical curves of with values obtained by computer simulations.(III) Finally, we use the Lifson-Jackson formula (8), along with different forms of —the same set of expressions as before—to obtain mathematical expressions of the mean effective diffusion coefficient, , as a function of geometrical parameters of the system. This last system is a periodic channel, made of juxtaposed elliptic tubes, as seen in Figure 2 (from now on called* periodic elliptic tube*). The various analytical expressions of , , elliptic tube, and , periodic elliptic tube, obtained along with the graphical comparisons between theory and computer simulated data, constitute the main results of this paper.