Hydrodynamically enforced entropic trapping of Brownian particles

Dispersive transport dynamics in porous media emerge from local correlations

Understanding and controlling transport through complex media is central for a plethora of processes ranging from technical to biological applications. Yet, the effect of micro-scale manipulations on macroscopic transport dynamics still poses conceptual conundrums. Here, we demonstrate the predictive power of a conceptual shift in describing complex media by local micro-scale correlations instead of an assembly of uncorrelated minimal units. Specifically, we show that the non-linear dependency between microscopic morphological properties and macroscopic transport characteristics in porous media is captured by transport statistics on the level of pore junctions instead of single pores. Probing experimentally and numerically transport through two-dimensional porous media while gradually increasing flow heterogeneity, we find a non-monotonic change in transport efficiency. Using analytic arguments, we built physical intuition on how this non-monotonic dependency emerges from junction statistics. The shift in paradigm presented here broadly affects our understanding of transport within the diversity of complex media.

Dispersive transport through complex media, relevant for semiconductors, liquid crystals, and biological soft matter, is influenced by their microscopic, porous structure. The authors consider the statistics of pore-junction units, in contrast to individual pores, to link morphology and macroscopic transport characteristics.

How Stickiness Can Speed Up Diffusion in Confined Systems

The paradigmatic model for heterogeneous media used in diffusion studies is built from reflecting obstacles and surfaces. It is well known that the crowding effect produced by these reflecting surfaces slows the dispersion of Brownian tracers. Here, using a general adsorption desorption model with surface diffusion, we show analytically that making surfaces or obstacles attractive can accelerate dispersion. In particular, we show that this enhancement of diffusion can exist even when the surface diffusion constant is smaller than that in the bulk. Even more remarkably, this enhancement effect occurs when the effective diffusion constant, when restricted to surfaces only, is lower than the effective diffusivity with purely reflecting boundaries. We give analytical formulas for this intriguing effect in periodic arrays of spheres as well as undulating microchannels. Our results are confirmed by numerical calculations and Monte Carlo simulations.

Active microrheology in corrugated channels: Comparison of thermal and colloidal baths

HYPOTHESIS

The dynamics of colloidal suspension confined within porous materials strongly differs from that in the bulk. In particular, within porous materials, the presence of boundaries with complex shapes entangles the longitudinal and transverse degrees of freedom inducing a coupling between the transport of the suspension and the density inhomogeneities induced by the walls.

METHOD

Colloidal suspension confined within model porous media are characterized by means of active microrheology where a net force is applied on a single colloid (tracer particle) whose transport properties are then studied. The trajectories provided by active microrheology are exploited to determine the local transport coefficients. In order to asses the role of the colloid-colloid interactions we compare the case of a tracer embedded in a colloidal suspension to the case of a tracer suspended in an ideal bath.

FINDING

Our results show that the friction coefficient increases and the passage time distribution widens upon increasing the corrugation of the channel. These features are obtained for a tracer suspended in a (thermalized) colloidal bath as well as for the case of an ideal thermal bath. These results highlight the relevance of the confinement on the transport and show a mild dependence on the colloidal/thermal bath. Finally, we rationalize our numerical results with a semi-analytical model. Interestingly, the predictions of the model are quantitatively reliable for mild external forces, hence providing a reliable tool for predicting the transport across porous materials.

Entropic stochastic resonance induced by a transverse driving force

The phenomenon of entropic stochastic resonance (ESR) is investigated with the presence of a time-periodic force in the transverse direction. Simulation results manifest that the ESR can survive even if there is no static bias force in any direction, just if a transverse driving field is applied. In the weak noise region, the transverse driving force leads to a giant-suppression of the escape rate from one well to another, i.e. the entropic trapping. The increase in noise intensity will eliminate this suppression and induce the ESR phenomenon. An alternative quantity, called the mean free flying time, is also proposed to characterize the ESR as well as the conventional spectral power amplification. The ESR can be modulated conveniently by the transverse periodic force, which implies an alternative method for controlling the dynamics of small-scale systems. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 2)'.

Advection-enhanced diffusion in biased convection arrays

We numerically investigated the transport of a passive colloidal particle in a one-dimensional periodic array of planar counter-rotating convection rolls at high Péclet numbers. We show that advection-enhanced diffusion is drastically suppressed by an external transverse bias but strongly reinforced by a longitudinal drive of appropriate intensity. Both effects are magnified by imposing free-slip flows at the array's edges. The dependence of the diffusion constant on an external forcing is interpreted as a measure of the fluid-mechanical robustness of the flow boundary layer mechanism governing diffusion in convection rolls.

