Ratchet effect in an asymmetric two-dimensional system of Janus particles.

We consider a disk-like Janus particle self-driven by a force of constant magnitude f, but an arbitrary direction depending on the stochastic rotation of the disk. The particle diffuses in a two-dimensional channel of varying width 2h(x). We applied the procedure mapping the 2+1-dimensional Fokker-Planck equation onto the longitudinal coordinate x; the result is the Fick-Jacobs equation extended by the spatially dependent effective diffusion constant D(x) and an additional effective potential -γ(x), derived recursively within the mapping procedure.

Transverse dichotomic ratchet in a two-dimensional corrugated channel.

A particle diffusing in a two-dimensional channel of varying width h(x) is considered. It is driven by a force of constant magnitude f, but random orientation across the channel. We suggest the projection technique to study the ratchet effect appearing in this system.

UV-Cured Green Polymers for Biosensorics: Correlation of Operational Parameters of Highly Sensitive Biosensors with Nano-Volumes and Adsorption Properties.

The investigated polymeric matrixes consisted of epoxidized linseed oil (ELO), acrylated epoxidized soybean oil (AESO), trimethylolpropane triglycidyl ether (RD1), vanillin dimethacrylate (VDM), triarylsulfonium hexafluorophosphate salts (PI), and 2,2-dimethoxy-2-phenylacetophenone (DMPA). Linseed oil-based (ELO/PI, ELO/10RD1/PI) and soybean oil-based (AESO/VDM, AESO/VDM/DMPA) polymers were obtained by cationic and radical photopolymerization reactions, respectively. In order to improve the cross-linking density of the resulting polymers, 10 mol.

Dichotomic ratchet in a two-dimensional corrugated channel.

We consider a particle diffusing in a two-dimensional (2D) channel of varying width h(x). It is driven by a force of constant magnitude f but random orientation there or back along the channel. We derive the effective generalized Fick-Jacobs equation for this system, which describes the dynamics of such a particle in the longitudinal coordinate x.

Reduced dynamics of a one-dimensional Janus particle.

A Janus particle diffusing on a line is considered. Aside from its own driving force f acting forward or backward according to its stochastic orientation, it moves in a position-dependent potential U(x). We propose here the mapping scheme generating the effective generalized Fick-Jacobs equation, describing motion of the particle in the spatial coordinate x only; the orientation is understood as the transverse coordinate.

Taylor dispersion in Poiseuille flow in three-dimensional tubes of varying diameter.

Diffusion of particles carried by Poiseuille flow of the surrounding solvent in a three-dimensional (3D) tube of varying diameter is considered. We revisit our mapping technique [F. Slanina and P.

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.

Dimensional reduction of a general advection-diffusion equation in 2D channels.

Diffusion of point-like particles in a two-dimensional channel of varying width is studied. The particles are driven by an arbitrary space dependent force. We construct a general recurrence procedure mapping the corresponding two-dimensional advection-diffusion equation onto the longitudinal coordinate x.

One-dimensional description of driven diffusion in periodic channels.

Diffusion of point-like particles driven by a constant longitudinal force in two-dimensional channels of periodically varying width is studied. The dynamics of such systems can be effectively described by the one-dimensional Smoluchowski(-Fick-Jacobs) equation in the longitudinal coordinate x, extended by a space dependent effective diffusion coefficient D(x). Our paper is focused on calculation of this function for an arbitrary channel shaping function h(x).

Nonscaling calculation of the effective diffusion coefficient in periodic channels.

An algorithm calculating the effective diffusion coefficient D(x) in 2D and 3D channels with periodically varying cross section along the longitudinal coordinate x is presented. Unlike other methods, it is not based on scaling of the transverse coordinates, or the smallness of the width of the channel. The result is expressed as an integral of specific contributions to D(x) coming from the positions neighboring to x.

Integral formula for the effective diffusion coefficient in two-dimensional channels.

The effective one-dimensional description of diffusion in two-dimensional channels of varying cross section is revisited. The effective diffusion coefficient D(x), extending Fick-Jacobs equation, depending on the longitudinal coordinate x, is derived here without use of scaling of the transverse coordinates. The result of the presented method is an integral formula for D(x), calculating its value at x as an integral of contributions from the neighboring positions x^{'} depending on h(x^{'}), a function shaping the channel.

Generalized method calculating the effective diffusion coefficient in periodic channels.

