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. Unlike the entropic potential ∼lnh(x), γ(x) becomes an increasing or decreasing function also in periodic channels, depending on the asymmetry of h(x) and thus it visualizes the net force driving the ratchet current. We demonstrate the appearance of the ratchet effect on a trial asymmetric channel; our theory is verified by a numerical solution of the corresponding Fokker-Planck equation. Isotropic driving force f results in the monotonic decrease of the ratchet current with a growing ratio α=D_{R}/D_{T} of the rotation and the translation diffusion constants; asymptotically going ∼1/α^{2}. If we allow anisotropy of the force, we can observe the current reversal depending on α.

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. Reducing the transverse coordinate, as well as the orientation of the force in the full-dimensional Fokker-Planck equation, we arrive at the generalized Fick-Jacobs equation, describing dynamics of the system in the longitudinal coordinate x only. The additional effective potential -γ(x), calculated within the mapping procedure, exhibits an increasing or decreasing part in the channel shaped by an asymmetric periodic h(x), which determines the appearing ratchet current. As shown on a specific example, random driving in the transverse direction is much more effective than that in the longitudinal direction, at least for quickly flipping orientation of the force. Also, the transverse and the longitudinal driving push the rectified current in opposite directions along the same channel.

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.% of RD1 was used as a reactive diluent in the cationic photopolymerization of ELO. In parallel, VDM was used as a plasticizer in AESO radical photopolymerization reactions. Positron annihilation lifetime spectroscopy (PALS) was used to characterize vegetable oil-based UV-cured polymers regarding their structural stability in a wide range of temperatures (120-320 K) and humidity. The polymers were used as laccase immobilization matrixes for the construction of amperometric biosensors. A direct dependence of the main operational parameters of the biosensors and microscopical characteristics of polymer matrixes (mostly on the size of free volumes and water content) was established. The biosensors are intended for the detection of trace water pollution with xenobiotics, carcinogenic substances with a very negative impact on human health. These findings will allow better predictions for novel polymers as immobilization matrixes for biosensing or biotechnology applications.

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. The self-propulsion, driving the system out of equilibrium, is reflected as an additional effective potential -γ(x) in the reduced picture. It enables us to understand peculiarities of this system in a handy way. The additionally corrected potential redistributes the confined particles in quasiequilibrium causing their piling at the walls. In periodic asymmetric channels, it acquires a growing contribution, responsible for driving the ratchet effect.

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. Aside from the effective diffusion coefficient D(x), our mapping also generates an additional effective potential -γ(x) added to the entropic potential -log[h(x)]. It acquires an increasing or decreasing component in asymmetric periodic channels, and thus it explains appearance of the ratchet current. We study this effect on a trial example and compare the results of our true 2D theory with a commonly used effective one-dimensional description; the data are verified by the numerical solution of the full 2D problem.

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. Kalinay, Phys. Rev. E 100, 032606 (2019)2470-004510.1103/PhysRevE.100.032606], projecting the corresponding 3D advection-diffusion equation onto the longitudinal coordinate and generating an effective one-dimensional modified Fick-Jacobs (or Smoluchowski) equation. A different scaling of the transverse forces by a small auxiliary parameter ε is used here. It results in a recurrence scheme enabling us to derive the corrections of the effective diffusion coefficient and the averaged driving force up to higher orders in ε. The new scaling also preserves symmetries of the stationary solution in any order of ε. Finally we show that Reguera-Rubí's formula, widely applied for description of diffusion in corrugated tubes, can be systematically corrected by the strength of the flow Q; we give here the first two terms in the form of closed analytic formulas.

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.

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. Unlike the previous specific cases, the presented procedure enables us to find the one-dimensional description of the confined diffusion even for non-conservative (vortex) forces, e.g. caused by flowing solvent dragging the particles. We show that the result is again the generalized Fick-Jacobs equation. Despite of non existing scalar potential in the case of vortex forces, the effective one-dimensional scalar potential, as well as the corresponding quasi-equilibrium and the effective diffusion coefficient [Formula: see text] can be always found.

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. The method avoids the hierarchy of derivatives of the channel shaping function h(x), so it is also suitable for the channels with cusps or jumps of their width. The method describes correctly D(x) in wide channels, giving the expected behavior in the limit of infinite width (no confinement).

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). Unlike the previous algorithms based on scaling of the transverse lengths, the method presented here uses periodicity of the channel. Instead of complicated expansion containing higher order derivatives of h(x), the proposed algorithm results in an integral formula for D(x), enabling us to study the system for wide range of the driving force and various (periodic) shaping functions h(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. Unlike the standard formulas based on the scaling, the new proposed formula also describes D(x) correctly near the cusps, or in wider channels.

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. Phys. 141, 144101 (2014)], is generalized to 3D channels of cylindrical symmetry, as well as to 2D or 3D channels with particles driven by a constant longitudinal external driving force. The next possible extensions are also indicated. The former calculation was based on calculus in the complex plane, suitable for the stationary diffusion in 2D domains. The method is reformulated here using standard tools of functional analysis, enabling the generalization.

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. [J. Chem. Phys. 133, 034707 (2010)].

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. The method presented enables us to solve the problem analytically for a sawtooth channel and also to look for approximative formulas applicable in a wide range of frequencies ω. Even the simplest approximation qualitatively reproduces the current reversal at higher frequencies as the result of growing phase lag of the rocking density behind the driving force.

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. Phys. Chem. 96, 3926 (1992); D. Reguera and J. M. Rubí, Phys. Rev. E 64, 061106 (2001)] is derived and tested on the exactly solvable model, diffusion in a 2D linear cone. The effects of such correction are demonstrated and discussed on elementary nonstationary processes, a time dependent perturbation of the stationary flow and calculation of the mean first passage time.

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. In the limit of stationary flow, an extended Fick-Jacobs equation is reached, which is corrected by an effective diffusion coefficient D(x) that depends on the longitudinal coordinate x. The result is an approximate formula for D(x) that also involves the influence of the transverse force. The calculations are verified by the stationary diffusion in a linear cone, which is exactly solvable.

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. Kalinay and J. K. Percus, Phys. Rev. E 74, 041203 (2006)]; i.e., we arrive at a one-dimensional evolution equation of the Fick-Jacobs type. On the other hand, the calculation presented serves as a prototype for mapping of the Smoluchowski equation with a wide class of potentials U(x,y) varying in both the longitudinal as well as the transverse directions, which is necessary for understanding, e.g., stochastic resonance.

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. We verify the surmise of Berezhkovskii [J. Chem. Phys. 131, 224110 (2009)] that one-dimensional diffusion behaves as free in such channels everywhere except at the point of change, which looks like a local trap for the particles. Applying the method of "sewing" of solutions, we show that this picture is valid also for three-dimensional symmetric channels.

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. We test this result on a model of stationary diffusion in a linear cone in a homogeneous potential, which is exactly solvable. Extension of the approximate formulas for D(x) derived for the diffusion alone is discussed.

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. We show that two different definitions of this quantity, based either on the mean first passage time calculated from solution of the Fick-Jacobs equation, or coming from transition state theory, are incompatible. They have different physical interpretation and also, they give different dependencies of the hopping time on the width of the channel.