Colloid particles in microfluidic inertial hydrodynamic ratchet at moderate Reynolds number

The movement of spherical Brownian particle carried by an alternating fluid flow in a tube of periodically variable diameter is investigated. On the basis of our previous results [Phys. Rev. E 99, 012604 (2019)10.1103/PhysRevE.99.012604] on the hydrodynamics of the problem, we look at the competition of hydrodynamics and diffusion. We use the method of Fick-Jacobs mapping on an effective one-dimensional problem. We calculate the ratchet current and show that is is strictly related to finite size of the particles. The ratchet current grows quadratically with particle radius. We also show that the dominant contribution to the ratchet current is due to inertial hydrodynamic effects. This means that Reynolds number must be at least of order one. We discuss the possible use for separation of particles by size and perspectives of optimization of the tube shape.

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.

Coupling effects in electromechanical ion transport in graphene nanochannels

In this work, we use molecular dynamics simulations to study the transport of ions in electromechanical flows in slit-like graphene nanochannels. The variation of ionic currents indicates a nonlinear coupling between pressure-driven and electroosmotic flows, which enhances the ionic currents for electromechanical flows compared with the linear superposition of pressure-driven and electroosmotic flows. The nonlinear coupling is attributed to the reduction of the total potential energy barrier due to the density variations of ions and water molecules in the channel. The numerical results may offer molecular insights into the design of nanofluidic devices for energy conversion.

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.

Movement of spherical colloid particles carried by flow in tubes of periodically varying diameter

We provide analytical formulas for the movement of spherical particles in a corrugated tube, in the approximation of small amplitude of the tube diameter variation. We calculate how the particle is pushed toward the wall at some places and pulled off the wall at others. We show that this effect causes rectification of the particle movement, when the direction of the fluid flow is alternated, thus leading to the hydrodynamic ratchet effect. We propose such scheme as a particle-separation device.

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.

Entropic Trapping of a Singly Charged Molecule in Solution

We demonstrate the ability to confine a single molecule in solution by spatial modulation of its local configurational entropy. Previously we established electrostatic trapping of a charged macromolecule by geometric tailoring of a repulsive electrical interaction potential in a parallel plate system. However, since the lifetime of the trapped state depends exponentially on the electrical charge of the molecule, the electrostatic interaction alone is often insufficient in magnitude to stably confine molecules carrying a net charge of magnitude ≤5 e. Here we show that the configurational entropy of a thermally fluctuating molecule in a geometrically modulated system can be exploited to spatially confine weakly charged molecules in solution. Measurement of the configurational entropy contribution reveals good agreement with theoretical expectations. This additional translational contribution to the total free energy facilitates direct optical imaging and measurement of the effective charge of molecules on the size scale of ∼1 nm and a charge as low as 1 e, physical properties comparable with those of a monovalent ion in solution.

Hydrodynamic and entropic effects on colloidal diffusion in corrugated channels

In the absence of advection, confined diffusion characterizes transport in many natural and artificial devices, such as ionic channels, zeolites, and nanopores. While extensive theoretical and numerical studies on this subject have produced many important predictions, experimental verifications of the predictions are rare. Here, we experimentally measure colloidal diffusion times in microchannels with periodically varying width and contrast results with predictions from the Fick-Jacobs theory and Brownian dynamics simulation. While the theory and simulation correctly predict the entropic effect of the varying channel width, they fail to account for hydrodynamic effects, which include both an overall decrease and a spatial variation of diffusivity in channels. Neglecting such hydrodynamic effects, the theory and simulation underestimate the mean and standard deviation of first passage times by 40% in channels with a neck width twice the particle diameter. We further show that the validity of the Fick-Jacobs theory can be restored by reformulating it in terms of the experimentally measured diffusivity. Our work thus shows that hydrodynamic effects play a key role in diffusive transport through narrow channels and should be included in theoretical and numerical models.

Fine separation of particles via the entropic splitter

We investigate the fine separation of particles with different sizes in an asymmetric confined channel by directing them moving to the opposite directions. Besides redesigning the geometry of the channel, we add a general rectangular wave oscillating force to enlarge the velocity differences between particles with different radii, which is important to increase the separation speed and sort particles of similar radii. The separation process is guaranteed by choosing a small period of the oscillating force and a proper partition strategy of the device length sifting particles to the left and right. The optimal set of parameters for a fixed amplitude of the oscillating force is found by the above regime. We show that by this regime the separation efficiency is significantly improved compared to the classic square wave force.