The method calculating the effective diffusion coefficient in an arbitrary periodic two-dimensional channel, presented in our previous paper [P. Kalinay, J. Chem.

Effective diffusion coefficient in 2D periodic channels.

Calculation of the effective diffusion coefficient D(x), depending on the longitudinal coordinate x in 2D channels with periodically corrugated walls, is revisited. Instead of scaling the transverse lengths and applying the standard homogenization techniques, we propose an algorithm based on formulation of the problem in the complex plane. A simple model is solved to explain the behavior of D(x) in the channels with short periods L, observed by Brownian simulations of Dagdug et al.

Rectification of confined diffusion driven by a sinusoidal force.

A particle diffusing in an asymmetric periodic channel, driven by a sinusoidal force F(t)=F0cosωt (the rocking ratchet) is considered. The asymptotic solution of the generalized Fick-Jacobs equation describing the system is studied in the nonadiabatic regime. The leading term of the rectified current, appearing in the order ∼F02, is derived.

When is the next extending of Fick-Jacobs equation necessary?

Applicability of the effective one-dimensional equations, such as Fick-Jacobs equation and its extensions, describing diffusion of particles in 2D or 3D channels with varying cross section A(x) along the longitudinal coordinate x, is studied. The leading nonstationary correction to Zwanzig-Reguera-Rubí equation [R. Zwanzig, J.

Effective one-dimensional description of confined diffusion biased by a transverse gravitational force.

Diffusion of pointlike noninteracting particles in a two-dimensional channel of varying cross section is considered. The particles are biased by a constant force in the transverse direction. A recurrence mapping procedure is applied, which enables the derivation of an effective one-dimensional (1D) evolution equation that governs the 1D density of the particles in the channel.

Mapping of diffusion in a channel with soft walls.

We study diffusion of pointlike particles biased toward the x axis by a quadratic potential U(x,y)=κ(x)y². This system mimics a channel with soft walls of some varying (effective) cross section A(x), depending on the stiffness κ(x). We show that diffusion in this geometry can also be mapped rigorously onto the longitudinal coordinate x by a procedure known for channels with hard walls [P.

Mapping of diffusion in a channel with abrupt change of diameter.

Mapping of the diffusion equation in a channel of varying cross section onto the longitudinal coordinate is already a well studied procedure for a slowly changing radius. We examine here the mapping of diffusion in a channel with abrupt change of diameter. In two dimensions, our considerations are based on solution of the exactly solvable geometry with abruptly doubled width at x=0.

Mapping of forced diffusion in quasi-one-dimensional systems.

Diffusion in an external potential in a two-dimensional channel of varying cross section is considered. We show that a rigorous mapping procedure applied on the corresponding Smoluchowski equation yields a one-dimensional evolution equation of the Fick-Jacobs type corrected by an effective coefficient D(x). The procedure enables us to derive this function within a recurrence scheme.

Two definitions of the hopping time in a confined fluid of finite particles.

We consider a fluid of hard disks diffusing in a flat long narrow channel of width approaching from above the doubled diameter of the disks. In this limit, the disks can pass their neighbors only rarely, in a mean hopping time growing to infinity, so the disks start by diffusing anomalously. We study the hopping time, which is the crucial parameter of the theory describing the subsequent transition to normal diffusion.

Approximations of the generalized Fick-Jacobs equation.

We analyze the generalized Fick-Jacobs equation, obtained by a rigorous mapping of the diffusion equation in a quasi-one-dimensional (quasi-1D) (narrow 2D or 3D) channel with varying cross section A(x) onto the longitudinal coordinate x . We show that for constructing approximations and understanding their applicability in practice, it is crucial to study the 2D (3D) density inside the channel in the regime of stationary flow. We present algorithms enabling us to derive approximate formulas for the effective diffusion coefficient involving derivatives of A(x) higher than A'(x) and give examples for 2D channels.

Calculation of the mean first passage time tested on simple two-dimensional models.

A particle diffusing in a two-dimensional (2D) container, shaped as a simplified configuration space of two passing 2D circular particles in a flat channel, is considered. The mean first passage time through one absorbing boundary is calculated using the one-dimensional Fick-Jacobs equation and its modification; both derived by mapping the 2D diffusion equation onto the longitudinal ("reaction") coordinate. The obtained results are compared with the hopping time, defined as the inverted lowest eigenvalue of the full 2D problem.