Diffusive escape through a narrow opening: new insights into a classic problem

We study the mean first exit time (T_ε) of a particle diffusing in a circular or a spherical micro-domain with an impenetrable confining boundary containing a small escape window (EW) of an angular size ε.

Inertial hydrodynamic ratchet: Rectification of colloidal flow in tubes of variable diameter

We investigate analytically a microfluidic device consisting of a tube with a nonuniform but spatially periodic diameter, where a fluid driven back and forth by a pump carries colloidal particles. Although the net flow of the fluid is zero, the particles move preferentially in one direction due to the ratchet mechanism, which occurs due to the simultaneous effect of inertial hydrodynamics and Brownian motion. We show that the average current is strongly sensitive to particle size, thus facilitating colloidal particle sorting.

Flow of colloidal solids and fluids through constrictions: dynamical density functional theory versus simulation

Using both dynamical density functional theory and particle-resolved Brownian dynamics simulations, we explore the flow of two-dimensional colloidal solids and fluids driven through a linear channel with a constriction. The flow is generated by a constant external force acting on all colloids. The initial configuration is equilibrated in the absence of flow and then the external force is switched on instantaneously. Upon starting the flow, we observe four different scenarios: a complete blockade, a monotonic decay to a constant particle flux (typical for a fluid), a damped oscillatory behaviour in the particle flux, and a long-lived stop-and-go behaviour in the flow (typical for a solid). The dynamical density functional theory describes all four situations but predicts infinitely long undamped oscillations in the flow which are always damped in the simulations. We attribute the mechanisms of the underlying stop-and-go flow to symmetry conditions on the flowing solid. Our predictions are verifiable in real-space experiments on magnetic colloidal monolayers which are driven through structured microchannels and can be exploited to steer the flow throughput in microfluidics.

Colloidal particles driven across periodic optical-potential-energy landscapes

We study the motion of colloidal particles driven by a constant force over a periodic optical potential energy landscape. First, the average particle velocity is found as a function of the driving velocity and the wavelength of the optical potential energy landscape. The relationship between average particle velocity and driving velocity is found to be well described by a theoretical model treating the landscape as sinusoidal, but only at small trap spacings. At larger trap spacings, a nonsinusoidal model for the landscape must be used. Subsequently, the critical velocity required for a particle to move across the landscape is determined as a function of the wavelength of the landscape. Finally, the velocity of a particle driven at a velocity far exceeding the critical driving velocity is examined. Both of these results are again well described by the two theoretical routes for small and large trap spacings, respectively. Brownian motion is found to have a significant effect on the critical driving velocity but a negligible effect when the driving velocity is high.

Wetting Driven by Thermal Fluctuations on Terraced Nanostructures

Theoretical analysis and fully atomistic molecular dynamics simulations reveal a Brownian ratchet mechanism by which thermal fluctuations drive the net displacement of immiscible liquids confined in channels or pores with micro- or nanoscale dimensions. The thermally driven displacement is induced by surface nanostructures with directional asymmetry and can occur against the direction of action of wetting or capillary forces. Mean displacement rates in molecular dynamics simulations are predicted via analytical solution of a Smoluchowski diffusion equation for the position probability density. The proposed physical mechanisms and derived analytical expressions can be applied to engineer surface nanostructures for controlling the dynamics of diverse wetting processes such as capillary filling, wicking, and imbibition in micro- or nanoscale systems.

Rectified motion in an asymmetric channel: the role of hydrodynamic interactions with walls

Dynamics of a Brownian particle in an asymmetric microchannel that is subjected to an external oscillating force is numerically analyzed. In addition to the elastic collisions with the walls that are kind of short range interactions, the long range hydrodynamic influences of the walls have been considered in an approximate way. We demonstrate how the geometrical parameters of the channel change the rectified current of the particle. As a result of numerical calculations, we show that long range hydrodynamic interactions with walls decrease the efficiency of the Brownian ratchet.

Front propagation in channels with spatially modulated cross section

Propagation of traveling fronts in a three-dimensional channel with spatially varying cross section is reduced to an equivalent one-dimensional reaction-diffusion-advection equation with boundary-induced advection term. Treating the advection term as a weak perturbation, an equation of motion for the front position is derived. We analyze channels whose cross sections vary periodically with L along the propagation direction of the front. Taking the Schlögl model as a representative example, we calculate analytically the nonlinear dependence of the front velocity on the ratio L/l where l denotes the intrinsic front width. In agreement with finite-element simulations of the three-dimensional reaction-diffusion dynamics, our theoretical results predicts boundary-induced propagation failure for a finite range of L/l values. In particular, the existence of the upper bound of L/l can be completely understood based on the linear eikonal equation. Last, we demonstrate that the front velocity is determined by the suppressed diffusivity of the reactants for L≪l.

Brownian motion in confined geometries

In a great number of technologically and biologically relevant cases, transport of micro- or nanosized objects is governed by both omnipresent thermal fluctuations and confining walls or constrictions limiting the available phase space. The present Topical Issue covers the most recent applications and theoretical findings devoted to studies of Brownian motion under confinement of channel-like geometries.

Entropic electrokinetics: recirculation, particle separation, and negative mobility

We show that when particles are suspended in an electrolyte confined between corrugated charged surfaces, electrokinetic flows lead to a new set of phenomena such as particle separation, mixing for low-Reynolds micro- and nanometric devices, and negative mobility. Our analysis shows that such phenomena arise, for incompressible fluids, due to the interplay between the electrostatic double layer and the corrugated geometrical confinement and that they are magnified when the width of the channel is comparable to the Debye length. Our characterization allows us to understand the physical origin of such phenomena, therefore, shedding light on their possible relevance in a wide variety of situations ranging from nano- and microfluidic devices to biological systems.

Deformable microswimmer in a swirl: capturing and scattering dynamics

Inspired by the classical Kepler and Rutherford problem, we investigate an analogous setup in the context of active microswimmers: the behavior of a deformable microswimmer in a swirl flow. First, we identify new steady bound states in the swirl flow and analyze their stability. Second, we study the dynamics of a self-propelled swimmer heading towards the vortex center, and we observe the subsequent capturing and scattering dynamics. We distinguish between two major types of swimmers, those that tend to elongate perpendicularly to the propulsion direction and those that pursue a parallel elongation. While the first ones can get caught by the swirl, the second ones were always observed to be scattered, which proposes a promising escape strategy. This offers a route to design artificial microswimmers that show the desired behavior in complicated flow fields. It should be straightforward to verify our results in a corresponding quasi-two-dimensional experiment using self-propelled droplets on water surfaces.

Channel-facilitated diffusion boosted by particle binding at the channel entrance

We investigate single-file diffusion of Brownian particles in arrays of closely confining microchannels permeated by a variety of attractive optical potentials and connecting two baths with equal particle concentration. We simultaneously test free diffusion in the channel, diffusion in optical traps coupled in the center of the channel, and diffusion in traps extending into the baths. We found that both classes of attractive optical potentials enhance the translocation rate through the channel with respect to free diffusion. Surprisingly, for the latter class of potentials we measure a 40-fold enhancement in the translocation rate with respect to free diffusion and find a sublinear power law dependence of the translocation rate on the average number of particles in the channel. Our results reveal the function of particle binding at the channel entrances for diffusive transport and open the way to a better understanding of membrane transport and design of synthetic membranes with enhanced diffusion rate.

Protein folding: complex potential for the driving force in a two-dimensional space of collective variables

Using the Helmholtz decomposition of the vector field of folding fluxes in a two-dimensional space of collective variables, a potential of the driving force for protein folding is introduced. The potential has two components. One component is responsible for the source and sink of the folding flows, which represent respectively, the unfolded states and the native state of the protein, and the other, which accounts for the flow vorticity inherently generated at the periphery of the flow field, is responsible for the canalization of the flow between the source and sink. The theoretical consideration is illustrated by calculations for a model β-hairpin protein.

Diffusion of finite-size particles in two-dimensional channels with random wall configurations

For the diffusion of finite-size particles in channels with randomised walls and reactive boundary conditions we find transient anomalous diffusion.

Viscosity control of the dynamic self-assembly in ferromagnetic suspensions

Recent studies of dynamic self-assembly in ferromagnetic colloids suspended in liquid-air or liquid-liquid interfaces revealed a rich variety of dynamic structures ranging from linear snakes to axisymmetric asters, which exhibit novel morphology of the magnetic ordering accompanied by large-scale hydrodynamic flows. Based on controlled experiments and first principles theory, we argue that the transition from snakes to asters is governed by the viscosity of the suspending liquid where less viscous liquids favor snakes and more viscous, asters. By obtaining analytic solutions of the time-averaged Navier-Stokes equations, we gain insight into the role of mean hydrodynamic flows and an overall balance of forces governing the self-assembly. Our results illustrate that the viscosity can be used to control the outcome of the dynamic self-assembly in magnetic colloidal suspensions.